Exact Overlap in the Lieb-Liniger Model from Coordinate Bethe Ansatz

In the paper arXiv:2002.12065, the authors developed a new method to compute the exact overlap formula between integrable boundary states and on-shell Bethe states in integrable spin chains. This method utilizes the coordinate Bethe ansatz representation of wave functions and singularity property of the off-shell overlaps. In this paper, we use this new method to derive the formula for overlap between the Lieb-Liniger Bethe states and the Bose-Einstein condensate (BEC) state. Our formula coincides with the earlier result.


Introduction
In the study of non-equilibrium statistical physics and high energy physics, in particular in the homogeneous quantum quench problem [2] and one-point function in defect CFT in the AdS/CF T correspondence [3], the overlaps of on-shell Bethe states with integrable boundary states play a very important role. It is also related to exact g-function or boundary entropy [4,5] in integrable quantum field theory. Recently, people find that under some interesting set-up, three-point function in AdS 5 /CF T 4 can be view as an exact g-function in some integrable quantum field theory on the string world sheet [6].
The first on-shell Bethe states with some particular initial state overlap formula was found in [7], it has a factorized form and is similar to Gaudin formula for the norm of the Bethe state. People found that this factorizable property still survives in a large set of states-integrable boundary states [8].
In [9], people studied the Bose-Einstein condensate (BEC) to the Lieb-Liniger quench using the quench action method where a conjectured exact overlap formula for the BEC state with the Lieb-Liniger energy eigenstate is a starting point. This conjectured overlap formula was rigorously proved in [10], using the fact that Lieb-Liniger model quantities can be obtained from a scaling limit of XXZ spin chain [11] where a generalized Quantum Transfer Matrix (QTM) method can be applied [12].
Although this overlap formula is rather important, a direct proof is still missing. In a very recent paper [1], the author proposed a new method to directly calculate the overlap between the on-shell Bethe states with the integrable boundary states in integrable spin chain models using the Coordinate Bethe Ansatz (CBA) formalism. This method is very simple and inspiring and can directly apply to the Lieb-Liniger model, where the Bethe wave function is known. In fact, using coordinate Bethe ansatz to calculate the overlaps between integrable boundary states and Bethe states were firstly proposed in [3].
In this paper, we use this new method to derive the overlap for the BEC state with the Lieb-Liniger energy eigenstate. We obtained the same result as before. The remaining part of this paper is organized as follows: In section 2, we review some basic facts about the Lieb-Liniger model and fix some notations. In section 3, we study the overlap formula using coordinate Bethe ansatz method started with the one-and two-particle cases which serve as warm-up examples and give hints to deal with generic cases. In the next subsection, we derive the overlap formula of the BEC state and the generic parity symmetric on-shell Bethe state. Finally, we conclude in section 4 and the proof of the vanishing of the overlap of generic on-shell Bethe states with the BEC state is given in appendix A.

The Bethe equations
We consider one-dimensional Boson gas on a ring of circumference L with δ-function repulsive potential and impose periodic boundary condition. The Hamiltonian is We can write down the energy eigenstates in the coordinate space as and |x N and |λ N is the shorthand notations of |x 1 , x 2 , · · · , x N and |λ 1 , λ 2 , · · · , λ N respectively.
The requirement of the wave function x N |λ N should be periodic in each of its arguments x j results in the Bethe equations The corresponding energy eigenvalue is When the Bethe equations eq. (2.4) are not satisfied, the wave function eq. (2.2) still defines a state which we call it off-shell. On the contrary, when the Bethe equations are instead satisfied, we call the state on-shell.

The parity symmetric Bethe state
The norm of the Bethe state is given by the Gaudin formula [13] where G is an N × N matrix whose elements are and the function f (λ 1 , λ 2 ) is defined as The function ϕ(λ) is given by (2.9) In this paper, we will consider the overlap between the BEC state |BEC with x N |BEC = 1 L N/2 and the Lieb-Liniger energy eigenstate or the Bethe state |λ N . More precisely, we will compute the overlap between the BEC state and the on-shell Bethe state with parity symmetry: where N is an even number. The norm of an on-shell Bethe state which has the above paired structure can be factorized further (2.14) For future use, we introduce the variables Thus the Bethe equations eq. 2.4 can be written as 3 Overlap from CBA 3.1 One-particle states As a warm-up, let's first consider the one-particle case where the definition eq. (2.15) has been used. For on-shell state, which means the Bethe equation a 1 = e iλ 1 L = 1 is satisfied, the above overlap is non-vanishing only when λ = 0. The integral is trivially integrated to give the result BEC|λ 1 = 0 = √ cL.
In order to manipulate more complicated cases, we can obtain the same result using a limiting procedure:

Two-particle states
We now consider the two-particle case. For the readers' convenience and to gain more experiences for computing higher particles cases, we work out the details. Firstly, we write In order to evaluate the above integral, it's useful to introduce the basic integral in the region 0 < x 1 < x 2 < L: where the a variables dependence of the function B 2 is obtained only after the integral is worked out and the definition eq. (2.15) is substituted. Other integral can be obtained by permutating the arguments of this function. Note that the last two lines in eq. (3.3) give the same result. It's easy to see that this property also holds in the multi-particle case. Thus where It's easy to check that when the Bethe state is on-shell, i.e. when we do the replacement in above two-particle off-shell overlap formula eq. (3.5) we get identically zero.
To obtain non-vanishing on-shell overlaps, we need to consider the parity-invariant states with λ 2 = −λ 1 . It has a pole at λ 2 = −λ 1 . Around this pole, we have Note that when the state is on-shell, we have a 1 a 2 = 1. Then the non-vanishing on-shell overlap only comes from this singular part. In fact, the exact parity-invariant two-particle overlap can be obtained by taking the following limiting procedure (3.9)

Multi-particle states
Having the above explicit calculation in mind. Now we turn to the multi-particle case. It's obvious that we have the following relation where we have defined the basic integral in the region 0 < x 1 < x 2 < · · · < x N < L as before For lower N, B N can be integrated out directly. For example . (3.12) For general N, it's not easy to directly work out this integral. However, we have the following recursion relation where we view B N (λ N ; a N ) as a function of the system size L inexplicitly as the L-dependence is encoded in the definition of the variables a i through eq. (2.15). We also have the initial conditions for each N B N (λ N ; a N )| L=0 = 0, N = 1, 2, 3, · · · . (3.14) For example, from we can get B 2 as (3. 16) which gives the same result as eq. (3.6). In a similar way, we can reproduce the result of eq. (3.12).
In the above calculation, it not hard to figure out the pattern. Using the recursion relation eq. (3.13) and the initial condition eq. (3.13) repeatedly, we obtain the general expressions for B N . . (

3.19)
These two formulas for B N in eq. (3.17) and eq. (3.18) have slight differences but are equivalent to each other. However, we found the later is more useful for our convenience.
The N-particle overlap can be written as (3.20) In above expression, we have defined the formal overlap It can be show that for generic (non-parity-symmetric) on-shell Bethe state, S N (λ N ; a N ) vanishes. Only for the parity symmetric on-shell states |λ + N/2 , −λ + N/2 , we have finite overlaps. See Appendix A for a proof of this statement.

Determining the singular part
In order to find out the non-vanishing overlap of on-shell parity symmetric Bethe states with the BEC state, we only need to figure out the singular part of S N (λ N ; a N ) and taking the limiting procedure. Now we try to find the residue of the pole of S N , say 1/(λ 1 + λ 2 ) for example. Firstly, we must try to find out the residue It's easy to find that there are two terms B N,m−1 and B N,m+1 contribute. After some computation, we find the following relation (3.23) For the same reason explained in detail in [1], after the summation of all permutations that make particles 1 and 2 in neighbouring position, we find the singularity of the overlap at is the formal overlap for N − 2 particles which do not include particles 1 and 2, and is evaluated with the modified a-variables 2 : The function F (λ) can be found (3.26)

Taking the limit
The exact on-shell parity symmetric Bethe state with the BEC state overlap is obtained only after the following limiting procedure is token As demonstrate in [1], it's useful to introduce new variables (3.28) We denote D(λ + N/2 , m + N/2 ) as the limit of S N (λ N ; a N ) describe in eq. (3.27).
Note that (3.29) It's easy to show that the function D satisfies the recursion where the modification rule for the m-parameters is The solution of the recursion relation eq. (3.30) can be found as [1] Then the exact on-shell parity symmetric Bethe state with the BEC state overlap is After the substitution of the expression eq. (3.26) and eq. (2.8) into eq. (3.33), we obtained the final result This formula agrees with the earlier results [9,10].

Conclusion
In this short note, we derived the exact overlap formula between the Lieb-Liniger Bethe states and the Bose-Einstein condensate state using a recently developed method. This method is based on the coordinate Bethe ansatz which is available for the Lieb-Liniger model and does not relies on the complicated "rotation trick" which is not known in our case. This overlap formula is of great importance in the study of the BEC to Lieb-Liniger quench, but a transparent derivation is lacking. This paper gives a rather concise computation procedure based on the newly proposed method. We hope more insight can be found in this new method and it's very interesting to find more applications such as the nest integrable system where similar exact overlap formula also exist (see for example [14][15][16]).
In integrable spin chain models, there also exist exact overlap formula for Matrix Product States (MPS) [17]. In [18], the author introduce the continuous Matrix Product States (cMPS) in the Lieb-Liniger model. It would be very interesting to find out exact overlap formula for these cMPS.
As you will see in the appendix, the proof of the overlap vanishes for generic (non-paritysymmetric) on-shell Bethe is very specific and is not inspiring (for a earlier proof, see [19]). A more transparent proof is needed.
It turns out that the coordinate Bethe ansatz and the calculation of exact overlap have some applications in the physics of the stochastic non-linear Kardar-Parisi-Zhang (KPZ) equation, where the computation of the exact generating function of the Kardar-Parisi-Zhang height requires to calculate the overlap between the Bethe wave functions and the initial condition of the equation [20]. We think our result and all the classification obtained in [8] could help to classify the integrable initial conditions in the KPZ equation.

Acknowledgments
I would like to thank Hao Ouyang for very helpful discussions, especially for the proof of the vanishing of the overlap for generic on-shell Bethe states with the BEC state.
A Proof of S N = 0 for generic on-shell Bethe state In this appendix, let's prove that the overlap of a generic on-shell Bethe state with the BEC state vanishes.
For a given σ, let's consider terms in eq. (3.21) containing the factor F σ ≡ j<k f (λ σ j , λ σ k ). (A.1) If we define λ σ j = ip σ j − ip σ j−1 , then we have The terms in eq. (A.5) with j = 0 and j = N come from the summand in eq. (3.21) corresponding to the permutation σ. and the term with 0 < j < N comes from the summand in eq. (3.21) corresponding to the permutation τ −1 j σ.