Irrelevant deformations with boundaries and defects

We will be interested in studying deformations generated by conserved charges. Hence, in particular in the context of integrability the space of deformations is very rich.

Bulk charges. Let us first briefly recall how integrability is realized when there are no boundaries. It is convenient to introduce complex coordinates ⁵

$$\begin{equation}z=x+\mathrm{i}y,\qquad \bar{z}=x-\mathrm{i}y,\end{equation} \tag{ 2.2 }$$

such that

$$\begin{equation}{\partial }_{z}=\frac{1}{2}({\partial }_{x}-\mathrm{i}{\partial }_{y}),\qquad {\partial }_{\bar{z}}=\frac{1}{2}({\partial }_{x}+\mathrm{i}{\partial }_{y}).\end{equation} \tag{ 2.3 }$$

Assuming Lorentz invariance, the conserved charges associated with the spacetime symmetries are fully encoded in the stress–energy tensor. Let us introduce some notation for the components of the stress tensor in complex coordinates:

$$\begin{equation}T=-{T}_{zz},\qquad \bar{T}=-{T}_{\bar{z}\bar{z}},\qquad {\Theta}=\bar{{\Theta}}={T}_{z\bar{z}}={T}_{\bar{z}z}.\end{equation} \tag{ 2.4 }$$

Here T_μν is the stress tensor of the theory.

In addition to the charges associated with the stress tensor, integrable theories possess an infinite set of mutually commuting integrals of motion, which can be understood as higher spin generalizations of the stress tensor charges. Conventionally, one can construct those conserved charges in terms of some local spin-s fields T_s and Θ_s , which satisfy

$$\begin{equation}{\partial }_{\bar{z}}{T}_{s+1}={\partial }_{z}{{\Theta}}_{s-1}.\end{equation} \tag{ 2.5 }$$

Here the allowed spins take values in a subset of integers and are fully determined by the theory. Assuming parity invariance, operators with negative spin s are related to the operators with positive spin by parity, so it is convenient to define the barred charges as

$$\begin{equation}{\bar{T}}_{s+1}={{\Theta}}_{-s-1},\qquad {\bar{{\Theta}}}_{s-1}={T}_{-s+1}.\end{equation} \tag{ 2.6 }$$

With this definition, the conservation equations can now be summarized as

$$\begin{equation}{\partial }_{\bar{z}}{T}_{s+1}={\partial }_{z}{{\Theta}}_{s-1}.\qquad {\partial }_{z}{\bar{T}}_{s+1}={\partial }_{\bar{z}}{\bar{{\Theta}}}_{s-1},\end{equation} \tag{ 2.7 }$$

where the spin label s is assumed to be non-negative. We shall employ this convention in the following.

Using the local fields, we can construct conserved charges by means of contour integrals. Defining

$$\begin{equation}{I}_{s}={\int }_{\mathcal{C}}\ \left(\mathrm{d}z\ {T}_{s+1}+\mathrm{d}\bar{z}\,{{\Theta}}_{s-1}\right),\qquad {\bar{I}}_{s}={\int }_{\mathcal{C}}\ \left(\mathrm{d}\bar{z}\,{\bar{T}}_{s+1}+\mathrm{d}z\ {\bar{{\Theta}}}_{s-1}\right),\end{equation} \tag{ 2.8 }$$

one immediately sees that I_s and ${\bar{I}}_{s}$ are independent of the choice of the contour because of (2.7). One recovers the charges in Euclidean coordinates by taking linear combinations of I_s and ${\bar{I}}_{s}$. For instance, the momentum and Hamiltonian are expressed as

$$\begin{equation}P=-\mathrm{i}\left({I}_{1}-{\bar{I}}_{1}\right),\qquad H={I}_{1}+{\bar{I}}_{1}.\end{equation} \tag{ 2.9 }$$

Similarly one obtains higher H- and P-type charges:

$$\begin{equation}{H}_{s}={I}_{s}+{\bar{I}}_{s}=\int {\mathcal{H}}_{s}(x)\mathrm{d}x,\qquad {P}_{s}=-\mathrm{i}\left({I}_{s}-{\bar{I}}_{s}\right)=\int {\mathcal{P}}_{s}(x)\mathrm{d}x.\end{equation} \tag{ 2.10 }$$

Here the local densities ${\mathcal{H}}_{s}(x)$ and ${\mathcal{P}}_{s}(x)$ are defined as

$$\begin{equation}{\mathcal{H}}_{s}={T}_{s+1}+{{\Theta}}_{s-1}+{\bar{T}}_{s+1}+{\bar{{\Theta}}}_{s-1},\end{equation} \tag{ 2.11 }$$

$$\begin{equation}{\mathcal{P}}_{s}=-\mathrm{i}\left({T}_{s+1}+{{\Theta}}_{s-1}-{\bar{T}}_{s+1}-{\bar{{\Theta}}}_{s-1}\right).\end{equation} \tag{ 2.12 }$$

Conservation of the above charges follows from current conservation

$$\begin{equation}{\partial }_{y}{\mathcal{H}}_{s}=-{\partial }_{x}{\mathcal{J}}_{{\mathcal{H}}_{s}},\qquad {\partial }_{y}{\mathcal{P}}_{s}=-{\partial }_{x}{\mathcal{J}}_{{\mathcal{P}}_{s}},\end{equation} \tag{ 2.13 }$$

where the generalized current densities take the form

$$\begin{equation}{\mathcal{J}}_{{\mathcal{H}}_{s}}=-\mathrm{i}\left({T}_{s+1}-{{\Theta}}_{s-1}-{\bar{T}}_{s+1}+{\bar{{\Theta}}}_{s-1}\right),\end{equation} \tag{ 2.14 }$$

$$\begin{equation}{\mathcal{J}}_{{\mathcal{P}}_{s}}=-{T}_{s+1}+{{\Theta}}_{s-1}-{\bar{T}}_{s+1}+{\bar{{\Theta}}}_{s-1}.\end{equation} \tag{ 2.15 }$$

Note that the P-type charges are only conserved in the bulk, while conserved H-type charges can also be defined in the boundary model as described in the following paragraph. It will also be useful to employ the universal notation

$$\begin{equation}{Q}_{2s}={H}_{2s-1},\qquad {Q}_{2s-1}={P}_{2s-1},\quad s=1,2,\dots \end{equation} \tag{ 2.16 }$$

Similarly we define the shifted current densities as

$$\begin{equation}{\mathcal{J}}_{2s}={\mathcal{J}}_{{\mathcal{H}}_{2s-1}},\qquad {\mathcal{J}}_{2s-1}={\mathcal{J}}_{{\mathcal{P}}_{2s-1}},\quad s=1,2,\dots ,\end{equation} \tag{ 2.17 }$$

such that the conservation equation takes the form

$$\begin{equation}{\partial }_{y}{\mathcal{Q}}_{s}=-{\partial }_{x}{\mathcal{J}}_{s}.\end{equation} \tag{ 2.18 }$$

Thus, for the boundary model the even charges Q_2s are conserved, while the odd charges Q_2s+1 will only be conserved in the bulk. The Heisenberg equation for the charges, which will be used in some of the following derivations, reads

$$\begin{equation}{\partial }_{y}{\mathcal{Q}}_{r}=\left[H,{\mathcal{Q}}_{r}\right].\end{equation} \tag{ 2.19 }$$

Finally, the one-particle eigenvalues of the different types of charges are denoted according to

$$\begin{equation}{Q}_{s}\to {q}_{s}(u),\qquad {H}_{s}\to {e}_{s}(u),\qquad {P}_{s}\to {p}_{s}(u),\end{equation} \tag{ 2.20 }$$

where the eigenvalues of the Q_s split up into even and odd parts:

$$\begin{equation}{q}_{2s}(u)={e}_{2s-1}(u),\qquad {q}_{2s-1}(u)={p}_{2s-1}(u).\end{equation} \tag{ 2.21 }$$

Boundary effects. In the presence of a boundary, the bulk conservation law (2.7) is not sufficient to guarantee the conservation of ${I}_{s},{\bar{I}}_{s}$. For notational simplicity, we consider a system with only one boundary and we set s_L = −∞, s_R = 0. Let us compare the conserved charges at different times. Consider say I₁ defined as an integral over two different contours ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$, which are parallel to the x-axis but end on different points y₁, y₂ on the y axis. Taking the difference we find

$$\begin{equation}\begin{aligned}\hfill {I}_{1}^{{\mathcal{C}}_{1}}-{I}_{1}^{{\mathcal{C}}_{2}}& ={\left.\mathrm{i}{\int }_{{y}_{1}}^{{y}_{2}}\mathrm{d}y\ (T-{\Theta})\right\vert }_{x=0}\ne 0,\hfill \\ \hfill {\bar{I}}_{1}^{{\mathcal{C}}_{1}}-{\bar{I}}_{1}^{{\mathcal{C}}_{2}}& ={\left.\mathrm{i}{\int }_{{y}_{1}}^{{y}_{2}}\mathrm{d}y\ (\bar{{\Theta}}-\bar{T})\right\vert }_{x=0}\ne 0,\hfill \end{aligned}\end{equation} \tag{ 2.22 }$$

for generic T_s , Θ_s , y₁, y₂, i.e. the charges are a priori not conserved. A simple fix is to choose an appropriate boundary function θ as given in (2.1), such that

$$\begin{equation}{\left.{T}_{xy}\right\vert }_{x=0}={\left.-\mathrm{i}(T-\bar{T})\right\vert }_{x=0}=\frac{\mathrm{d}}{\mathrm{d}y}\theta (y),\end{equation} \tag{ 2.23 }$$

where θ(y) = θ(x = 0, y) is some local boundary field. Physically, the original conformal boundary condition T_xy = 0 just means that there is no energy/momentum flow passing through the impenetrable boundary at x = 0. The generalized boundary condition with T_xy ≠ 0 then implies that the energy flow can be absorbed by a 'potential term' on the boundary. Adding θ(y), the new bulk-boundary Hamiltonian of the form

$$\begin{equation}H={\int }_{-\infty }^{0}\mathrm{d}x\ (T+\bar{T}+{\Theta}+\bar{{\Theta}})+\theta (y)\end{equation} \tag{ 2.24 }$$

is a conserved quantity since the additional θ-term compensates the boundary contribution.

Similarly, we can introduce generalized boundary conditions for higher spin fields,

$$\begin{equation}{\left.-\mathrm{i}({T}_{r+1}-{\bar{T}}_{r+1}+{\bar{{\Theta}}}_{r-1}-{{\Theta}}_{r-1})\right\vert }_{x=0}=\frac{\mathrm{d}}{\mathrm{d}y}{\theta }_{r}(y),\end{equation} \tag{ 2.25 }$$

and the higher conserved charges are now given by

$$\begin{equation}{H}_{r}={\int }_{-\infty }^{0}\mathrm{d}x\ ({T}_{r+1}+{\bar{T}}_{r+1}+{\bar{{\Theta}}}_{r-1}+{{\Theta}}_{r-1})+{\theta }_{r}(y).\end{equation} \tag{ 2.26 }$$

The discussion above immediately generalizes to the case with two boundaries: one only needs to impose the boundary condition for both, the left (L) and right (R) boundary. Conservation of the charges requires that the boundary field obeys

$$\begin{equation}{\partial }_{y}{\theta }_{r\mathrm{L}/\mathrm{R}}(t)={\mathcal{J}}_{{\mathcal{H}}_{r}}({s}_{\mathrm{L}/\mathrm{R}}).\end{equation} \tag{ 2.27 }$$

To summarize, we can construct conserved charges in the presence of the boundaries by adding a boundary function θ_r to the Hamiltonian for each boundary. At the same time, only particular (H-type, see (2.10)) linear combinations of the two types of conserved charges I_s and ${\bar{I}}_{s}$ are conserved. The odd P-type charges are not conserved in the presence of boundaries since translation symmetry and its higher spin generalizations are broken by the boundaries.

The boundary can be placed either in the spatial or temporal direction as is shown in figure 1. These choices give different but equivalent descriptions of the same theory. In the open channel, the boundary is placed in the spatial direction. Without loss of generality, we can put it at x = 0 and define the system on the left half line, as is shown in the right panel of figure 1. The Hamiltonian in the open channel reads

$$\begin{equation}{H}_{\text{open}}={\int }_{-\infty }^{0}\mathrm{d}x\,{T}_{yy}(x).\end{equation} \tag{ 2.28 }$$

Correlation functions in this channel are computed by

$$\begin{equation}\langle {O}_{1}({x}_{1},{y}_{1})\dots {O}_{N}({x}_{N},{y}_{N})\rangle =\frac{{}_{\mathrm{B}}\langle 0\vert {\mathcal{T}}_{y}[{O}_{1}({x}_{1},{y}_{1})\dots {O}_{N}({x}_{N},{y}_{N})]\vert {0\rangle }_{\mathrm{B}}}{{}_{\mathrm{B}}\langle 0\vert {0\rangle }_{\mathrm{B}}},\end{equation} \tag{ 2.29 }$$

where |0⟩_B is the ground state of H_open and

$$\begin{equation}{O}_{i}(x,y)={\mathrm{e}}^{-y{H}_{\text{open}}}{O}_{i}(x,0){\mathrm{e}}^{y{H}_{\text{open}}}.\end{equation} \tag{ 2.30 }$$

Here, ${\mathcal{T}}_{y}$ means ordering with respect to the y direction.

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In the closed channel, the boundary is placed in the Euclidean time direction, see the left panel in figure 1. The Hamiltonian is the same as for a QFT without boundaries and given by

$$\begin{equation}{H}_{\text{closed}}={\int }_{-\infty }^{\infty }\mathrm{d}y\,{T}_{xx}(y),\end{equation} \tag{ 2.31 }$$

where T_xx (x) is one of the components of the stress energy tensor T_μν (x) on some constant x slice.

Since the boundary is placed in the temporal direction, it should be understood as a boundary state, which we denote by |B⟩. Correlation functions of local operators in the closed channel with one boundary at x = 0 are given by

$$\begin{equation}\langle O({x}_{1},{y}_{1})\dots O({x}_{N},{y}_{N})\rangle =\frac{\langle 0\vert {\mathcal{T}}_{x}[O({x}_{1},{y}_{1})\dots O({x}_{N},{y}_{N})]\vert B\rangle }{\langle 0\vert B\rangle },\end{equation} \tag{ 2.32 }$$

where |0⟩ is the ground state of H_closed and ${\mathcal{T}}_{x}$ is the time ordering. We have

$$\begin{equation}{O}_{i}(x,y)={\mathrm{e}}^{x{H}_{\text{closed}}}{O}_{i}(0,y){\mathrm{e}}^{-x{H}_{\text{closed}}}.\end{equation} \tag{ 2.33 }$$

In this section, we will discuss how the usual scattering picture arises in integrable theories. Deformations of the respective bulk and boundary scattering matrices will then be discussed in the subsequent sections.

Consider a multi-particle scattering process in two dimensions. Each particle is specified by its energy and momentum, which satisfies the relativistic dispersion relation e² − p² = m², where m is the mass of the particle. It is convenient to parametrize the energy e and momentum p by the rapidity variable u, defined via (e, p) = (m cosh u, m sinh u). We thus describe scattering processes in terms of the rapidities of the particles.

Bulk scattering. Integrability imposes strong constraints on a multi-particle scattering process. All multi-particle processes factorize into consecutive two-to-two scattering events. The compatibility condition is that the order of this factorization does not affect the physical amplitudes, which leads to the Yang–Baxter equation

$$\begin{equation}{S}_{12}{S}_{13}{S}_{23}={S}_{23}{S}_{13}{S}_{12},\end{equation} \tag{ 2.34 }$$

where S_ij = S_ij (u_i , u_j ) denotes the S-matrix of the ij two-to-two scattering process. Here u_j represents the rapidity of the scattered particle j.

Boundary scattering. In the presence of boundaries, we have to add new integredients to the Yang–Baxter equation to retain integrability. Let us first recall the ordinary quantum mechanical scattering picture in the presence of boundaries. Consider an incoming plane wave moving towards the boundary. If the boundary is impenetrable, this wave must be fully reflected. For a unitary theory with a single type of particle, the reflection amplitude can only differ by a phase from the incoming amplitude. For a system with more than one particle type, the reflection matrix is a unitary matrix. This boundary scattering matrix or boundary scattering phase, respectively, will be denoted by S_L(u) or S_R(u). Here u stands for the rapidity of the reflected particle and L or R denotes the left or right boundary, respectively.

Integrability imposes non-trivial constraints on the boundary S-matrix. Consider a two-particle scattering process, where the two particles scatter through each other in the bulk, hit the boundary and then return into the bulk. Demanding that this process is independent of the order of scattering, we obtain the boundary Yang–Baxter equation for e.g. the left boundary scattering matrix S_L:

$$\begin{equation}{S}_{\text{L}}({u}_{2}){S}_{21}({u}_{1}+{u}_{2}){S}_{\text{L}}({u}_{1}){S}_{12}({u}_{1}-{u}_{2})={S}_{12}({u}_{1}-{u}_{2}){S}_{\text{L}}({u}_{1}){S}_{21}({u}_{1}+{u}_{2}){S}_{\text{L}}({u}_{2}).\end{equation} \tag{ 2.35 }$$

In addition to that, the boundary S-matrix must also satisfy the crossing and unitarity constraints [45]. The boundary S-matrices can be obtained non-perturbatively by solving these constraints.

Faddeev–Zamolodchikov algebra. Before discussing the more general cases, let us first review a convenient way of describing integrable scattering processes. Consider the asymptotic states of an integrable field theory, which can be expressed in terms of creation operators A^†(u) via

$$\begin{equation}{\vert {u}_{1},\dots ,{u}_{n}\rangle }_{\text{in/out}}={A}^{{\dagger}}({u}_{1})\dots {A}^{{\dagger}}({u}_{n})\vert 0\rangle .\end{equation} \tag{ 2.36 }$$

The in/out states are distinguished by the relative ordering of the particle rapidities: if u₁ > u₂ > ...u_n it is understood to be an 'in-state'; if instead the rapidities are ordered as u₁ < u₂ < ... < u_n it is understood as an 'out-state'. It is worth mentioning that the creation operators are not the creation operators of the free theory, instead they take into account all interactions. However, a nice feature of those creation operators is that we can describe the scattering process as in a free theory, where the creation operators satisfy

$$\begin{equation}{A}^{{\dagger}}({u}_{1}){A}^{{\dagger}}({u}_{2})=S({u}_{1},{u}_{2}){A}^{{\dagger}}({u}_{2}){A}^{{\dagger}}({u}_{1}).\end{equation} \tag{ 2.37 }$$

Here the coefficient S represents the two-to-two scattering matrix. Since asymptotic states diagonalize the local charges, we can write

$$\begin{equation}\left[{I}_{s},{A}^{{\dagger}}(u)\right]={\gamma }^{(s)}\,{\mathrm{e}}^{su}{A}^{{\dagger}}(u),\qquad \left[{\bar{I}}_{s},{A}^{{\dagger}}(u)\right]={\gamma }^{(s)}\,{\mathrm{e}}^{-su}{A}^{{\dagger}}(u),\end{equation} \tag{ 2.38 }$$

where the γ^(s) are constants determined by the theory.

Defect theory. A defect is a generalization of an impenetrable boundary. Quantum mechanically, the physics is identical to the scattering off a potential barrier, where we are allowed to have both, transmitted and reflected waves. For a unitary theory, the sum of the modulus of the reflection and transmission amplitude is 1. The study of integrable line defects was initiated in [46, 47].

In integrable theories it is convenient to describe integrable defects by the Faddeev–Zamolodchikov (FZ) algebra. The reason is that the generalized FZ algebra in the presence of a defect has the same structure as the usual asymptotic quantum mechanical scattering picture.

The line defect separates the space into two parts, which will be called the left and right part. We denote the FZ operators in the two parts by A^†(u) and B^†(u), respectively, and we assume that the action of the defect can be described by a defect creating operator D^†. The most general defect algebra is then

$$\begin{align}\hfill {A}^{{\dagger}}(u){\mathbf{D}}^{{\dagger}}=& R(u){A}^{{\dagger}}(-u){\mathbf{D}}^{{\dagger}}+{T}_{-}(u){\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}(u),\hfill \\ \hfill {\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}(u)=& R(u){\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}(-u)+{T}_{+}(u){A}^{{\dagger}}(u){\mathbf{D}}^{{\dagger}}.\hfill \end{align} \tag{ 2.39 }$$

The physical meaning of these equations is quite clear: for instance, the first equation describes the scattering process of the left particle off the defect.

It has been proven that the only integrable defects in an interacting theory are topological ones [48]. These defects are purely transmissive. Setting the reflection coefficient to zero (R(u) = 0) in the previous equations, the algebra satisfied by these operators reads

$$\begin{equation}{A}^{{\dagger}}(u){\mathbf{D}}^{{\dagger}}={T}_{-}(u)\,{\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}(u),\qquad {\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}(-u)={T}_{+}(-u){A}^{{\dagger}}(-u){\mathbf{D}}^{{\dagger}},\end{equation} \tag{ 2.40 }$$

where T_±(u) are the transition amplitudes. For a parity symmetric theory, we have T₋(u) = T₊(u). The asymptotic states are given by

$$\begin{equation}\vert {u}_{1},\dots ,{u}_{M};{v}_{1},\dots ,{v}_{N}\rangle \equiv {A}^{{\dagger}}({u}_{1})\dots {A}^{{\dagger}}({u}_{M}){\mathbf{D}}^{{\dagger}}{B}^{{\dagger}}({v}_{1})\dots {B}^{{\dagger}}({v}_{N})\vert 0\rangle .\end{equation} \tag{ 2.41 }$$

Here, we have introduced an evident ket notation |u; v⟩ to denote the rapidities of the left/right side of space

One can have non-topological defects, but the price to pay is that the theory has to be free, namely the bulk S-matrices are simply S = ±1. In this case, the creation operators A^†, B^† are identical, and the most general asymptotic state can be written as

$$\begin{equation}\vert {u\rangle }_{\mathrm{D}}=a(u)\vert u;\varnothing \rangle +b(u)\vert \varnothing ;u\rangle +c(u)\vert -u;\varnothing \rangle ,\end{equation} \tag{ 2.42 }$$

where we have

$$\begin{equation}T(u)=\frac{b(u)}{a(u)},\qquad R(u)=\frac{c(u)}{a(u)}.\end{equation} \tag{ 2.43 }$$

This completes our introduction to two-dimensional quantum field theories whose deformations will be discussed in the following.

In order to study classical deformations it will be convenient to use the Lagrangian description. Recall that our field theory is defined on the −x axis, by taking s_L = −∞, s_R = 0, with the fields vanishing at x = s_L. Performing a Legendre transformation of the Hamiltonian (2.1), the action of the system can be written as

$$\begin{equation}S(\phi )={\int }_{-\infty }^{\infty }\mathrm{d}y{\int }_{-\infty }^{0}\mathrm{d}x\,\mathcal{L}(\phi ,{\partial }_{\mu }\phi )+\int \mathrm{d}y\,{\mathcal{L}}_{\mathrm{b}\mathrm{d}\mathrm{r}}({\phi }_{\mathrm{b}\mathrm{d}\mathrm{r}}),\end{equation} \tag{ 3.1 }$$

where ϕ_bdr(y) = ϕ(x = 0, y). Here, the boundary Lagrangian ${\mathcal{L}}_{\mathrm{b}\mathrm{d}\mathrm{r}}$ is essentially the boundary potential θ_2R(x = 0, y) defined in (2.1).

As usual, the classical equation of motion can be obtained by taking the functional variation. In the bulk, the equation of motion is given by the usual Euler–Lagrange equation, which does not depend on the boundary Lagrangian. On the contrary, the equation of motion for the boundary field ϕ_bdr depends on both, the bulk and the boundary Lagrangian. The dependence on the bulk Lagrangian comes from the total spatial derivative terms in the bulk (recall that these are x-derivatives in our setup). For instance, suppose under a field variation ϕ → ϕ + δϕ the variation of the bulk Lagrangian contains a spatial derivative term $\frac{\partial \mathcal{L}}{\partial ({\partial }_{x}\phi )}{\partial }_{x}\delta \phi$. After integration by parts, a boundary term ${\left.\frac{\partial \mathcal{L}}{\partial ({\partial }_{x}\phi )}\delta \phi \right\vert }_{x=0}$ remains. This term, together with the usual boundary variation, will determine the boundary condition for the field variable ϕ.

Integrable boundary conditions. In order to preserve integrability, the boundary potential has to satisfy the non-trivial constraint (2.25), which depends on the form of the bulk Lagrangian. Hence, for a given bulk Lagrangian, the integrability constraint restricts the form of the boundary Lagrangian. In fact, following the procedure of Ghoshal and Zamolodchikov for the Sine-Gordon model [45], we have found that in the case of the bulk $T\bar{T}$-Lagrangian the existence of a higher conserved charge requires the boundary potential to be zero. Our results are summarized in table 2.

Table 2. Given a certain bulk theory, we display the boundary conditions that are compatible with the existence of a higher conserved charge, which is a necessary requirement for integrability. Here in the last two rows we have performed the analysis up to the leading deformation at $\mathcal{O}(\lambda )$. The θ(ϕ) or θ(ϕ, ∂_y ϕ) in the right column indicates our assumptions on the dependence of the boundary Hamiltonian θ on the fields.

Tested	Bulk	Boundary
Generic	Free theory	Mass term type: θ = gϕ2/2
		Sine-Gordon type: $\theta ={\mathfrak{c}}_{1}\,\mathrm{cosh}({\mathfrak{c}}_{2}\phi -{\phi }_{0})$
Order $\mathcal{O}(\lambda )$	$T\bar{T}$-deformed free theory	θ(ϕ) ⇒ θ(ϕ) = 0, θ(ϕ, ∂y ϕ) ⇒ θ(ϕ, ∂y ϕ) ≠ 0
Order $\mathcal{O}(\lambda )$	$T\bar{T}$-deformed Sine-Gordon	θ(ϕ) ⇒ θ(ϕ) = 0

In order to illustrate how to obtain constraints on the boundary function using (2.25), let us first study the free theory for a single massless scalar field, defined on the −x axis:

$$\begin{equation}S(\phi )=\frac{1}{2}{\int }_{-\infty }^{\infty }\mathrm{d}y{\int }_{-\infty }^{0}\mathrm{d}x\,{\partial }_{\mu }\phi {\partial }^{\mu }\phi +\int \mathrm{d}y\,\theta ({\phi }_{\mathrm{b}\mathrm{d}\mathrm{r}}).\end{equation} \tag{ 3.2 }$$

Here ϕ_bdr = ϕ(x = 0). Our goal is to find the most general boundary potential θ that is compatible with (2.25).

In order to obtain the equations of motion we consider a field variation ϕ → ϕ + δϕ:

$$\begin{equation}\begin{aligned}\hfill \delta S& ={\int }_{-\infty }^{\infty }\mathrm{d}y{\int }_{-\infty }^{0}\mathrm{d}x\,{\partial }_{\mu }\phi {\partial }^{\mu }\delta \phi +\int \mathrm{d}y\frac{\delta \theta }{\delta \phi }\delta \phi \hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\mathrm{d}y{\int }_{-\infty }^{0}\mathrm{d}x\,(-{\partial }^{2}\phi ){\left.\delta \phi +{\int }_{-\infty }^{\infty }\mathrm{d}y\,{\partial }_{x}\phi \delta \phi \right\vert }_{x=0}+\int \mathrm{d}y\frac{\delta \theta }{\delta \phi }\delta \phi .\hfill \end{aligned}\end{equation} \tag{ 3.3 }$$

As usual, we assume vanishing fields at infinity, such that the bulk and boundary equation of motion are given by

$$\begin{equation}{\left.{\partial }^{2}\phi =0,\qquad {\partial }_{x}\phi +\frac{\delta \theta }{\delta \phi }\right\vert }_{x=0}=0.\end{equation} \tag{ 3.4 }$$

Conserved higher charges. For the free scalar the definition of higher charges is actually ambiguous. For instance, using the equation of motion $\partial \bar{\partial }\phi =0$, where we remind of our complex coordinates with $\partial =\frac{1}{2}({\partial }_{x}-\mathrm{i}{\partial }_{y})$ and $\bar{\partial }=\frac{1}{2}({\partial }_{x}+\mathrm{i}{\partial }_{y})$, one immediately sees that any linear combination of the form

$$\begin{equation}{T}_{s}=\sum\limits _{j=0}^{s}{c}_{s,j}({\partial }^{s-j}\phi ){(\partial \phi )}^{j}\end{equation} \tag{ 3.5 }$$

defines an on-shell spin-s conserved current in the bulk. Here the c_s,j are constant coefficients. However, not all of these linear combinations are conserved for any boundary Lagrangian. We shall see how different choices of the boundary potential fix this ambiguity.

Among the conserved charges (3.5), the T₂ term is the stress–energy tensor, while T₃ amounts to a total z-derivative. Therefore, the first non-trivial conserved higher charge is T₄. Comparing with (3.5), one finds that the most general form of T₄ can be obtained from the integer partition of 4 by replacing the following sets by the respective derivative terms:

$$\begin{equation}\left\{4\right\},\left\{3,1\right\},\left\{2,2\right\},\left\{2,1,1\right\},\left\{1,1,1,1\right\}.\end{equation} \tag{ 3.6 }$$

Thus, we obtain the ansatz

$$\begin{equation}{T}_{4}={c}_{1}{\partial }^{4}\phi +{c}_{2}(\partial \phi ){\partial }^{3}\phi +{c}_{3}{({\partial }^{2}\phi )}^{2}+{c}_{4}{(\partial \phi )}^{2}{\partial }^{2}\phi +{c}_{5}{(\partial \phi )}^{4},\end{equation} \tag{ 3.7 }$$

where the c_j denote general coefficients. The conjugate ${\bar{T}}_{4}$ is obtained by the replacement $\partial \to \bar{\partial }$. Since we are dealing with free theories, the conservation of T₄ immediately follows from the equation of motion, $\partial \bar{\partial }\phi =0$.

Before going into further details, let us discuss the structure of T₄. The c₁ and c₄ terms are different from the others, since they are total z-derivatives. Therefore, it suffices to consider the case where c₁ = c₄ = 0. (We will see that those terms drops out automatically, if we take them into account.) The c₂, c₃ terms are not independent, but related by an integration by parts, ${({\partial }^{2}\phi )}^{2}=\partial (\partial \phi {\partial }^{2}\phi )-\partial \phi {\partial }^{3}\phi$, so the only physical parameter is c₃ − c₂. We shall see that this is indeed the case.

Integrability constraints on the boundary potential. For free theories, the ${{\Theta}}_{2},{\bar{{\Theta}}}_{2}$ terms vanish (cf (2.5)). Based on this, we can compute the boundary contribution (2.25) when r = 3,

$$\begin{equation}-8\mathrm{i}({T}_{4}-{\bar{T}}_{4})=A(\phi ){({\partial }_{y}\phi )}^{3}+B(\phi ){\partial }_{y}\phi {\partial }_{y}^{2}\phi +C(\phi ){\partial }_{y}^{3}\phi +(\text{functions}\;\text{of}\ \phi )\ {\partial }_{y}\phi ,\end{equation} \tag{ 3.8 }$$

where (for ${\theta }^{(j)}(\phi )={\partial }_{\phi }^{j}\theta (\phi )$)

$$\begin{equation}\begin{aligned}\hfill A(\phi )& =-8{c}_{1}{\theta }^{(4)}(\phi )-4{c}_{4}{\theta }^{(3)}(\phi )-2{c}_{4}{\theta }^{{\prime\prime}}(\phi )-4{c}_{5}{\theta }^{\prime }(\phi ),\hfill \\ \hfill B(\phi )& =-24{c}_{1}{\theta }^{(3)}(\phi )-4{c}_{2}{\theta }^{{\prime\prime}}(\phi )-8{c}_{3}{\theta }^{{\prime\prime}}(\phi )-4{c}_{4}{\theta }^{\prime }(\phi ),\hfill \\ \hfill C(\phi )& =-8{c}_{1}{\theta }^{{\prime\prime}}(\phi )-4{c}_{2}{\theta }^{\prime }(\phi ).\hfill \end{aligned}\end{equation} \tag{ 3.9 }$$

Here, we have replaced ${\partial }_{x}^{2}$ using the bulk equation of motion, and we have replaced ∂_x using the boundary equation of motion, see (3.4). Employing those equations we can thus eliminate all x-derivatives at the boundary.

The last term of (3.8) is automatically a total y-derivative. For the first three terms, we can perform integration by parts on the A, C terms to bring them into the form of the B term:

$$\begin{align}\hfill A(\phi ){({\partial }_{y}\phi )}^{3}& =\frac{\mathrm{d}}{\mathrm{d}y}\left(\int \mathrm{d}\phi \,A(\phi )\right){({\partial }_{y}\phi )}^{2}\hfill \\ \hfill & =-2\left(\int \mathrm{d}\phi A(\phi )\right){\partial }_{y}\phi {\partial }^{2}\phi +\text{totaly}\mbox{-}\text{derivative}\;\text{terms},\hfill \\ \hfill C(\phi ){\partial }_{y}^{3}\phi & =C(\phi )\frac{\mathrm{d}}{\mathrm{d}y}({\partial }_{y}^{2}\phi )=-{C}^{\prime }(\phi ){\partial }_{y}\phi {\partial }_{y}^{2}\phi +\text{total}\mbox{-}\text{derivative}\;\text{terms}.\hfill \end{align} \tag{ 3.10 }$$

Therefore, the condition of being a total y-derivative is equivalent to

$$\begin{equation}2\int \mathrm{d}\phi \left(A(\phi )-B(\phi )+{C}^{\prime }(\phi )\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 3.11 }$$

We allow for a constant since

$$\begin{equation}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\times {\partial }_{y}\phi ({\partial }_{y}^{2}\phi )=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\times \frac{1}{2}{\partial }_{y}\left[{({\partial }_{y}\phi )}^{2}\right]\end{equation} \tag{ 3.12 }$$

can also be expressed as a total derivative. Using the explicit expressions for A(ϕ), B(ϕ), and C(ϕ), we find

$$\begin{equation}2\int \mathrm{d}\phi \ A(\phi )-B(\phi )+{C}^{\prime }(\phi )=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}{\Rightarrow}({c}_{3}-{c}_{2}){\theta }^{{\prime\prime}}(\phi )-{c}_{5}\theta (\phi )=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 3.13 }$$

We see that the dependence on c₁ and c₄ drops out automatically, and indeed the result only depends on c₃ − c₂. The solutions of this differential equation for the boundary function θ fall into three categories:

• c₃ − c₂, c₅ ≠ 0: the constant term on the right-hand side just shifts the boundary potential by const/c₅, which has no physical consequences. Therefore, it suffices to take this constant to be zero, and the solution reads
  $$\begin{equation}\theta (\phi )={\mathfrak{c}}_{1}\,\mathrm{cosh}\left({\mathfrak{c}}_{2}\phi -{\phi }_{0}\right),\qquad {\mathfrak{c}}_{2}=\sqrt{\frac{{c}_{5}}{{c}_{3}-{c}_{2}}},\end{equation} \tag{ 3.14 }$$
  where ${\mathfrak{c}}_{1},{\phi }_{0}$ denote constants of integration. This result essentially represents the boundary potential for the Sine-Gordon theory as found by Ghoshal and Zamolodchikov [45].
• c₅ = 0, c₃ − c₂ ≠ 0: the constant term on right-hand side is now important, since it yields a quadratic contribution to the potential:
  $$\begin{equation}\theta (\phi )=\frac{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}{2({c}_{3}-{c}_{2})}{\phi }^{2}+{\mathfrak{c}}_{3}+{\mathfrak{c}}_{4}\phi ,\end{equation} \tag{ 3.15 }$$
  where ${\mathfrak{c}}_{3},{\mathfrak{c}}_{4}$ denote constants of integration. Physically they are not important, since we can absorb them by a constant field shift. This is the boundary function discussed in [49].
• c₃ − c₂ = 0, c₅ ≠ 0 case: the boundary potential is a constant. Consequently, the bulk field satisfies Neumann boundary conditions.

This completes the story of the free theory. In summary, based on the analysis of the first non-trivial higher charge T₄, there are only two options to choose a boundary potential (or boundary Lagrangian), which preserves integrability of the free bulk scalar:

• (a)  
  If on the one hand θ(ϕ) = gϕ²/2, we find for the higher conserved charges

  $$\begin{equation}\theta (\phi )=g\frac{{\phi }^{2}}{2}{\Rightarrow}{T}_{2s}={({\partial }^{s}\phi )}^{2}.\end{equation} \tag{ 3.16 }$$
• (b)  
  If on the other hand $\theta (\phi )={\mathfrak{c}}_{1}\,\mathrm{cosh}\left({\mathfrak{c}}_{2}\phi -{\phi }_{0}\right)$, the higher conserved charges take a different form:

  $$\begin{equation}\theta (\phi )={\mathfrak{c}}_{1}\,\mathrm{cosh}\left({\mathfrak{c}}_{2}\phi -{\phi }_{0}\right){\Rightarrow}{T}_{2s}={(\partial \phi )}^{2s}+\frac{1}{{\mathfrak{c}}_{2}^{2}}\ {({\partial }^{s}\phi )}^{2}.\end{equation} \tag{ 3.17 }$$

In the following it will be convenient to distinguish between charges of the form (a) ${({\partial }^{s}\phi )}^{2}$ or (b) (∂ϕ)^2s . Statements about their linear combinations, as e.g. in (3.17) follow straightforwardly.

Now we proceed with a similar investigation in the context of deformed theories. We shall restrict to the $T\bar{T}$ case in this section.

Bulk $T\bar{T}$ deformation. As a starting point for the $T\bar{T}$ deformation consider the stress tensor. For the free theory, using the general formula

$$\begin{equation}{T}_{\mu \nu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\phi )}{\partial }_{\nu }\phi -{\delta }_{\mu \nu }\mathcal{L},\end{equation} \tag{ 3.18 }$$

we find

$$\begin{equation}\mathrm{det}\,{T}_{\mu \nu }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=\frac{1}{4}{[{({\partial }_{x}\phi )}^{2}+{({\partial }_{y}\phi )}^{2}]}^{2}.\end{equation} \tag{ 3.19 }$$

Equivalently, in complex coordinates we have

$$\begin{equation}\mathrm{det}\,{T}_{\mu \nu }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}={(\partial \phi \bar{\partial }\phi )}^{2}.\end{equation} \tag{ 3.20 }$$

We assume that in the bulk the action is deformed by the conventional $T\bar{T}$ deformation, i.e. the deformation is defined by the equation

$$\begin{equation}\frac{\mathrm{d}{S}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}}{\mathrm{d}\lambda }=-{\int }_{-\infty }^{\infty }\mathrm{d}y{\int }_{-\infty }^{0}\mathrm{d}x\,\mathrm{det}\,{T}_{\mu \nu }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}.\end{equation} \tag{ 3.21 }$$

The deformed bulk Lagrangian is the Nambu–Goto Lagrangian [2]

$$\begin{equation}{\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=\frac{1}{2\lambda }\left(\sqrt{1+4\lambda (\partial \phi \bar{\partial }\phi )}-1\right)=(\partial \phi \bar{\partial }\phi )-\lambda {(\partial \phi )}^{2}{(\bar{\partial }\phi )}^{2}+\mathcal{O}({\lambda }^{2}).\end{equation} \tag{ 3.22 }$$

It induces the deformed bulk equation of motion

$$\begin{equation}\partial \bar{\partial }\phi =\lambda \frac{{\bar{\partial }}^{2}\phi {(\partial \phi )}^{2}+{\partial }^{2}\phi {(\bar{\partial }\phi )}^{2}}{1+2\lambda \partial \phi \bar{\partial }\phi },\end{equation} \tag{ 3.23 }$$

and the deformed boundary equation of motion ⁶

$$\begin{equation}0=\frac{{\partial }_{x}\phi }{\sqrt{1+\lambda [{({\partial }_{x}\phi )}^{2}+{({\partial }_{y}\phi )}^{2}]}}+{\left.\frac{\mathrm{d}{\theta }_{\lambda }}{\mathrm{d}\phi }\right\vert }_{x=0}.\end{equation} \tag{ 3.24 }$$

Here θ_λ represents the deformed boundary potential, and we assume it is still only a function of the bulk field, but not of its derivatives.

In the following we will identify the choices of the boundary potential which preserve integrability, i.e. which imply that all higher conserved charges satisfy the modified boundary condition (2.25).

Deformed higher charges. Using the deformed equation of motion and the different possibilities for the initial (undeformed) higher charges T_2s (λ = 0), one can obtain the deformed higher charges:

• (a)  
  If the initial charges are T_2s (0) = (∂ϕ)^2s and Θ_s (0) = 0, the all-order deformations are known and given by [2]

  $$\begin{equation}{T}_{s}(\lambda )=\frac{{(\partial \phi )}^{s}}{\mathcal{S}}{\left(\frac{2}{\mathcal{S}+1}\right)}^{s-2},\qquad {{\Theta}}_{s-2}(\lambda )=\frac{\lambda {(\partial \phi )}^{s}{(\bar{\partial }\phi )}^{2}}{\mathcal{S}}{\left(\frac{2}{\mathcal{S}+1}\right)}^{s},\end{equation} \tag{ 3.25 }$$

  where $\mathcal{S}=\sqrt{1+4\lambda \partial \phi \bar{\partial }\phi }$.
• (b)  
  On the other hand, if the initial charges are ${T}_{2s}(0)={({\partial }^{s}\phi )}^{2}$ and Θ_s (0) = 0, the all order deformations can in principle be obtained, but there is no simple formula for a generic deformed charge. The leading higher charges T₄(λ) and Θ₂(λ) are given by [22]

  $$\begin{equation}\begin{aligned}\hfill {T}_{4}(\lambda )& =\frac{{(\partial \phi )}^{2}}{\mathcal{S}}{\left(\frac{{(\mathcal{S}-1)}^{4}{\bar{\partial }}^{2}\phi -16{\lambda }^{2}{(\bar{\partial }\phi )}^{4}{\partial }^{2}\phi }{4\lambda (\mathcal{S}-1)({\mathcal{S}}^{2}+1){(\bar{\partial }\phi )}^{3}}\right)}^{2},\hfill \\ \hfill {{\Theta}}_{2}(\lambda )& =\frac{{(\mathcal{S}-1)}^{2}}{4\lambda \mathcal{S}}{\left(\frac{{(\mathcal{S}-1)}^{4}{\bar{\partial }}^{2}\phi -16{\lambda }^{2}{(\bar{\partial }\phi )}^{4}{\partial }^{2}\phi }{4\lambda (\mathcal{S}-1)({\mathcal{S}}^{2}+1){(\bar{\partial }\phi )}^{3}}\right)}^{2}.\hfill \end{aligned}\end{equation} \tag{ 3.26 }$$

The conjugates ${\bar{T}}_{s}$ and ${\bar{{\Theta}}}_{s}$ are obtained by replacing z by $\bar{z}$.

Clearly, if the initial charges are linear combinations of both types (a) and (b) presented above, as in the case (3.17), the deformed charges are given by the same (deformed) linear combinations.

Boundary $T\bar{T}$ deformation for θ =θ (ϕ) . Let us now investigate the following boundary integrablity condition (2.25) for the first few charges:

$$\begin{equation}{\left.-\mathrm{i}({T}_{r+1}-{\bar{T}}_{r+1}+{\bar{{\Theta}}}_{r-1}-{{\Theta}}_{r-1})\right\vert }_{x=0}=\frac{\mathrm{d}}{\mathrm{d}y}{\theta }_{r}(y).\end{equation} \tag{ 3.27 }$$

Even though conservation of T₂ does not correspond to integrability, we first discuss this case for completeness. For r = 1 the above equation involves the deformed stress tensor T₂. Note that T₂ is unique, such that we can use (3.25). We immediately see that Θ₀(λ) and ${\bar{{\Theta}}}_{0}(\lambda )$ cancel out, and the non-trivial contribution at the boundary is given by

$$\begin{align}\hfill {\left.-\mathrm{i}({T}_{2}+{\bar{{\Theta}}}_{0}-{\bar{T}}_{2}-{{\Theta}}_{0})\right\vert }_{x=0}& =\frac{\mathrm{i}}{\sqrt{1+4\lambda \partial \phi \bar{\partial }\phi }}\left({(\partial \phi )}^{2}-{(\bar{\partial }\phi )}^{2}\right)\hfill \\ \hfill & ={\left.\frac{{\partial }_{x}\phi {\partial }_{y}\phi }{\sqrt{1+4\lambda \partial \phi \bar{\partial }\phi }}\right\vert }_{x=0}\hfill \\ \hfill & =-{\partial }_{y}\phi \ \frac{\mathrm{d}{\theta }_{\lambda }}{\mathrm{d}\phi }.\hfill \end{align} \tag{ 3.28 }$$

It is now clear that as long as θ_λ is a function of ϕ only, the expression above is a total y-derivative, regardless of the functional form of θ_λ .

The first non-trivial constraint on the boundary function θ_λ comes from the leading higher charge T₄. A generic all-order analysis seems rather involved, so we restrict to a perturbative analysis at $\mathcal{O}(\lambda )$. The bulk equation of motion is then given by

$$\begin{equation}{\partial }_{x}^{2}\phi =-{\partial }_{y}^{2}\phi +\lambda \left({\partial }_{y}^{2}\phi \left[{({\partial }_{y}\phi )}^{2}-{({\partial }_{x}\phi )}^{2}\right]+2{\partial }_{x}\phi {\partial }_{y}\phi ({\partial }_{x}{\partial }_{y}\phi )\right)+\mathcal{O}({\lambda }^{2}).\end{equation} \tag{ 3.29 }$$

Similarly, we can solve the deformed boundary equation of motion. Expanding the boundary potential in λ according to

$$\begin{equation}{\theta }_{\lambda }={\theta }_{(0)}+\lambda {\theta }_{(1)}+\mathcal{O}({\lambda }^{2}),\end{equation} \tag{ 3.30 }$$

we find

$$\begin{equation}{\left.{\partial }_{x}\phi \right\vert }_{x=0}=-{\theta }_{(0)}^{\prime }-\lambda \left({\theta }_{(1)}^{\prime }-\frac{1}{2}{\theta }_{(0)}^{\prime }[{({\theta }_{(0)}^{\prime })}^{2}+{({\partial }_{y}\phi )}^{2}]\right)+\mathcal{O}({\lambda }^{2}).\end{equation} \tag{ 3.31 }$$

Using those deformed equations of motion, we can analyze the deformed T₄ for different undeformed charges:

• (a)  
  If the undeformed charges are T₄ = (∂ϕ)⁴, using the general expressions (3.25), we find that at $\mathcal{O}(\lambda )$ the expression only contains single x-derivatives. Substituting ∂_x ϕ, we obtain the $\mathcal{O}(\lambda )$ contributions

  $$\begin{equation}\begin{aligned}\hfill & {\left.-2\mathrm{i}({T}_{4}+{\bar{{\Theta}}}_{2}-{\bar{T}}_{4}-{{\Theta}}_{2})\right\vert }_{x=0,\mathcal{O}(\lambda )}\hfill \\ \hfill & \qquad =\left(\frac{3}{4}{({\theta }_{(0)}^{\prime })}^{3}-{\theta }_{(1)}^{\prime }\right){({\partial }_{y}\phi )}^{3}+\frac{3}{8}{\theta }_{(0)}^{\prime }{({\partial }_{y}\phi )}^{5}+(\text{functions}\;\text{of}\ \phi )\ {\partial }_{y}\phi .\hfill \end{aligned}\end{equation} \tag{ 3.32 }$$
• (b)  
  On the other hand, if the undeformed charge reads ${T}_{4}={({\partial }^{2}\phi )}^{2}$, using (3.26) and substituting ${\partial }_{x}\phi ,{\partial }_{x}^{2}\phi$, we find that the additional $\mathcal{O}(\lambda )$ contribution is given by

  $$\begin{align}\hfill & {\left.-2\mathrm{i}({T}_{4}+{\bar{{\Theta}}}_{2}-{\bar{T}}_{4}-{{\Theta}}_{2})\right\vert }_{x=0,\mathcal{O}(\lambda )}\hfill \\ \hfill & \qquad =(\text{functions}\;\text{of}\ \phi )\ {\partial }_{y}\phi +\left(\frac{3}{2}{\theta }_{(0)}^{\prime }{({\theta }_{(0)}^{{\prime\prime}})}^{2}{({\partial }_{y}\phi )}^{3}-\left[\frac{3}{2}{({\theta }_{(0)}^{\prime })}^{2}{\theta }_{(0)}^{{\prime\prime}}+2{\theta }_{(1)}^{{\prime\prime}}\right]{\partial }_{y}\phi {\partial }_{y}^{2}\phi \right)\hfill \\ \hfill & \quad \qquad +\frac{3}{2}\left({\theta }_{(0)}^{{\prime\prime}}{({\partial }_{y}\phi )}^{3}{\partial }_{y}^{2}\phi -{\theta }_{(0)}^{\prime }({\partial }_{y}\phi ){({\partial }_{y}^{2}\phi )}^{2}\right).\hfill \end{align} \tag{ 3.33 }$$

From the expressions for both of the above cases one immediately sees that the five-derivative terms, namely the terms that are proportional to ${({\partial }_{y}\phi )}^{5},({\partial }_{y}\phi ){({\partial }_{y}^{2}\phi )}^{2}$, or ${({\partial }_{y}\phi )}^{3}({\partial }_{y}^{2}\phi )$ only depend on the undeformed boundary potential θ₍₀₎. Since we are missing a term ${\partial }_{y}^{3}\phi {\partial }_{y}^{2}\phi$ to combine with ${\partial }_{y}\phi {({\partial }_{y}^{2}\phi )}^{2}$, we are unable to turn those five-derivative terms into a total y-derivative. The only solution which preserves integrability is to set θ₍₀₎ to zero.

If we set θ₍₀₎ = 0, the θ₍₁₎ dependent terms now have exactly the same structure as those for the boundary function in the undeformed theory. Hence, the most general solution at this order is identical to the undeformed boundary function. In other words, the boundary potential would be delayed by one order. However, since we have found that the $\mathcal{O}(\lambda )$ analysis forces θ₍₀₎ to vanish, it is conceivable that going to the next perturbative order $\mathcal{O}({\lambda }^{2})$ implies that also θ₍₁₎ has to vanish in order to preserve integrability. This phenomenon is actually quite general. We have verified that for the Sine-Gordon theory, the deformed boundary potential also gets delayed, see appendix A for details. This suggests that the most general boundary potential which is compatible with integrability and the bulk $T\bar{T}$ deformation is zero.

Boundary $T\bar{T}$ deformation for θ =θ (ϕ, ∂ _y ϕ) . The result in the previous paragraph does not necessarily exclude the possibility that the boundary potential depends on the derivatives of ϕ. In fact, we shall see that at leading order, the deformed boundary Lagrangian can be non-trivial, if it depends also on ∂_y ϕ.

Drawing inspiration from the $T\bar{T}$ deformed charges, we now assume that the deformed boundary potential is a function of ϕ and ∂_y ϕ. The deformed Lagrangian takes the form

$$\begin{equation}{\theta }_{\lambda }={\theta }_{(0)}(\phi )+\lambda \left({\theta }_{(1,1)}(\phi )+{\theta }_{(1,2)}(\phi ){({\partial }_{y}\phi )}^{2}\right)+\mathcal{O}({\lambda }^{2}),\end{equation} \tag{ 3.34 }$$

where we assume the undeformed boundary Lagrangian θ₍₀₎ is still of the potential type, i.e. does not depend on derivatives of ϕ. Here, we have assumed that the $\mathcal{O}(\lambda )$ terms only contain y-derivatives up to second order ${\partial }_{y}^{2}\phi$, because higher order derivative terms would contribute to (2.25) with more than five-derivative terms, which would be inconsistent with the structure we have found. Now, the deformed boundary equation of motion becomes

$$\begin{align}\hfill {\left.{\partial }_{x}\phi \right\vert }_{x=0}& =-{\theta }_{(0)}^{\prime }-\lambda \left({\theta }_{(1,1)}{(\phi )}^{\prime }-{\theta }_{(1,2)}{(\phi )}^{\prime }{({\partial }_{y}\phi )}^{2}-2{\theta }_{(1,2)}(\phi ){({\partial }_{y}\phi )}^{2}\right.\hfill \\ \hfill & \quad \left.+\frac{1}{2}{\theta }_{(0)}^{\prime }[{({\theta }_{(0)}^{\prime })}^{2}+{({\partial }_{y}\phi )}^{2}]\right)+\mathcal{O}({\lambda }^{2}).\hfill \end{align} \tag{ 3.35 }$$

With the additional θ_(1,2) term, there will be a new five-derivative contribution proportional to $({\partial }_{y}^{3}\phi )({\partial }_{y}^{2}\phi )$ in (3.33). Combined with the problematic ${\partial }_{y}\phi {({\partial }_{y}^{2}\phi )}^{2}$ term, they can become a total y-derivative. Explicitly, they read

$$\begin{equation}({\partial }_{y}\phi ){({\partial }_{y}^{2}\phi )}^{2}\underset{={\mathcal{A}}_{1,2,2}}{\underbrace{\left(8{\theta }_{(1,2)}^{\prime }-\frac{3}{2}{\theta }_{(0)}^{\prime }\right)}}+({\partial }_{y}^{3}\phi )({\partial }_{y}^{2}\phi )\underset{={\mathcal{A}}_{2,3}}{\underbrace{4{\theta }_{(1,2)}}}.\end{equation} \tag{ 3.36 }$$

Thus, the coefficients must satisfy

$$\begin{equation}{\mathcal{A}}_{2,3}(\phi )-2\int \mathrm{d}\phi \ {\mathcal{A}}_{1,2,2}(\phi )=\text{const}{\Rightarrow}{\theta }_{(1,2)}={\mathfrak{c}}_{1,2}+\frac{1}{4}{\theta }_{(0)},\end{equation} \tag{ 3.37 }$$

where ${\mathfrak{c}}_{1,2}$ denotes an arbitrary constant. What remains in (3.33) is simply a term of the form $2{\theta }_{(0)}^{{\prime\prime}}{({\partial }_{y}\phi )}^{3}({\partial }_{y}^{2}\phi )$. Now, we can use the explicit forms of the undeformed boundary potential, which are compatible with integrability (see (3.16) and (3.17)):

• (a)  
  If the undeformed boundary potential reads gϕ²/2, the remaining term $2{\theta }_{(0)}^{{\prime\prime}}{({\partial }_{y}\phi )}^{3}({\partial }_{y}^{2}\phi )$ is automatically a total y-derivative. Demanding the three-derivative terms to be a total y-derivative, we find

  $$\begin{equation}{\theta }_{(1,1)}=-\frac{5{g}^{3}}{32}{\phi }^{4}+{\mathfrak{d}}_{1}{\phi }^{2}+{\mathfrak{d}}_{2}\phi +{\mathfrak{d}}_{3},\end{equation} \tag{ 3.38 }$$

  where the ${\mathfrak{d}}_{j}$ denote constants.
• (b)  
  If the undeformed boundary potential reads ${\mathfrak{c}}_{1}\,\mathrm{cosh}({\mathfrak{c}}_{2}\phi -{\phi }_{0})$, the term $2{\theta }_{(0)}^{{\prime\prime}}{({\partial }_{y}\phi )}^{3}({\partial }_{y}^{2}\phi )$ must combine with other five-derivative terms coming from the deformed (∂ϕ)⁴ charge. For the five-derivative terms the condition of being a total derivative is then automatically satisfied. Plugging this into the third-order terms, we find a second-order differential equation for θ_(1,1):

  $$\begin{equation}0=\frac{3}{2}{\mathfrak{c}}_{1}^{2}{\mathfrak{c}}_{2}^{2}\,{\mathrm{sinh}}^{2}\left({\mathfrak{c}}_{2}\phi -{\phi }_{0}\right)\left({\mathfrak{c}}_{1,2}-2{\mathfrak{c}}_{1}\,\mathrm{cosh}\left({\mathfrak{c}}_{2}\phi -{\phi }_{0}\right)\right)-\frac{2{\theta }_{1,1}^{{\prime\prime}}(\phi )}{{\mathfrak{c}}_{2}^{2}}.\end{equation} \tag{ 3.39 }$$

  The solution to this equation is easily obtained, but it is not illuminating. Therefore we shall not present it here.

Summary. To summarize, given the $T\bar{T}$ deformed bulk Lagrangian our $\mathcal{O}(\lambda )$ perturbative analysis shows the following:

• If the boundary function only depends on ϕ and not on its derivatives, the deformed boundary potential is delayed by one order. This suggests that for the full $T\bar{T}$ deformed bulk Lagrangian, the only integrability-preserving boundary potential is zero.
• If instead we allow for derivative corrections at $\mathcal{O}(\lambda )$ in the boundary function, then a non-trivial, integrability-preserving solution for the deformed boundary Lagrangian exists. This suggests that a non-trivial, integrability-preserving boundary Lagrangian necessarily depends on ∂_y ϕ.

Let us review the construction of [9, 10] adapted to the field theory context. We are interested in deformations of a Hamiltonian, or more generically, a set of conserved charges, which preserve locality ⁷ in the sense that the undeformed as well as the deformed Hamiltonian can be written as an integral over a local density $\mathcal{H}(x)$:

$$\begin{equation}H=\int \mathrm{d}x\,\mathcal{H}(x).\end{equation} \tag{ 4.1 }$$

In the following we will focus on continuous models but similar considerations apply to discrete spin chains, where integrals are replaced by lattice sums. A general class of deformed Hamiltonians H_λ is defined via a parallel transport equation of the following form:

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\lambda }{H}_{\lambda }=\left[X,{H}_{\lambda }\right].\end{equation} \tag{ 4.2 }$$

where the deformation operator X also depends on λ, but we omit it systematically for simplicity. For an integrable model, such deformations preserve integrability if the integrable charges Q_r are deformed by means of the same deformation equation as the Hamiltonian. Deformations that preserve locality in the above sense are conveniently introduced using the notion of bilocal operators defined as

Here A and B represent two local operators in the above sense and s_L and s_R denote the positions of the left and right boundary, respectively. For the infinite line we have s_L/R = ∓∞. The label 'closed' indicates that we will slightly refine this definition of a bilocal operator in the context of open boundary systems. While the introduction of the above bilocal operators may seem ad hoc at first sight, it is motivated by the following fact: for spin chains it has been shown that this class of deformation generators exhausts the complete space of integrability preserving deformations found for closed [51] and open [52] $\mathfrak{g}\mathfrak{l}(N)$ chains, as well as in the XXZ case [27].

Importantly, locality of the Hamiltonian H defined via (4.2) is preserved if the local operators A and B both commute with H, e.g. for two conserved charges A = Q_r and B = Q_s . In that case, i.e. for X = [Q_r |Q_s ], the only non-vanishing contributions to the commutator $\left[[A\vert B],H\right]$ originate from the local term ${\mathcal{Q}}_{r}(x){\mathcal{Q}}_{s}(x)$ in the definition of the bilocal operator (4.3), which yields a local result, see figure 2. ⁸ Would either Q_r or Q_s not commute with H, the result of the commutator were bilocal. This subtlety arises in the case of open boundaries where the parity-odd charges are typically not conserved but still required to generate the full space of admissible deformations [11], see section 4.2. Note that the charges Q_r can be taken to be the spacetime P- or H-type charges described in section 2.1 or some internal commuting charge, see [27] for an example of latter. While the $T\bar{T}$-deformation belongs to the class of deformations induced by bilocal spacetime charges, the combination of spacetime and internal charge is referred to as $J\bar{T}$-deformations in the field theory context, see [12–19]. Similar deformations for spin chains were considered in [27].

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Note that also the constituent charges Q_r and Q_s of the bilocal charge [Q_r |Q_s ] should be deformed via an equation of the form (4.2), in order to preserve $\left[H(\lambda ),{Q}_{r}(\lambda )\right]=0$ for λ ≠ 0. That is, strictly speaking all operators in this paragraph carry an argument λ, which we have omitted to avoid clutter.

Relation to $T\bar{T}$ -like deformations. Let us relate the above bilocal deformations to bilinear deformations expressed in terms of currents, see [25, 26, 53]. Consider the deformation equation (4.2) with

$$\begin{equation}X=[{Q}_{r}\vert {Q}_{s}]={\int }_{{x}_{1}< {x}_{2}}\mathrm{d}{x}_{1}\,\mathrm{d}{x}_{2}\,{\mathcal{Q}}_{r}({x}_{1}){\mathcal{Q}}_{s}({x}_{2}).\end{equation} \tag{ 4.4 }$$

Using the Heisenberg equation (2.19) given by ${\partial }_{y}{\mathcal{Q}}_{r}=\left[\mathcal{H},{\mathcal{Q}}_{r}\right]$ as well as the conservation equation (2.18) given by ${\partial }_{y}{\mathcal{Q}}_{r}=-{\partial }_{x}{\mathcal{J}}_{r}$, we find

$$\begin{align}\hfill \left[X,H\right]& ={\int }_{{x}_{1}< {x}_{2}}\mathrm{d}{x}_{1}\,\mathrm{d}{x}_{2}\left(\left[{\mathcal{Q}}_{r}({x}_{1}),H\right]{\mathcal{Q}}_{s}({x}_{2})+{\mathcal{Q}}_{r}({x}_{1})\left[{\mathcal{Q}}_{s}({x}_{2}),H\right]\right)\hfill \\ \hfill & ={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\mathrm{d}{x}_{2}\left({\mathcal{J}}_{r}({x}_{2})-{\mathcal{J}}_{r}({s}_{\text{L}})\right){\mathcal{Q}}_{s}({x}_{2})+{\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\mathrm{d}{x}_{1}\,{\mathcal{Q}}_{r}({x}_{1})\left({\mathcal{J}}_{s}({s}_{\text{R}})-{\mathcal{J}}_{s}({x}_{1})\right).\hfill \end{align} \tag{ 4.5 }$$

We introduce the operator

$$\begin{equation}{\mathcal{O}}_{rs}=-{{\epsilon}}_{\mu \nu }{J}_{r}^{\mu }{J}_{s}^{\nu }={\mathcal{J}}_{r}{\mathcal{Q}}_{s}-{\mathcal{Q}}_{r}{\mathcal{J}}_{s},\end{equation} \tag{ 4.6 }$$

such that we can write

$$\begin{equation}\left[X,H\right]={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}{\mathcal{O}}_{rs}(x)\mathrm{d}x-{\mathcal{J}}_{r}({s}_{\text{L}}){Q}_{s}+{Q}_{r}{\mathcal{J}}_{s}({s}_{\text{R}}).\end{equation} \tag{ 4.7 }$$

On the infinite line with s_L = −∞ and s_R = +∞, where the last two terms drop out, we thus find

$$\begin{equation}\frac{\mathrm{d}{H}_{\lambda }}{\mathrm{d}\lambda }={\int }_{-\infty }^{\infty }{\mathcal{O}}_{rs}(x)\mathrm{d}x.\end{equation} \tag{ 4.8 }$$

In the special case of s = 1, r = 2 with T = T₂, Θ = Θ₀, we have

$$\begin{equation}{\mathcal{O}}_{21}=4\left(T\bar{T}-{\Theta}\bar{{\Theta}}\right)=\mathrm{det}\,{T}_{\mu \nu }.\end{equation} \tag{ 4.9 }$$

Note that the densities ${\mathcal{Q}}_{r}$ and ${\mathcal{J}}_{s}$ also receive deformations in λ, and that the deformation equation (4.2) is formally solved by

$$\begin{equation}{H}_{\lambda }={U}_{\lambda }^{-1}{H}_{0}{U}_{\lambda },\qquad {U}_{\lambda }=\mathcal{P}\,\mathrm{exp}\left[-{\int }_{0}^{\lambda }X({\lambda }^{\prime })\,\mathrm{d}{\lambda }^{\prime }\right].\end{equation} \tag{ 4.10 }$$

For systems with open boundary conditions there are important differences in the construction of locality and integrability preserving deformations to the closed case discussed above, see [11].

Firstly, for open systems merely the parity-even charges Q_2r = H_2r−1 are conserved, while conservation of the odd charges Q_2r−1 = P_2r−1 is typically broken by boundary terms. In particular, the odd momentum operator Q₁ = P is not conserved. Nevertheless, we can still define odd 'charge operators', see (2.16), by the requirement that these are conserved in the bulk of the theory. In particular, we have

$$\begin{equation}\left[{Q}_{r},{Q}_{2s+1}\right]=\text{boundary}\;\text{terms},\end{equation} \tag{ 4.11 }$$

where boundary terms may act nontrivially on the boundary but vanish in the bulk. Note that formally we may include boundary terms like the boundary Hamiltonian θ into the bulk density of the operators by writing them as total derivatives, e.g.

$$\begin{equation}\theta (x={s}_{\text{R}},y)={\int }_{-\infty }^{{s}_{\text{R}}}\mathrm{d}x\,{\partial }_{x}\theta (x,y).\end{equation} \tag{ 4.12 }$$

Note, however, that the classical analysis of the $T\bar{T}$-deformed model in the previous section 3 suggests to set the boundary functions θ of the considered charges to zero.

Secondly, deformations with bilocal charges [Q_r |Q_s ] will generically result in bilocal deformations of the even conserved charges Q_2t of the open model, if one of the charges Q_r or Q_s is odd. Therefore the order of the local operators entering the bilocal operator is crucial when applied to semi-infinite systems. We will distinguish such systems with either an open boundary on the left (left-open) or on the right (right-open). For a left-open model for instance, we obtain local deformations only when using the bilocal operator [Q_2r |Q_2s+1], but not for [Q_2s+1|Q_2r ].

Finally, in the case of open boundaries nontrivial deformations can be induced by local operators in addition to the above bilocal charges. In particular, this means that the precise choice of local regularization of bilocal operators becomes important. In the open case we define the bilocal operators as ⁹

$$\begin{align}\hfill [A\vert B]& {:=}{\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\mathrm{d}{x}_{2}{\int }_{{s}_{\text{L}}}^{{x}_{2}}\mathrm{d}{x}_{1}\,\frac{1}{2}\left\{\mathcal{A}({x}_{1}),\mathcal{B}({x}_{2})\right\}-\frac{1}{4}{\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\mathrm{d}x\,\left\{\mathcal{A}(x),\mathcal{B}(x)\right\}\hfill \\ \hfill & ={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\mathrm{d}{x}_{2}{\int }_{{s}_{\text{L}}}^{{x}_{2}}\mathrm{d}{x}_{1}\,\frac{1}{2}\left(1-\frac{1}{2}\delta ({x}_{1}-{x}_{2})\right)\left\{\mathcal{A}({x}_{1}),\mathcal{B}({x}_{2})\right\}.\hfill \end{align} \tag{ 4.13 }$$

Here $\left\{\cdot ,\cdot \right\}$ denotes the anti-commutator. The above definition yields

$$\begin{equation}[A\vert B]+[B\vert A]=\frac{1}{2}\left\{A,B\right\}.\end{equation} \tag{ 4.14 }$$

In particular, this regularization implies that the sum of conserved bilocal charges

$$\begin{equation}[{Q}_{r}\vert {Q}_{s}]+[{Q}_{s}\vert {Q}_{r}]={Q}_{r}{Q}_{s}\end{equation} \tag{ 4.15 }$$

commutes with the Hamiltonian (and higher integrable charges) in the bulk, i.e. the only non-trivial bulk deformations are induced by the difference of bilocal charges X = [Q_r |Q_s ] − [Q_s |Q_r ] when inserted into (4.13). Moreover, the bulk projection of the commutator of the form $\left[[{Q}_{2r}\vert {Q}_{2s+1}],{Q}_{2t}\right]$ equals the bulk projection of the commutator $-\left[[{Q}_{2s+1}\vert {Q}_{2r}],{Q}_{2t}\right]$, i.e.

$$\begin{equation}{\left.\left[[{Q}_{2r}\vert {Q}_{2s+1}],{Q}_{2t}\right]+\left[[{Q}_{2s+1}\vert {Q}_{2r}],{Q}_{2t}\right]\right\vert }_{\text{bulk}}={\left.\frac{1}{2}\left[\left\{{Q}_{2r},{Q}_{2s+1}\right\},{Q}_{2t}\right]\right\vert }_{\text{bulk}}=0,\end{equation} \tag{ 4.16 }$$

which will be important for the below construction.

Following [11], in this section, we present more details of the deformations for open boundaries which are the main focus of this paper. In particular, we will review how deformations generated in a left- and right-open model can be combined into deformed charge operators for systems with two boundaries.

First of all we refine the above notion of boundary terms by introducing left boundary terms ${A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}$ and right boundary terms ${A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}$, which only act on the left or the right boundary, respectively. Acting for instance with a right boundary term ${A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}$ on a state in a left-open model yields zero:

$$\begin{equation}{A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\left\vert \psi \right\rangle }_{\text{L}}=0.\end{equation} \tag{ 4.17 }$$

Here we denote states in the left- or right-open model by ${\left\vert \cdot \right\rangle }_{\text{L}}$ or ${\left\vert \cdot \right\rangle }_{\text{R}}$, respectively. Moreover, it will be useful to introduce a notion of setting boundary terms to zero. We employ the notation |_L to indicate that we set right boundary terms to zero and |_R to set left boundary terms to zero. More explicitly, if we apply the boundary conditions of a left- or right-open model denoted by |_L or |_R, respectively, we have

$$\begin{equation}{A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\vert }_{\text{L}}={A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}},\qquad {A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\vert }_{\text{R}}=0,\end{equation} \tag{ 4.18 }$$

$$\begin{equation}{A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\vert }_{\text{L}}=0,\qquad {A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\vert }_{\text{R}}={A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}.\end{equation} \tag{ 4.19 }$$

The bulk part of a local operator A can thus be defined as

$$\begin{equation}{A}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=A{\vert }_{\text{L}}{\vert }_{\text{R}}\equiv A{\vert }_{\mathrm{L}\mathrm{R}}.\end{equation} \tag{ 4.20 }$$

Note that for a system with open boundaries, the odd charges commute up to boundary terms according to

$$\begin{equation}\left[{Q}_{r},{Q}_{2s+1}\right]={A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}+{A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}.\end{equation} \tag{ 4.21 }$$

Semi-infinite systems. We now want to deform a set of even charge operators Q_2r,L/R = Q_2r,L/R(λ = 0), which are conserved in the left or right open model, respectively. For a non-integrable model this set may only contain the Hamiltonian Q_2,L/R = H_L/R. Accordingly, we introduce two sets of charges deformed in the parameter λ labelled by L and R and defined by the equation ¹⁰

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\lambda }{Q}_{2r,\mathrm{L}/\mathrm{R}}(\lambda )={\left.\left[{X}_{\mathrm{L}/\mathrm{R}}(\lambda ),{Q}_{2r,\mathrm{L}/\mathrm{R}}(\lambda )\right]\right\vert }_{\mathrm{L}/\mathrm{R}}.\end{equation} \tag{ 4.22 }$$

This equation guarantees that charges, which commute for λ = 0, will also commute for a non-vanishing deformation parameter λ ≠ 0. Here the requirement of locality for the deformed charges implies that the bilocal deformation generators have to be chosen as

$$\begin{equation}{X}_{\text{L}}=+[{Q}_{2r}\vert {Q}_{2s+1}],\qquad {X}_{\text{R}}=-[{Q}_{2s+1}\vert {Q}_{2r}].\end{equation} \tag{ 4.23 }$$

The two sets of (deformed) charges defined by (4.22) take the form

$$\begin{equation}{Q}_{2r,\mathrm{L}/\mathrm{R}}={Q}_{2r}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}+{Q}_{2r,\mathrm{L}/\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}},\end{equation} \tag{ 4.24 }$$

with a bulk term and a left or right boundary term, respectively. Here the building blocks are defined in terms of the solutions of (4.22) according to

$$\begin{equation}{Q}_{2r}^{\text{bulk}}={Q}_{2r,\mathrm{L}}{\vert }_{\mathrm{L}\mathrm{R}}={Q}_{2r,\mathrm{R}}{\vert }_{\mathrm{L}\mathrm{R}},\end{equation} \tag{ 4.25 }$$

$$\begin{equation}{Q}_{2r,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}={Q}_{2r,\mathrm{L}}-{Q}_{2r}^{\text{bulk}},\end{equation} \tag{ 4.26 }$$

$$\begin{equation}{Q}_{2r,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}={Q}_{2r,\mathrm{R}}-{Q}_{2r}^{\text{bulk}},\end{equation} \tag{ 4.27 }$$

cf (4.16) for the second equality in the first line. The deformed charges commute by definition in the left- or right-open model, respectively, e.g. in the left-open case we have

$$\begin{equation}\left[{Q}_{2r,\mathrm{L}},{Q}_{2s,\mathrm{L}}\right]={A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}{\vert }_{\text{L}}=0.\end{equation} \tag{ 4.28 }$$

Here ${A}_{\text{R}}^{\text{bdr}}$ denotes some boundary term that only acts on a right boundary. In the left-open model, however, the right boundary is absent and thus the boundary term vanishes when imposing the respective boundary conditions as denoted by |_L. The above equation (4.28) can be expanded according to

$$\begin{equation}\left[{Q}_{2r}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}},{Q}_{2s}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\right]={A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}+{A}_{\text{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}},\end{equation} \tag{ 4.29 }$$

$$\begin{equation}\left[{Q}_{2r}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}},{Q}_{2s,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}\right]+\left[{Q}_{2r,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}},{Q}_{2s}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\right]+\left[{Q}_{2r,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}},{Q}_{2s,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}\right]=-{A}_{\text{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}.\end{equation} \tag{ 4.30 }$$

Finite systems. The next important step is to proceed to a finite open system with boundaries on the left and on the right. Using the above building blocks from both half-open systems, we define deformed charges as

$$\begin{equation}{Q}_{2r}(\lambda )={Q}_{2r}^{\text{bulk}}(\lambda )+{Q}_{2r,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}(\lambda )+{Q}_{2r,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}(\lambda ).\end{equation} \tag{ 4.31 }$$

The charges defined in this way obey

$$\begin{equation}\left[{Q}_{2r},{Q}_{2s}\right]=\left[{Q}_{2r,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}},{Q}_{2s,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}\right]+\left[{Q}_{2r,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}},{Q}_{2s,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}\right]={A}_{\mathrm{L}\& \mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}.\end{equation} \tag{ 4.32 }$$

Here the terms ${A}_{\mathrm{L}\& \mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}$ act on both boundaries at the same time and are referred to as spanning terms, cf [11, 52]. In the spin chain case it is clear that the interaction range of the charge deformations increases with increasing order in λ. Hence, for a given chain of finite length, spanning terms arise at a finite order of the deformation parameter λ. In the field theory case the deformations of the charge operators at a given perturbative order in λ are naively localized at some point x and would thus never act on both boundaries at the same time, i.e. contributions ${A}_{\mathrm{L}\& \mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}$ would be zero in the field theory. In a non-perturbative context, however, interactions seeing both boundaries may arise.

Finally we note that deformations with bilocal operators [Q_2r |Q_2s ] composed of two even charges can as well be performed within the finite model and thus correspond to trivial similarity transformations. Deformations with [Q_2r+1|Q_2s+1] do not result in local deformations.

Expressions in terms of currents. A feature that did so far not appear in the context of field theory $T\bar{T}$-like deformations for closed systems are the above deformations generated by odd charges, see [11] for the spin chain case. Let us thus translate these deformations into expressions in terms of currents. For the open model the Hamiltonian does not commute with the odd charges and we have

$$\begin{align}\hfill 0\ne \left[H,{Q}_{2r+1}\right]& ={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}\left[H,{\mathcal{Q}}_{2r+1}(x)\right]\mathrm{d}x={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}{\partial }_{y}{\mathcal{Q}}_{2r+1}(x)\hfill \\ \hfill & ={\int }_{{s}_{\text{L}}}^{{s}_{\text{R}}}(-{\partial }_{x}{\mathcal{J}}_{2r+1}(x))=-{\mathcal{J}}_{2r+1}({s}_{\text{R}})+{\mathcal{J}}_{2r+1}({s}_{\text{L}}).\hfill \end{align} \tag{ 4.33 }$$

Hence, we find

$$\begin{equation}\left[+{Q}_{2r+1},H\right]{\vert }_{\text{L}}=-{\mathcal{J}}_{2r+1}({s}_{\text{L}}),\end{equation} \tag{ 4.34 }$$

$$\begin{equation}\left[-{Q}_{2r+1},H\right]{\vert }_{\text{R}}=-{\mathcal{J}}_{2r+1}({s}_{\text{R}}),\end{equation} \tag{ 4.35 }$$

such that via (4.22) the deformed Hamiltonian becomes

$$\begin{equation}H(\lambda )=H+\lambda \left({H}_{\lambda }^{\text{bulk}}+{H}_{2r,\lambda ,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}+{H}_{2r,\lambda ,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}\right)+\mathcal{O}({\lambda }^{2}),\end{equation} \tag{ 4.36 }$$

with

$$\begin{equation}{H}_{\lambda }^{\text{bulk}}=0,\qquad {H}_{2r,\lambda ,\mathrm{L}}^{\mathrm{b}\mathrm{d}\mathrm{r}}=-{\mathcal{J}}_{2r+1}({s}_{\text{L}}),\qquad {H}_{2r,\lambda ,\mathrm{R}}^{\mathrm{b}\mathrm{d}\mathrm{r}}=-{\mathcal{J}}_{2r+1}({s}_{\text{R}}).\end{equation} \tag{ 4.37 }$$

The fact that these deformations merely act on the boundary is in agreement with the observation that they only deform the boundary scattering matrix as shown in the following.

Consider the two-particle scattering state

$$\begin{equation}\left\vert u,{u}^{\prime }\right\rangle \simeq a(u,{u}^{\prime })\left\vert u< {u}^{\prime }\right\rangle +a({u}^{\prime },u)\left\vert {u}^{\prime }< u\right\rangle ,\end{equation} \tag{ 5.1 }$$

which is an asymptotic eigenstate of the Hamiltonian:

$$\begin{equation}H\left\vert u,{u}^{\prime }\right\rangle =\left(h(u)+h({u}^{\prime })\right)\left\vert u,{u}^{\prime }\right\rangle .\end{equation} \tag{ 5.2 }$$

Here $\left\vert u< {u}^{\prime }\right\rangle$ represents a partial momentum eigenstate (as opposed to the in/out states in (2.36)) with the particle with rapidity u(p) (or momentum p) being on the left of the particle with rapidity u'(p') (or momentum p'):

$$\begin{equation}\left\vert p< {p}^{\prime }\right\rangle ={\int }_{x\ll {x}^{\prime }}{\mathrm{e}}^{\mathrm{i}px+\mathrm{i}{p}^{\prime }{x}^{\prime }}\left\vert x,{x}^{\prime }\right\rangle .\end{equation} \tag{ 5.3 }$$

The ≃ in (5.1) signals that we ignore contributions where both particles forming the two-particle state are close to each other. Such contributions will not affect the two-particle scattering factor which is defined as

$$\begin{equation}S(u,{u}^{\prime })=\frac{a({u}^{\prime },u)}{a(u,{u}^{\prime })}.\end{equation} \tag{ 5.4 }$$

We deform the Hamiltonian via the equation

$$\begin{equation}\frac{\mathrm{d}{H}_{\lambda }}{\mathrm{d}\lambda }=\left[X,{H}_{\lambda }\right],\end{equation} \tag{ 5.5 }$$

where

$$\begin{equation}X\left\vert u,{u}^{\prime }\right\rangle =g(u,{u}^{\prime })\left\vert u,{u}^{\prime }\right\rangle ,\end{equation} \tag{ 5.6 }$$

for some eigenvalue function g; for the moment we do not specify g or X, but we will do so in due course. Differentiating the eigenvalue equation (5.2) with respect to λ and using ¹¹

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\lambda }h(u)=0,\qquad \frac{\mathrm{d}}{\mathrm{d}\lambda }\left\vert u< {u}^{\prime }\right\rangle =0,\end{equation} \tag{ 5.7 }$$

we obtain

$$\begin{align}\hfill 0& =\frac{\mathrm{d}}{\mathrm{d}\lambda }\left[{H}_{\lambda }-h(u)-h({u}^{\prime })\right]\left\vert u,{u}^{\prime }\right\rangle \hfill \\ \hfill & =\left[X,{H}_{\lambda }\right]\left\vert u,{u}^{\prime }\right\rangle +\left[{H}_{\lambda }-h(u)-h({u}^{\prime })\right]\left(\frac{\mathrm{d}a(u,{u}^{\prime })}{\mathrm{d}\lambda }\left\vert u< {u}^{\prime }\right\rangle +\frac{\mathrm{d}a({u}^{\prime },u)}{\mathrm{d}\lambda }\left\vert {u}^{\prime }< u\right\rangle \right)\hfill \\ \hfill & =\left[{H}_{\lambda }-h(u)-h({u}^{\prime })\right]\left(-X\left\vert u,{u}^{\prime }\right\rangle +\frac{\mathrm{d}a(u,{u}^{\prime })}{\mathrm{d}\lambda }\left\vert u< {u}^{\prime }\right\rangle +\frac{\mathrm{d}a({u}^{\prime },u)}{\mathrm{d}\lambda }\left\vert {u}^{\prime }< u\right\rangle \right).\hfill \end{align} \tag{ 5.8 }$$

Reading off the coefficients of $\left\vert u< {u}^{\prime }\right\rangle$ and $\left\vert {u}^{\prime }< u\right\rangle$ yields the equation

$$\begin{equation}0=-g(u,{u}^{\prime })a(u,{u}^{\prime })+\frac{\mathrm{d}a(u,{u}^{\prime })}{\mathrm{d}\lambda },\end{equation} \tag{ 5.9 }$$

which is solved by

$$\begin{equation}a(u,{u}^{\prime })={\mathrm{e}}^{\lambda g(u,{u}^{\prime })}{a}_{0}(u,{u}^{\prime }).\end{equation} \tag{ 5.10 }$$

Hence, the deformed two-particle scattering factor reads

$$\begin{equation}{S}_{\lambda }(u,{u}^{\prime })={\mathrm{e}}^{\lambda (g({u}^{\prime },u)-g(u,{u}^{\prime }))}S(u,{u}^{\prime }).\end{equation} \tag{ 5.11 }$$

Now we may specify the deformation generator X to the bilocal charge operator, such that

$$\begin{equation}X=[{Q}_{r}\vert {Q}_{s}],\qquad g(u,{u}^{\prime })=\mathrm{i}{q}_{r}(u){q}_{s}({u}^{\prime })+{f}_{rs}(u)+{f}_{rs}({u}^{\prime }).\end{equation} \tag{ 5.12 }$$

Here f_rs denotes a local contribution that originates from both constituent charges of the bilocal operator acting on the same particle. Hence, we find the following deformation of the bulk scattering matrix:

$$\begin{equation}{S}_{\lambda }(u,{u}^{\prime })={\text{e}}^{-\text{i}\lambda ({q}_{r}(u){q}_{s}({u}^{\prime })-{q}_{s}(u){q}_{r}({u}^{\prime }))}S(u,{u}^{\prime }).\end{equation} \tag{ 5.13 }$$

Consider the left boundary scattering state ¹²

$$\begin{equation}{\left\vert u\right\rangle }_{\text{L}}\simeq a(u)\left\vert u\right\rangle +a(-u)\left\vert -u\right\rangle ,\end{equation} \tag{ 5.14 }$$

which is an eigenstate of the Hamiltonian:

$$\begin{equation}H{\left\vert u\right\rangle }_{\text{L}}=h(u){\left\vert u\right\rangle }_{\text{L}}.\end{equation} \tag{ 5.15 }$$

Similarly as above we ignore contributions to the boundary scattering state for which the particle is close to the boundary and which do not affect the scattering factors. The boundary scattering factor is defined as

$$\begin{equation}{S}_{\text{L}}(u)=\frac{a(u)}{a(-u)}.\end{equation} \tag{ 5.16 }$$

We deform the Hamiltonian via the equation

$$\begin{equation}\frac{\mathrm{d}{H}_{\lambda }}{\mathrm{d}\lambda }=\left[X,{H}_{\lambda }\right],\end{equation} \tag{ 5.17 }$$

where

$$\begin{equation}X\left\vert u\right\rangle =f(u)\left\vert u\right\rangle ,\end{equation} \tag{ 5.18 }$$

for some eigenvalue function f and again, for the moment we do not specify f or X. Differentiating the eigenvalue equation (5.15) with respect to λ and using

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\lambda }h(u)=0,\qquad \frac{\mathrm{d}}{\mathrm{d}\lambda }\left\vert u\right\rangle =\frac{\mathrm{d}}{\mathrm{d}\lambda }\left\vert -u\right\rangle =0,\end{equation} \tag{ 5.19 }$$

we obtain

$$\begin{align}\hfill 0& =\frac{\mathrm{d}}{\mathrm{d}\lambda }\left[{H}_{\lambda }-h(u)\right]{\left\vert u\right\rangle }_{\text{L}}\hfill \\ \hfill & =\left[X,{H}_{\lambda }\right]{\left\vert u\right\rangle }_{\text{L}}+\left[{H}_{\lambda }-h(u)\right]\left(\frac{\mathrm{d}a(u)}{d\lambda }\left\vert u\right\rangle +\frac{\mathrm{d}a(-u)}{d\lambda }\left\vert -u\right\rangle \right)\hfill \\ \hfill & =\left[{H}_{\lambda }-h(u)\right]\left(-X{\left\vert u\right\rangle }_{\text{L}}+\frac{\mathrm{d}a(u)}{\mathrm{d}\lambda }\left\vert u\right\rangle +\frac{\mathrm{d}a(-u)}{\mathrm{d}\lambda }\left\vert -u\right\rangle \right).\hfill \end{align} \tag{ 5.20 }$$

Reading off the coefficients of $\left\vert u\right\rangle$ and $\left\vert -u\right\rangle$ yields the equations

$$\begin{align}\hfill 0& =-f(u)a(u)+\frac{\mathrm{d}a(u)}{\mathrm{d}\lambda },\hfill \\ \hfill 0& =-f(-u)a(-u)+\frac{\mathrm{d}a(-u)}{\mathrm{d}\lambda },\hfill \end{align} \tag{ 5.21 }$$

which are solved by

$$\begin{equation}a(u)={\mathrm{e}}^{\lambda f(u)}{a}_{0}(u).\end{equation} \tag{ 5.22 }$$

Hence, the deformed boundary scattering factor reads

$$\begin{equation}{S}_{\mathrm{L},\lambda }(u)={\mathrm{e}}^{\lambda (f(u)-f(-u))}{S}_{\text{L}}(u).\end{equation} \tag{ 5.23 }$$

Now we may specify the deformation generator X to one of the two cases which induce deformations of the left boundary scattering matrix:

$$\begin{equation}1\left.\!\right):\quad X=[{Q}_{2r}\vert {Q}_{2s+1}],\qquad f(u)=\frac{\mathrm{i}}{2}{q}_{2r}(u){q}_{2s+1}(u),\end{equation} \tag{ 5.24 }$$

$$\begin{equation}2\left.\!\right):\quad X={Q}_{2r+1},\qquad f(u)=\mathrm{i}{q}_{2r+1}(u).\end{equation} \tag{ 5.25 }$$

Note that the factor 1/2 in (5.24) originates from the 1/2 in front of the local contribution q_2r (x)q_2s+1(x) to the bilocal operator as prescribed by the definition (4.13). Analogously we can proceed for the right boundary where we use X = [Q_2s+1|Q_2r ].

For deformations generated by X = [Q_2r |Q_2s+1], the deformed bulk S-matrix takes the form (cf (5.13))

$$\begin{equation}{S}_{\lambda }(u,{u}^{\prime })={\text{e}}^{-\text{i}\lambda ({q}_{2r}(u){q}_{2s+1}({u}^{\prime })-{q}_{2s+1}(u){q}_{2r}({u}^{\prime }))}S(u,{u}^{\prime }),\end{equation} \tag{ 5.26 }$$

and for the boundary scattering factor deformed by (5.24) we have

$$\begin{equation}{S}_{\mathrm{L},\lambda }(u)={S}_{\lambda }(u,-u){S}_{\mathrm{L},\lambda }(-u),\end{equation} \tag{ 5.27 }$$

which is the boundary cross-unitarity condition of [45].

The derivation of the deformed scattering phase parallels the boundary case. We first consider the topological case and then the non-topological one.

Topological defects. The topological defect is purely transmissive. To determine the deformed transmissive amplitude, we consider the following one-particle state

$$\begin{equation}\vert {u\rangle }_{\mathrm{D}}=a(u)\vert u;\varnothing \rangle +b(u)\vert \varnothing ;u\rangle .\end{equation} \tag{ 5.28 }$$

The transmissive amplitude is given by

$$\begin{equation}{T}_{-}(u)={T}_{+}(u)=T(u)=\frac{b(u)}{a(u)}.\end{equation} \tag{ 5.29 }$$

Consider the bilocal deformation (5.17) of the Hamiltonian. We denote the deformed Hamiltonian by H_λ . In the infinite volume limit, we have

$$\begin{equation}{H}_{\lambda }\vert {u\rangle }_{\mathrm{D}}=h(u)\vert {u\rangle }_{\mathrm{D}}.\end{equation} \tag{ 5.30 }$$

As before, the asymptotic states diagonalize the operator X:

$$\begin{equation}X\vert u;\varnothing \rangle =f(u)\vert u;\varnothing \rangle ,\qquad X\vert \varnothing ;u\rangle =f(u)\vert \varnothing ;u\rangle .\end{equation} \tag{ 5.31 }$$

Taking the derivative of (5.30) with respect to λ, we obtain

$$\begin{equation}[X,{H}_{\lambda }]\vert {u\rangle }_{\mathrm{D}}+{H}_{\lambda }\frac{\mathrm{d}}{\mathrm{d}\lambda }\vert {u\rangle }_{\mathrm{D}}=h(u)\frac{\mathrm{d}}{\mathrm{d}\lambda }\vert {u\rangle }_{\mathrm{D}},\end{equation} \tag{ 5.32 }$$

where we have used the fact that

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\lambda }\vert u;\varnothing \rangle =\frac{\mathrm{d}}{\mathrm{d}\lambda }\vert \varnothing ;u\rangle =0,\qquad \frac{\mathrm{d}}{\mathrm{d}\lambda }h(u)=0.\end{equation} \tag{ 5.33 }$$

Equation (5.32) can be brought to the form

$$\begin{equation}[{H}_{\lambda }-h(u)]\left(-X\vert {u\rangle }_{\mathrm{D}}+\frac{\mathrm{d}a(u)}{\mathrm{d}\lambda }\vert u;\varnothing \rangle +\frac{\mathrm{d}b(u)}{\mathrm{d}\lambda }\vert \varnothing ;u\rangle \right)=0,\end{equation} \tag{ 5.34 }$$

which implies

$$\begin{equation}-f(u)+\frac{\mathrm{d}a(u)}{\mathrm{d}\lambda }=0,\qquad -f(u)+\frac{\mathrm{d}b(u)}{\mathrm{d}\lambda }=0.\end{equation} \tag{ 5.35 }$$

These equations can be solved by

$$\begin{equation}{a}_{\lambda }(u)={\mathrm{e}}^{\lambda f(u)}a(u),\qquad {b}_{\lambda }(u)={\mathrm{e}}^{\lambda f(u)}b(u).\end{equation} \tag{ 5.36 }$$

This leads to the conclusion that the transmission amplitude is not affected:

$$\begin{equation}{T}_{\lambda }(u)=\frac{{a}_{\lambda }(u)}{{b}_{\lambda }(u)}=\frac{a(u)}{b(u)}=T(u).\end{equation} \tag{ 5.37 }$$

Physically, this is expected for the $T\bar{T}$ deformation. Topological defects, by definition, are invariant under variations of the metric. As a result, they are not sensitive to the stress energy tensor. Since the $T\bar{T}$ operator is constructed from the stress energy tensor, it is natural that topological defects are not affected by the $T\bar{T}$ deformation. From our derivation, we see that the topological defect is not affected by these deformations.

Non-topological defects. Integrable defects that are non-topological are only allowed in free theories with the bulk S-matrix being S = ±1. In this case, the one-particle state is given by

$$\begin{equation}\vert {u\rangle }_{\mathrm{D}}=a(u)\vert u;\varnothing \rangle +b(u)\vert \varnothing ;u\rangle +c(u)\vert -u;\varnothing \rangle .\end{equation} \tag{ 5.38 }$$

The transmission and reflection amplitudes are given by

$$\begin{equation}T(u)=\frac{b(u)}{a(u)},\qquad R(u)=\frac{c(u)}{a(u)}.\end{equation} \tag{ 5.39 }$$

Going through the same steps, we find that the corresponding deformed quantities are

$$\begin{equation}{R}_{\lambda }(u)={\mathrm{e}}^{\lambda f(u)-\lambda f(-u)}R(u),\qquad {T}_{\lambda }(u)=T(u).\end{equation} \tag{ 5.40 }$$

We see that the reflection amplitude is deformed in the same way as in the boundary case while the transmission amplitude is undeformed like the topological defect. Here we note similarities to the deformation factor obtained in [54].

We first consider the limit L ≫ 1. In this limit, the spectrum is determined by the asymptotic Bethe equations

$$\begin{equation}{\mathrm{e}}^{2\mathrm{i}p({u}_{j})L}{S}_{\mathrm{L}}({u}_{j}){S}_{\mathrm{R}}(-{u}_{j})\prod\limits _{k\ne j}^{N}S({u}_{j},{u}_{k})S({u}_{j},-{u}_{k})=1,\quad j=1,\dots ,N,\end{equation} \tag{ 6.4 }$$

where S(u, v) is the bulk S-matrix and S_L,R(u) denotes the left and right boundary S-matrix, respectively.

Bilinear deformations. For the bilinear deformations, the deformed bulk and boundary matrices are given by ¹³ (5.13) and (5.23), which we quote here

$$\begin{align}\hfill {S}_{\lambda }(u,v)=\;& S(u,v){\text{e}}^{-\text{i}\lambda ({p}_{r}(u){e}_{s}(v)-{e}_{s}(u){p}_{r}(v))},\hfill \\ \hfill {S}_{\mathrm{L},\lambda }(u)=\;& {S}_{\mathrm{L}}(u){\text{e}}^{\text{i}\lambda {p}_{r}(u){e}_{s}(u)},\hfill \\ \hfill {S}_{\mathrm{R},\lambda }(u)=\;& {S}_{\mathrm{R}}(u){\text{e}}^{-\text{i}\lambda {p}_{r}(u){e}_{s}(u)},\hfill \end{align} \tag{ 6.5 }$$

where

$$\begin{equation}{p}_{r}(u)={\gamma }_{r}\,\mathrm{sinh}(ru),\qquad {e}_{s}(u)={\gamma }_{s}\,\mathrm{cosh}(su),\end{equation} \tag{ 6.6 }$$

and both r, s are odd integers. We have

$$\begin{equation}{p}_{r}(-u)=-{p}_{r}(u),\qquad {e}_{s}(-u)={e}_{s}(u).\end{equation} \tag{ 6.7 }$$

For deformations involving higher charges, it is more convenient to consider the twisted Bethe equations

$$\begin{equation}{\mathrm{e}}^{2\mathrm{i}L(p({u}_{j})+{\nu }_{r}{p}_{r}({u}_{j}))}{S}_{\mathrm{L}}({u}_{j}){S}_{\mathrm{R}}(-{u}_{j})\prod\limits _{k\ne j}^{N}S({u}_{j},{u}_{k})S({u}_{j},-{u}_{k})=1,\end{equation} \tag{ 6.8 }$$

where ν_r is the twist that couples to the odd (P-type) charges. The deformed Bethe equations read

$$\begin{equation}{\mathrm{e}}^{2\mathrm{i}L(p({u}_{j})+{\nu }_{r}{p}_{r}({u}_{j}))}\prod\limits _{k=1}^{N}{\mathrm{e}}^{2\mathrm{i}\lambda {p}_{r}({u}_{j}){e}_{s}({u}_{k})}\,{S}_{\mathrm{L}}({u}_{j}){S}_{\mathrm{R}}(-{u}_{j})\prod\limits _{k\ne j}^{N}S({u}_{j},{u}_{k})S({u}_{j},-{u}_{k})=1.\end{equation} \tag{ 6.9 }$$

They can be recast as

$$\begin{equation}{\mathrm{e}}^{2\mathrm{i}L(p({u}_{j})+{\nu }_{r}{p}_{r}({u}_{j}))}\,{S}_{\mathrm{L}}({u}_{j}){S}_{\mathrm{R}}(-{u}_{j})\prod\limits _{k\ne j}^{N}S({u}_{j},{u}_{k})S({u}_{j},-{u}_{k})={\mathrm{e}}^{-2\mathrm{i}\lambda {Q}_{N}^{(s)}\,{p}_{r}({u}_{j})},\end{equation} \tag{ 6.10 }$$

where

$$\begin{equation}{Q}_{N}^{(s)}=\sum\limits _{k=1}^{N}{e}_{s}({u}_{k}),\qquad {E}_{N}={Q}_{N}^{(1)}.\end{equation} \tag{ 6.11 }$$

We see that the deformed Bethe equations take the same form as the undeformed ones, except that the twist ν_r is shifted according to

$$\begin{equation}{\nu }_{r}\to {\nu }_{r}+\frac{\lambda {Q}_{N}^{(s)}}{L}.\end{equation} \tag{ 6.12 }$$

This implies the following flow equation for the energy and total even charge:

$$\begin{equation}{\partial }_{\lambda }{E}_{N}=\frac{1}{L}{Q}_{N}^{(s)}{\partial }_{{\nu }_{r}}{E}_{N},\qquad {\partial }_{\lambda }{Q}_{N}^{(s)}=\frac{1}{L}{Q}_{N}^{(s)}{\partial }_{{\nu }_{r}}{Q}_{N}^{(s)}.\end{equation} \tag{ 6.13 }$$

The case r = 1 is special. In this case p_r (u_j ) coincides with the momentum and there is no need to introduce additional twist. The deformation has the effect of changing the length L as

$$\begin{equation}L\to L+\lambda {Q}_{N}^{(s)}.\end{equation} \tag{ 6.14 }$$

This leads to the following flow equations

$$\begin{equation}{\partial }_{\lambda }{E}_{N}={Q}_{N}^{(s)}{\partial }_{L}{E}_{N},\qquad {Q}_{N}^{(s)}={Q}_{N}^{(s)}{\partial }_{L}{Q}_{N}^{(s)}.\end{equation} \tag{ 6.15 }$$

The deformations triggered by ${\mathcal{O}}_{1s}$ have an interesting intuitive interpretation as is shown in figure 4. For λ < 0, the deformation turns point particles into finite size hard rods with length −λe_s (u_j ). Therefore, the effective length, which describes the 'free space' between the rods, is reduced and becomes $L+\lambda {Q}_{N}^{(s)}$. For λ > 0, the distances between the particles are increased and the effective length becomes larger. This interpretation was first proposed in non-relativistic models [28–30]. Here we see a natural generalization to the relativistic case.

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The odd charge deformations. For the odd charge deformations triggered by P_r , the deformed bulk and boundary S-matrices read

$$\begin{align}\hfill {S}_{\lambda }(u,v)=& \hspace{2pt}S(u,v),\hfill \\ \hfill {S}_{\mathrm{L},\lambda }(u)=& \hspace{2pt}{S}_{\mathrm{L}}(u){\text{e}}^{\text{i}\lambda {p}_{r}(u)},\hfill \\ \hfill {S}_{\mathrm{R},\lambda }(u)=& \hspace{2pt}{S}_{\mathrm{R}}(u){\text{e}}^{-\text{i}\lambda {p}_{r}(u)}.\hfill \end{align} \tag{ 6.16 }$$

Notice that the bulk S-matrix is undeformed. The deformed asymptotic Bethe equation takes the following form:

$$\begin{equation}{\mathrm{e}}^{2\mathrm{i}L(p({u}_{j})+{\nu }_{r}{p}_{r}({u}_{j}))}\,{S}_{\mathrm{L}}({u}_{j}){S}_{\mathrm{R}}(-{u}_{j})\prod\limits _{k\ne j}^{N}S({u}_{j},{u}_{k})S({u}_{j},-{u}_{k})={\mathrm{e}}^{-2i\lambda {p}_{r}({u}_{j})}.\end{equation} \tag{ 6.17 }$$

We see that it can be brought to the original form by shifting the chemical potential

$$\begin{equation}{\nu }_{r}\to {\nu }_{r}+\frac{\lambda }{L}.\end{equation} \tag{ 6.18 }$$

This implies the following flow equation for the energy and total even charge:

$$\begin{equation}{\partial }_{\lambda }{E}_{N}=\frac{1}{L}{\partial }_{{\nu }_{r}}{E}_{N}.\end{equation} \tag{ 6.19 }$$

The case r = 1 is again special. The deformed Bethe equation is simply obtained from the undeformed one by setting

$$\begin{equation}L\to L+\lambda ,\end{equation} \tag{ 6.20 }$$

which implies a linear flow equation for the spectrum:

$$\begin{equation}{\partial }_{\lambda }{E}_{N}={\partial }_{L}{E}_{N}.\end{equation} \tag{ 6.21 }$$

We also have an intuitive interpretation for this result, as is shown in figure 5. For λ < 0, the deformation makes the boundary 'thicker', which reduces the distances between the two boundaries by |λ|. For λ > 0, the distance between the boundaries is increased by |λ|.

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We make one comment before ending this subsection. Although the flow equations we have obtained so far are in the large volume limit, we expect that they should hold also in the finite volume, based on the experience on previous results in the bulk case. As we shall see, this will be confirmed by the boundary TBA computation below.

Now we consider the situation where L is finite. Due to finite size corrections, the asymptotic Bethe ansatz is no longer sufficient. To obtain the spectrum, we exploit the boundary TBA approach. The idea of this approach is to translate the calculation of the finite size spectrum to the calculation of the thermal free energy of the mirror theory. The mirror theory is defined in the infinite volume limit and the asymptotic description is valid. Because our deformations involve higher conserved charges, we need to consider a generalized partition function that contains additional chemical potentials and charges. In the mirror theory, these chemical potentials correspond to twists, which enter the quantization conditions of the mirror rapidities [23].

Double Wick rotation. The mirror theory is obtained from the physical theory by performing a double Wick rotation which swaps the role of space and time:

$$\begin{equation}H{\mapsto}\mathrm{i}\tilde{P},\qquad P{\mapsto}\mathrm{i}\tilde{H}.\end{equation} \tag{ 6.22 }$$

Here we use a tilde to denote quantities in the mirror theory. Under the double Wick rotation, the rapidity is transformed as $u{\mapsto}u+\frac{\mathrm{i}\pi }{2}$. For higher charges with s = 2r − 1, we have similarly

$$\begin{equation}{H}_{2r-1}{\mapsto}\mathrm{i}{(-1)}^{r-1}{\tilde{P}}_{2r-1},\qquad {P}_{2r-1}{\mapsto}\mathrm{i}{(-1)}^{r-1}{\tilde{H}}_{2r-1},\quad r=1,2,\dots \end{equation} \tag{ 6.23 }$$

The single-particle eigenvalues of the higher mirror charges ${\tilde{P}}_{2r-1}$, ${\tilde{H}}_{2r-1}$ take the same form as in the original theory

$$\begin{equation}{\tilde{e}}_{2r-1}(u)={\gamma }_{2r-1}\,\mathrm{cos}(2r-1)u,\qquad {\tilde{p}}_{2r-1}={\gamma }_{2r-1}\,\mathrm{sin}(2r-1)u.\end{equation} \tag{ 6.24 }$$

The mirror bulk S-matrix is given by the mirror transformation of the physical S-matrix. The boundary S-matrix is related to the two-particle form factor K(u) in the mirror channel. More explicitly, we have

$$\begin{equation}\tilde{S}(u,v)=S\left(u+\frac{\mathrm{i}\pi }{2},v+\frac{\mathrm{i}\pi }{2}\right),\qquad K(u)={S}_{\mathrm{R}}\left(u-\frac{\mathrm{i}\pi }{2}\right).\end{equation} \tag{ 6.25 }$$

Generalized mirror BTBA. Now we consider the generalized mirror boundary TBA. In the physical channel, the length of the space is L. We take the length of the periodic direction to be R. The generalized partition function takes the following form:

$$\begin{equation}{Z}_{ab}=\text{tr}\left[{\mathrm{e}}^{-R(H+{\mu }_{s}{H}_{s})}\right].\end{equation} \tag{ 6.26 }$$

Here we have introduced the additional chemical potentials μ_s for the even charges ¹⁴ . In the limit R ≫ 1, the partition function is dominated by the ground state charges:

$$\begin{equation}{Z}_{ab}\sim {\mathrm{e}}^{-R\left[{E}^{(0)}(L)+{\mu }_{s}{E}_{s}^{(0)}(L)\right]}.\end{equation} \tag{ 6.27 }$$

To compute the energy and the higher charge of the ground state, we go to the mirror kinematics, as is shown in figure 6. The partition function in the mirror kinematics takes the form

$$\begin{equation}{Z}_{ab}=\langle {B}_{a}\vert {\mathrm{e}}^{-L\left(\tilde{H}+{(-1)}^{\frac{r+1}{2}}{\nu }_{r}{\tilde{H}}_{r}\right)}\vert {B}_{b}\rangle ,\end{equation} \tag{ 6.28 }$$

where |B_j ⟩ (j = a, b) are the boundary states which correspond to the two boundaries in the open channel. We introduced an additional (−1)^(r+1)/2 in the definition of the chemical potential for later convenience. The chemical potential in (6.26) is implicitly contained in the mirror partition function. It becomes the twist in the mirror channel, which enters the quantization condition for the mirror rapidities. The mirror partition function (6.28) can be written as

$$\begin{equation}{Z}_{ab}=\sum\limits _{n}\frac{\langle {B}_{a}\vert n\rangle \langle n\vert {B}_{b}\rangle }{\langle n\vert n\rangle }{\mathrm{e}}^{-L{\mathcal{X}}_{n}(R)},\end{equation} \tag{ 6.29 }$$

where the sum is over all eigenstates of the Hamiltonian and ${\mathcal{X}}_{n}(R)$ is defined as

$$\begin{equation}\left(\tilde{H}+{(-1)}^{\frac{r+1}{2}}{\nu }_{r}{\tilde{H}}_{r}\right)\vert n\rangle ={\mathcal{X}}_{n}(R)\vert n\rangle .\end{equation} \tag{ 6.30 }$$

For integrable boundaries, the overlap ⟨B_a |n⟩ is only non-zero for the states with paired rapidities

$$\begin{equation}\vert n\rangle =\vert {\alpha }_{2N}\rangle =\vert {u}_{N},-{u}_{N},\dots ,{u}_{1},-{u}_{1}\rangle ,\end{equation} \tag{ 6.31 }$$

where u_N > u_N−1 > ... > u₁. For a state with paired rapidities {u₁, − u₁, ..., u_N , − u_N }, the eigenvalue is given by

$$\begin{equation}{\mathcal{X}}_{n}(R)=2\sum\limits _{j=1}^{N}{X}_{\nu }({u}_{j}),\end{equation} \tag{ 6.32 }$$

where

$$\begin{equation}{X}_{\nu }(u)=\tilde{e}(u)+{(-1)}^{\frac{r+1}{2}}{\nu }_{r}{\tilde{e}}_{r}(u).\end{equation} \tag{ 6.33 }$$

In the mirror channel for R ≫ 1, the spectrum can be described by the asymptotic Bethe ansatz. The rapidities satisfy the mirror Bethe ansatz equations. In the usual case, the mirror Bethe ansatz equations take the following form

$$\begin{equation}{\text{e}}^{\text{i}mR\,\mathrm{sinh}({u}_{i})}\tilde{S}({u}_{i},-{u}_{i})\prod\limits _{j\ne \mathrm{i}}^{N}\tilde{S}({u}_{i},{u}_{j})\tilde{S}({u}_{i},-{u}_{j})=1.\end{equation} \tag{ 6.34 }$$

As mentioned before, we have additional twists which come from the chemical potential in the physical channel. In this case, we need to modify the Bethe equations by the replacement

$$\begin{equation}{\mathrm{e}}^{\mathrm{i}mR\,\mathrm{sinh}(u)}{\mapsto}{\mathrm{e}}^{R{Y}_{\boldsymbol{\mu }}\left(u+\frac{\mathrm{i}\pi }{2}\right)},\end{equation} \tag{ 6.35 }$$

where

$$\begin{equation}{Y}_{\boldsymbol{\mu }}(u)=e(u)+{\mu }_{s}{e}_{s}(u).\end{equation} \tag{ 6.36 }$$

Following the standard procedure, we introduce the density of pair rapidities and holes $\tilde{\rho }(u)$, ${\tilde{\rho }}_{h}(u)$. From (6.34), we have

$$\begin{equation}\tilde{\rho }(u)+{\tilde{\rho }}_{h}(u)=\frac{{\partial }_{u}{Y}_{\boldsymbol{\mu }}\left(u+\frac{\mathrm{i}\pi }{2}\right)}{2\pi \mathrm{i}}+({\tilde{\varphi }}_{+}\ast \tilde{\rho })(u),\end{equation} \tag{ 6.37 }$$

where

$$\begin{equation}\tilde{\varphi }(u,v)=\frac{1}{2\pi \mathrm{i}}\frac{\partial }{\partial u}\,\mathrm{log}\,\tilde{S}(u,v),\end{equation} \tag{ 6.38 }$$

and

$$\begin{equation}{\tilde{\varphi }}_{+}(u,v)=\tilde{\varphi }(u,v)+\tilde{\varphi }(u,-v),\qquad ({\tilde{\varphi }}_{+}\ast \tilde{\rho })(u)={\int }_{0}^{\infty }\tilde{\varphi }(u,v)\tilde{\rho }(v)\mathrm{d}v.\end{equation} \tag{ 6.39 }$$

Therefore the partition function can be written as

$$\begin{equation}{Z}_{ab}=\int \mathcal{D}\tilde{\rho }\,\mathrm{exp}\left[R{\int }_{0}^{\infty }\left(\mathrm{log}[{\chi }_{ab}(u)]-2L{X}_{\nu }(u)\right)\tilde{\rho }(u)\mathrm{d}u+S[\tilde{\rho },{\tilde{\rho }}_{h}]\right],\end{equation} \tag{ 6.40 }$$

where

$$\begin{equation}{\chi }_{ab}(u)={\bar{K}}_{a}(u){K}_{b}(u),\end{equation} \tag{ 6.41 }$$

and $S[\tilde{\rho },{\tilde{\rho }}_{h}]$ is the Yang–Yang entropy

$$\begin{equation}S[\tilde{\rho },{\tilde{\rho }}_{h}]=R{\int }_{0}^{\infty }\left[(\tilde{\rho }+{\tilde{\rho }}_{h})\mathrm{log}(\tilde{\rho }+{\tilde{\rho }}_{h})-\tilde{\rho }\,\mathrm{log}\,\tilde{\rho }-{\tilde{\rho }}_{h}\mathrm{log}{\tilde{\rho }}_{h}\right]\mathrm{d}u.\end{equation} \tag{ 6.42 }$$

In the limit R → ∞, the partition function Z_ab (6.40) is dominated by the saddle point. The saddle-point equation is the boundary TBA equation

$$\begin{equation}{\epsilon}(u)=2L{X}_{\nu }(u)-\mathrm{log}[{\chi }_{ab}(u)]-\mathrm{log}(1+{\mathrm{e}}^{-{\epsilon}})\ast {\varphi }_{+},\end{equation} \tag{ 6.43 }$$

where (u) is the pseudo-energy given by ${\mathrm{e}}^{{\epsilon}(u)}={\tilde{\rho }}_{h}(u)/\tilde{\rho }(u)$. The free energy reads

$$\begin{equation}F=\frac{R}{2\pi \mathrm{i}}{\int }_{0}^{\infty }{\partial }_{u}{Y}_{\boldsymbol{\mu }}\left(u+\frac{\mathrm{i}\pi }{2}\right)\mathrm{log}\left(1+{\mathrm{e}}^{-{\epsilon}(u)}\right)\mathrm{d}u.\end{equation} \tag{ 6.44 }$$

Comparing with (6.27), we obtain the expressions for the finite volume charges in the ground state:

$$\begin{align}\hfill {E}^{(0)}(L,{\nu }_{r})=& -\frac{1}{2\pi \mathrm{i}}{\int }_{0}^{\infty }{\partial }_{u}e(u+\mathrm{i}\pi /2)\mathrm{log}\left(1+{\mathrm{e}}^{-{\epsilon}(u)}\right)\mathrm{d}u,\hfill \\ \hfill {Q}_{s}^{(0)}(L,{\nu }_{r})=& -\frac{1}{2\pi \mathrm{i}}{\int }_{0}^{\infty }{\partial }_{u}{e}_{s}(u+\mathrm{i}\pi /2)\mathrm{log}\left(1+{\mathrm{e}}^{-{\epsilon}(u)}\right)\mathrm{d}u.\hfill \end{align} \tag{ 6.45 }$$

The deformed TBA kernel is given by

$$\begin{align}\hfill {\tilde{\varphi }}_{+,\lambda }(u,v)=& {\tilde{\varphi }}_{+}(u,v)+\frac{\lambda }{\pi }{\partial }_{u}{e}_{s}(u+\frac{\mathrm{i}\pi }{2}){p}_{r}(v+\frac{\mathrm{i}\pi }{2})\hfill \\ \hfill =& {\tilde{\varphi }}_{+}(u,v)-{(-1)}^{\frac{r-1}{2}}\frac{\lambda }{\pi \mathrm{i}}{\partial }_{u}{e}_{s}(u+\frac{\mathrm{i}\pi }{2}){\tilde{e}}_{r}(v).\hfill \end{align} \tag{ 6.46 }$$

Plugging this into the TBA equation (6.43), we find that it takes the same form as the undeformed one, except for the shift of ν_r according to

$$\begin{equation}{\nu }_{r}\to {\nu }_{r}+\frac{\lambda }{L}{Q}_{s}^{(0)}.\end{equation} \tag{ 6.47 }$$

This implies the following flow equation for the ground state energy and charge:

$$\begin{equation}{\partial }_{\lambda }{E}_{\lambda }^{(0)}=\frac{1}{L}{Q}_{s,\lambda }^{(0)}{\partial }_{{\nu }_{r}}{E}_{\lambda }^{(0)},\qquad {\partial }_{\lambda }{Q}_{s,\lambda }^{(0)}=\frac{1}{L}{Q}_{s,\lambda }^{(0)}{\partial }_{{\nu }_{r}}{Q}_{s,\lambda }^{(0)}.\end{equation} \tag{ 6.48 }$$

This is precisely the same flow equation as we obtained in the large volume limit (6.15). Again the case r = 1 is special. In this case, we simply have

$$\begin{equation}L\to L+\lambda {Q}_{s}^{(0)}\end{equation} \tag{ 6.49 }$$

and the flow equation becomes

$$\begin{equation}\partial {E}_{\lambda }^{(0)}={Q}_{s,\lambda }^{(0)}{\partial }_{L}{E}_{\lambda }^{(0)},\qquad {\partial }_{\lambda }{Q}_{s,\lambda }^{(0)}={Q}_{s,\lambda }^{(0)}{\partial }_{L}{Q}_{s,\lambda }^{(0)}.\end{equation} \tag{ 6.50 }$$

The flow equations (6.48) and (6.50) are the same as in the large volume case (6.13) and (6.15), as expected.

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We can perform the same analysis for the odd charge deformations in the finite volume. The resulting flow equations of the spectrum again take the same form as the ones in the large volume limit (6.19) and (6.21).

For generic QFTs at thermal equilibrium, the quantum states with energy E are fully characterized by the Boltzmann weight exp(−βE). The same also holds for IQFTs with vanishing higher charges, because those charges will stay zero after the deformation.

Asymptotic limit. Before discussing the deformed partition function and the g-function, let us first study the general behavior of the partition function in the large volume limit, where the circumference R and the height L of the cylinder are both large. For massive IQFT, the precise meaning of large is R, L ≫ 1/m, where m is the mass gap of the theory. We remind that the open and closed channel description was briefly introduced in section 2.2.

Open channel. In the open channel, the partition function is given by

$$\begin{equation}{Z}_{ab}(R,L)=\mathrm{Tr}\,{\mathrm{e}}^{-{H}_{ab}R}=\sum\limits _{\psi }{\mathrm{e}}^{-{E}_{ab}^{\psi }(L)R},\end{equation} \tag{ 7.1 }$$

where H_ab is the open channel Hamiltonian and the sum is over all the eigenstates of H_ab , denoted here by ψ. In the limit mR ≫ 1, the partition function is dominated by the ground state energy ${E}_{ab}^{(0)}(L)$. In the large L limit, the ground state energy has the following universal behavior

$$\begin{equation}{E}_{ab}^{(0)}(L)={{\epsilon}}_{0}\,L+{f}_{a}+{f}_{b}+\mathcal{O}({\mathrm{e}}^{-mL}),\end{equation} \tag{ 7.2 }$$

where ₀ is the bulk energy density and f_a,b are the non-extensive contributions from the boundary. To sum up, in the limit mR, mL ≫ 1, we find

$$\begin{equation}{Z}_{ab}(R,L)={\mathrm{e}}^{-R({{\epsilon}}_{0}L+{f}_{a}+{f}_{b})}+\cdots ={\mathrm{e}}^{-RL{{\epsilon}}_{0}-R{f}_{a}-R{f}_{b}}+\cdots \end{equation} \tag{ 7.3 }$$

Closed channel. In the closed channel, the partition function is given by

$$\begin{equation}{Z}_{ab}(R,L)=\langle {B}_{a}\vert {\mathrm{e}}^{-H(R)L}\vert {B}_{b}\rangle =\sum\limits _{\phi }{({G}_{a}^{\phi }(R))}^{\ast }{G}_{b}^{\phi }(R){\mathrm{e}}^{-{E}_{\phi }(R)L},\end{equation} \tag{ 7.4 }$$

where the amplitudes ${G}_{j}^{\phi }(R)$ are defined as the normalized overlaps

$$\begin{equation}{G}_{j}^{\phi }(R)=\frac{\langle \phi \vert {B}_{j}\rangle }{\sqrt{\langle \phi \vert \phi \rangle }},\quad j=a,b.\end{equation} \tag{ 7.5 }$$

In the limit mL ≫ 1, the partition function is dominated by the ground state

$$\begin{equation}{Z}_{ab}(R,L)\sim {[{G}_{a}^{(0)}(R)]}^{\ast }{G}_{b}^{(0)}(R){\mathrm{e}}^{-{E}_{0}(R)L}.\end{equation} \tag{ 7.6 }$$

Taking the mR ≫ 1 limit of this expression, we find

$$\begin{equation}{E}_{0}(R)\approx {{\epsilon}}_{0}R+\mathcal{O}({\mathrm{e}}^{-mR}),\end{equation} \tag{ 7.7 }$$

where ₀ is the same quantity as the one which appears in (7.2). As opposed to (7.2), E₀(R) does not have $\mathcal{O}(1)$ contributions from the boundaries, since the system is closed. Therefore, in the mR, mL ≫ 1 limit, we have

$$\begin{equation}{Z}_{ab}(R,L)\sim {[{G}_{a}^{(0)}(R)]}^{\ast }{G}_{b}^{(0)}(R){\mathrm{e}}^{-RL{{\epsilon}}_{0}}+\cdots \end{equation} \tag{ 7.8 }$$

Definition of the g -function. Comparing the asymptotics in the closed channel with the open channel, (7.3), we must have

$$\begin{equation}{G}_{j}^{(0)}(R)\sim {\mathrm{e}}^{-{f}_{j}R}(1+\cdots \,).\end{equation} \tag{ 7.9 }$$

Namely, the overlap should be ${\mathrm{e}}^{-{f}_{j}R}$ multiplied by some order 1 quantity. We will define this quantity as the g-function:

$$\begin{equation}\mathrm{log}\,{g}_{a}(R)=\mathrm{log}\,{G}_{a}^{(0)}(R)+{f}_{a}R.\end{equation} \tag{ 7.10 }$$

To summarize, the exact g-function, or the boundary entropy, is defined as the overlap between the finite volume vacuum state and the boundary state. To extract important information about the boundary, we have subtracted a universal constant part f_a R from the naive overlap ${G}_{j}^{(0)}$, in the definition of g. The exact g-function is an important quantity of interest in QFTs with boundaries. It describes the boundary degrees of freedom [56] and has interesting properties along RG flow [57].

In this section, we present derivations of the flow equation of the $T\bar{T}$ deformed partition function and the g-function. Our strategy is as follows. The flow equation for the spectrum in the open channel is known from the exact deformed S-matrices. Since the spectrum is deformed universally, that is, all the energies obey the same flow equation, we can write down the equation for the partition function in the open channel. We then re-interpret the flow equation of the partition function in the closed channel. We also know the flow equation for the spectrum in the closed channel. This leads to the flow equation for the overlap, which gives the flow equation for the g-function after taking into account the exponential factor.

Open channel. Recall that the flow equation of the spectrum in the open channel is given by

$$\begin{equation}{\partial }_{\lambda }{E}_{ab}(L,\lambda )={E}_{ab}(L,\lambda ){\partial }_{L}{E}_{ab}(L,\lambda ).\end{equation} \tag{ 7.11 }$$

Applying it to the open channel partition function Z_ab ,

$$\begin{equation}{Z}_{ab}(R,L\vert \lambda )=\sum\limits _{\psi }{\mathrm{e}}^{-{E}_{ab}^{\psi }(L,\lambda )R},\end{equation} \tag{ 7.12 }$$

we find

$$\begin{align}\hfill {\partial }_{\lambda }{Z}_{ab}=& \sum\limits _{\psi }R\left(-{\partial }_{\lambda }{E}_{ab}^{\psi }\right){\mathrm{e}}^{-{E}_{ab}^{\psi }R}\hfill \\ \hfill =& \sum\limits _{\psi }R\left(-{E}_{ab}^{\psi }{\partial }_{L}{E}_{ab}^{\psi }\right){\mathrm{e}}^{-{E}_{ab}^{\psi }R}\hfill \\ \hfill =& -({\partial }_{R}-{R}^{-1}){\partial }_{L}{Z}_{ab}.\hfill \end{align} \tag{ 7.13 }$$

Therefore, we have the following flow equation

$$\begin{equation}{\partial }_{\lambda }{Z}_{ab}(R,L\vert \lambda )=-\left(\frac{\partial }{\partial R}-\frac{1}{R}\right){\partial }_{L}{Z}_{ab}(R,L\vert \lambda ).\end{equation} \tag{ 7.14 }$$

This flow equation has been obtained first by Cardy in [58] from random geometry considerations, which serves as a consistency check for our result. We can also apply the flow equation to the closed channel partition function. We define ${F}_{ab}^{\phi }(R)={({G}_{a}^{\phi }(R))}^{\ast }{G}_{b}^{\phi }(R)$; identifying each Boltzmann weight, we have

$$\begin{equation}{\partial }_{\lambda }\left({F}_{ab}^{\phi }(R,\lambda ){\mathrm{e}}^{-{E}_{\phi }(R,\lambda )L}\right)=-\left(\frac{\partial }{\partial R}-\frac{1}{R}\right){\partial }_{L}\left({F}_{ab}^{\phi }(R,\lambda ){\mathrm{e}}^{-{E}_{\phi }(R,\lambda )L}\right).\end{equation} \tag{ 7.15 }$$

Expanding both sides, we find that the $\mathcal{O}(L)$ piece reduces to the flow equation of the deformed spectrum E_ϕ (R, λ) in the closed channel

$$\begin{equation}{\partial }_{\lambda }{E}_{\phi }(R,\lambda )={E}_{\phi }(R,\lambda ){\partial }_{R}{E}_{\phi }(R,\lambda ),\end{equation} \tag{ 7.16 }$$

while the $\mathcal{O}(1)$ piece gives the flow equation for ${F}_{ab}^{\phi }$

$$\begin{equation}{\partial }_{\lambda }{F}_{ab}^{\phi }(R,\lambda )=\left(\frac{\partial }{\partial R}-\frac{1}{R}\right)({F}_{ab}^{\phi }\,{E}_{\phi }).\end{equation} \tag{ 7.17 }$$

Now we specify to the ground state and denote the corresponding quantity as F_ab . The product of g-functions is given by

$$\begin{equation}{g}_{ab}={g}_{a}^{\ast }{g}_{b}={\mathrm{e}}^{({f}_{a}(\lambda )+{f}_{b}(\lambda ))R}{F}_{ab}.\end{equation} \tag{ 7.18 }$$

To write down the flow equation for g_ab , we need to know the flow equation for f_a (λ) + f_b (λ). This can be derived by considering the large L limit of the flow equation (7.11). By comparing the coefficients of L, we find that

$$\begin{equation}{{\epsilon}}_{0}^{\prime }(\lambda )={{\epsilon}}_{0}{(\lambda )}^{2},\qquad {f}_{a}^{\prime }(\lambda )+{f}_{b}^{\prime }(\lambda )={{\epsilon}}_{0}(\lambda )({f}_{a}(\lambda )+{f}_{b}(\lambda )).\end{equation} \tag{ 7.19 }$$

These equations can be solved, which leads to

$$\begin{equation}{{\epsilon}}_{0}(\lambda )=\frac{{{\epsilon}}_{0}}{1-\lambda {{\epsilon}}_{0}},\qquad {f}_{a}(\lambda )+{f}_{b}(\lambda )=\frac{{f}_{a}+{f}_{b}}{1-\lambda {{\epsilon}}_{0}},\end{equation} \tag{ 7.20 }$$

where the quantities on the right-hand side of the above equations are undeformed. We can then derive the flow equation for the quantity g_ab (R, λ) as

$$\begin{align}\hfill {\partial }_{\lambda }{g}_{ab}=& R({f}_{a}^{\prime }(\lambda )+{f}_{b}^{\prime }(\lambda )){g}_{ab}+{\mathrm{e}}^{({f}_{a}+{f}_{b})R}{\partial }_{\lambda }{F}_{ab}\hfill \\ \hfill =& R{{\epsilon}}_{0}({f}_{a}(\lambda )+{f}_{b}(\lambda )){g}_{ab}+{\mathrm{e}}^{({f}_{a}+{f}_{b})R}{\partial }_{R}\left({F}_{ab}{E}_{\phi }\right)-\frac{1}{R}{g}_{ab}{E}_{\phi }.\hfill \end{align} \tag{ 7.21 }$$

Using the fact that

$$\begin{equation}{\mathrm{e}}^{({f}_{a}+{f}_{b})R}{\partial }_{R}{F}_{ab}={\partial }_{R}{g}_{ab}-({f}_{a}+{f}_{b}){g}_{ab},\end{equation} \tag{ 7.22 }$$

and the flow equation (7.19), we obtain

$$\begin{equation}{\partial }_{\lambda }{g}_{ab}=(R{{\epsilon}}_{0}-{E}_{\phi })({f}_{a}+{f}_{b}){g}_{ab}+({\partial }_{R}-{R}^{-1})\left({g}_{ab}{E}_{\phi }\right).\end{equation} \tag{ 7.23 }$$

Example: free theory and CFTs. We can easily verify the flow equation (7.17), or its equivalent form (7.23) for free theories and CFTs, where the exact g-function is known [2]:

$$\begin{equation}{G}_{a}^{\phi }(R,\lambda )={G}_{a}^{\phi ,\mathrm{C}\mathrm{F}\mathrm{T}}\sqrt{\frac{R}{R+\lambda {E}_{\phi }(R,\lambda )}}.\end{equation} \tag{ 7.24 }$$

Here ${G}_{a}^{\phi ,\mathrm{C}\mathrm{F}\mathrm{T}}$ is a constant that only depends on the boundary condition, but not on the size of the system. It implies that we could treat it as a constant, when studying the flow equation. Comparing with the definition of F_ab , we immediately see that

$$\begin{equation}{F}_{ab}^{\phi }={({G}_{a}^{\phi })}^{\ast }{G}_{b}^{\phi }={({G}_{a}^{\phi ,\mathrm{C}\mathrm{F}\mathrm{T}})}^{\ast }{G}_{b}^{\phi ,\mathrm{C}\mathrm{F}\mathrm{T}}\frac{R}{R+\lambda {E}_{\phi }(R,\lambda )}.\end{equation} \tag{ 7.25 }$$

Using the flow equation of the energy (7.16), one finds the deformed ${G}_{a}^{\phi }$ defined above solves the flow equation (7.17).

General solutions. We can simplify the structure of (7.17). Defining

$$\begin{equation}{F}_{ab}^{\phi }={\tilde{F}}_{ab}^{\phi }\frac{R}{R+\lambda {E}_{\phi }(R,\lambda )},\end{equation} \tag{ 7.26 }$$

the flow equation of ${\tilde{F}}_{ab}$ simplifies to

$$\begin{equation}{\partial }_{\lambda }\,\mathrm{log}\,{\tilde{F}}_{ab}={E}_{\phi }{\partial }_{R}\,\mathrm{log}\,{\tilde{F}}_{ab}+{\partial }_{R}{E}_{\phi }.\end{equation} \tag{ 7.27 }$$

Denoting $\mathrm{log}\,\tilde{F}$ by $\mathcal{F}$, we can solve this first order partial differential equation (PDE) for $\mathcal{F}$ by the method of characteristics. We refer to appendix C for a brief review of this method. Introducing an auxiliary 'time' parameter t, the equation above is equivalent to the following set of ordinary differential equations,

$$\begin{equation}\frac{\mathrm{d}\lambda (t)}{\mathrm{d}t}=1,\qquad \frac{\mathrm{d}R(t)}{\mathrm{d}t}=-{E}_{\phi }(R(t),\lambda (t)),\qquad \frac{\mathrm{d}\mathcal{F}}{\mathrm{d}t}={\partial }_{R}{E}_{\phi },\end{equation} \tag{ 7.28 }$$

where we treat λ, R and their functions as functions of t.

We choose the initial value of those equations in such way that when t = 0, the solutions correspond to the undeformed theory, where λ = 0. The first equation then immediately implies λ = t. For the second equation, notice that the total derivative of E_ϕ vanishes due to the flow equation (7.16):

$$\begin{equation}\frac{\mathrm{d}{E}_{\phi }}{\mathrm{d}t}=\frac{\mathrm{d}R}{\mathrm{d}t}{\partial }_{R}{E}_{\phi }+\frac{\mathrm{d}\lambda }{\mathrm{d}t}{\partial }_{\lambda }{E}_{\phi }={\partial }_{\lambda }{E}_{\phi }-{E}_{\phi }{\partial }_{R}{E}_{\phi }=0.\end{equation} \tag{ 7.29 }$$

Therefore, the rhs of the second equation is actually independent of t. The solution is then obtained by integrating once, R(t) = R₀ − tE₀(R₀), where E₀, R₀ denote the undeformed energy and the undeformed circumference, respectively. In order to solve the last equation, notice that

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}({\partial }_{R}{E}_{\phi })=\frac{\mathrm{d}R}{\mathrm{d}t}{\partial }_{R}^{2}{E}_{\phi }+\frac{\mathrm{d}\lambda }{\mathrm{d}t}{\partial }_{\lambda }{\partial }_{R}E={\partial }_{R}{\partial }_{\lambda }{E}_{\phi }-{E}_{\phi }{\partial }_{R}^{2}{E}_{\phi }={({\partial }_{R}{E}_{\phi })}^{2},\end{equation} \tag{ 7.30 }$$

where we take another ∂_R derivative of the flow equation (7.16) to simplify the rhs. Therefore, as a function of t, ∂_R E is easily found to read

$$\begin{equation}({\partial }_{R}{E}_{\phi })(t)=\frac{{\partial }_{{R}_{0}}{E}_{0}}{1-t{\partial }_{{R}_{0}}{E}_{0}},\end{equation} \tag{ 7.31 }$$

where the derivative ${\partial }_{{R}_{0}}{E}_{0}$ is evaluated at t = 0, namely in the undeformed theory. The solution for $\mathcal{F}$ is then easily obtained,

$$\begin{equation}\mathcal{F}(t,{R}_{0})={\mathcal{F}}_{0}({R}_{0})-\mathrm{log}\left(1-t\,{\partial }_{{R}_{0}}{E}_{0}({R}_{0})\right),\end{equation} \tag{ 7.32 }$$

where ${\mathcal{F}}_{0}=\mathrm{log}\,{F}_{ab}(\lambda =0,{R}_{0})$ is the undeformed g-function.

To summarize, we have found the general solution to the flow equation $\mathcal{F}=\mathrm{log}\,{F}_{ab}$, which depends on two parameters (R₀, t). In order to convert them into the original variables (R, λ), we have to solve them in terms of (R₀, t). Explicitly, they are given by the implicit solutions of the following equations,

$$\begin{equation}t=\lambda ,\qquad R={R}_{0}-t{E}_{0}({R}_{0})={R}_{0}-tE(R(t),\lambda (t)),\end{equation} \tag{ 7.33 }$$

where ${\mathcal{F}}_{0}=\mathrm{log}\,{F}_{ab}(\lambda =0)$ is the undeformed g-function. Substituting the solution t = t(R, λ), R₀ = R₀(R, λ) to $\mathcal{F}(t,{R}_{0})$, we find the final solution of log F_ab .

For the free theory, the ground state energy is given by the Casimir energy E₀(R₀) = −πc/6R₀, where c is the effective central charge. In this case, the coordinate transformation between (t, R₀) and (λ, R) can be made explicit,

$$\begin{equation}t=\lambda ,\qquad {R}_{0}=\frac{1}{2}\left(R+\sqrt{{R}^{2}-\frac{2c\pi \lambda }{3}}\right).\end{equation} \tag{ 7.34 }$$

One can easily solve for $\mathcal{F}$:

$$\begin{equation}\mathcal{F}(\lambda ,R)={\mathcal{F}}_{0}({R}_{0}(\lambda ,R))-\mathrm{log}\left(1-\frac{c\pi \lambda }{6{R}_{0}^{2}}\right).\end{equation} \tag{ 7.35 }$$

Substituting the function R₀(R, λ), one can verify that the flow equation (7.27) holds, regardless of the form of ${\mathcal{F}}_{0}$.