An angular dependent supersymmetric quantum mechanics with a Z2-invariant potential

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a ${\bf Z}_2$ angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a spontaneous symmetry breaking with two ground states above a critical scale. Unlike the ordinary $O(2)$-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.


Introduction
In [1] a solvable model for a superconformal quantum mechanical system was introduced, giving an example of a quantum topological theory with no ground state and no mass gap.
In [2] this model was extended to the case of a possibly infinite chain. This toy model was motivated by the search of an abstract definition of mechanical systems possessing epicycle descriptions, in an attempt to encode the robotic movements of rods presented in [3].
More refined models are expected to have a breaking of conformal symmetry. It is tempting to look for new classes of potentials that break other types of symmetries as well, like the rotational invariance, while keeping some of the properties of the topological models [1,2]. Here we generalize the topological invariant γ dθ that defines the topological quantum mechanics of [1] into another invariant, which has a further dependence on a point A distinct from the singular origin O of the punctured plane; this allows us to introduce a length a = | −→ OA| and a dimensionless order parameter. The model is defined by a potential which breaks both the conformal and the rotational symmetry, but keeps a discrete Z 2 symmetry. This discrete symmetry is reminiscent of the T -duality transformation in string theory, see [4]. It carries, however, an extra angular dependence.
The introduction of the length scale allows for the existence of a discrete spectrum and a normalizable ground state. A spontaneous symmetry breaking is found. Indeed, above a critical value for the order parameter, two ground states are found. Two main properties single out this potential with respect to the standard Higgs potential. The first one is the presence of an angular-dependence; the second one is the fact that, above the critical scale, the potential does not possess any local maximum, but only (two) saddle points. The possibility, following [5], of the existence of instantons is investigated. CBPF-NF-007/17

A reminder of conformal topological mechanics
The conformally invariant topological quantum mechanics of [1] is defined by the topological gauge-fixing following the BRSTQFT, i.e. localization, scheme of [6] for the topological invariant γ dθ on a punctured plane with real coordinates q i (i = 1, 2, x ≡ q 1 , y ≡ q 2 ) or complex coordinates q 1 + iq 2 = | q| 2 exp iθ.
By introducing a coupling constant g, the topological classical action [1] reads where t is a Euclidean time and the dot means the time derivative d dt .
This action must be gauge-fixed in a topological BRST invariant way. Using the complex coordinate notation z(t) ≡ q 1 (t) + iq 2 (t) (z = z * ), the topological gauge-fixing equation was obtained in [1] as:ż The BRSTQFT gauge invariant procedure provides a conformally invariant supersymmetric action which, once untwisted, gives a N = 2 supersymmetry.
The solutions of (2.2) are periodic instantons z N (t), where N is an integer. We have They minimize (to the zero value) the non-negative bosonic action The instantons are pseudo-particles that make N cycles per unit time at constant angular velocity around the singular origin on circles with radius |z N | = 1 √ 2πN g . The area Σ N of the surface that the instanton z N circles around is independent on the value of N : (2.5) The instanton structure follows from the identitẏ where the vector indices i, j are raised/lowered by the Euclidean metric δ ij .
The equation of motion, derived from the supersymmetric action, of the fermionic partner Ψ = Ψ 1 + iΨ 2 of z, corresponds to the supersymmetric variation Q of Eq. (2.2).
The representations of the matrices acting on vectors are cos 2θ sin 2θ . (2.8) In complex notation one has U θ z = iexp(−2iθ)z * .
The action of the rotation R ±θ on the instanton z N is a multiplication by exp(±2πiN t).
Therefore, the solution of the fermion equation (2.7) in the z N background is where Ψ 0 is time independent and It follows that Ψ 0 is just an eigenvector of the operator C which runs attached to the instanton. One has The fermionic zero modes can be thought as tangent vectors cycling at constant frequencies 2πN g around the circles of radius √ 2πN g. Their existence implies a non-trivial cohomology for the BRST symmetry of the theory. The paper [1] discusses in detail the model and the way it allows to compute some (simple) topological invariants of the punctured plane.

A new model and its topological gauge-fixing
Let us introduce a point A = O on the punctured plane with singular origin O. We define the orthonormal coordinates (x = q 1 , y = q 2 ) such that −→ OA = (a, 0). We also use the complex and polar coordinates z = q 1 + iq 2 = r exp(iθ) when needed.
Given any contour γ in the punctured plane, we add to the closed and not-exact form gdθ an exact-term that depends on the distance of the particle to the point A. The simplest possibility is offered by gdθ + ω 2 d(| q − −→ OA| 2 ). We therefore define the classical topological action I cl γ (g, ω) of the new model as The action I cl γ (g, ω) is independent of any infinitesimal shift of coordinates with appropriate boundary conditions.
Following the general BRSTQFT scheme of [6], we are interested in computing observables that satisfy the Ward identities corresponding to the above gauge symmetry (3.2). We will come back on this later, by showing the possibility of fermionic zero modes. Comparing the actions (2.1) and (3.1) one understands that the topological gauge-fixing (2.2) must be improved intȯ The above equation reads, in polar coordinates, aṡ and, in Cartesian coordinates, aṡ The parametrization expressed by a, ω, g can be simplified through redefinitions, as we will shortly see. On the other hand, to have the possibility of discussing several interesting limits, it is better to keep track of the explicit dependence on these parameters.
The supersymmetric Q-exact action that localizes the topological gauge fixing (3.3) is easier to compute in Cartesian coordinates. It transforms the topological term into a supersymmetric action. It gives, on the same time, a consistent (that is, BRSTQFTinvariant) information about the g, ω, a dependence of the theory. We get (and vanishing otherwise).
The computation of the Q-transformation of the topological gauge function provides a Dirac-type operator acting on fermions which, in Cartesian coordinates, reads as where R θ and C have been introduced in Eq. (2.8).
We postpone to Section 7 some discussion about the supersymmetry of the model given by the action I susy γ (g, ω). We are interested at first in the vacuum and therefore in the bosonic part of the Euclidean action I susy γ (g, ω). After expanding the Q-exact term in (3.6), eliminating the auxiliary fields H x , H y by their algebraic equations of motion and discarding the terms involving fermions, one obtains the following Euclidean bosonic action (written, for convenience, in polar coordinates). One has where the potential V (r, θ) is non-negative, being the sum of squares: The potential possesses the interesting Z 2 discrete symmetry T (T 2 = I), given by The symmetry can be expressed as where r 0 , given by can be set as a unit of length. One can thus define the dimensionless parameter The potential (4.2) can be expressed as the product of the dimensional factor gω times a dimensionless function: The action (4.1) is both conformally and rotationally invariant at α = 0. Both symmetries are broken for α = 0. On the other hand the discrete symmetry T in (4.4) of the potential is maintained. We will show that the potential exhibits a transition between a simple and a double well, with a critical value for the dimensionless parameter α.
If we use r 0 as the unit of length the only relevant parameter of the model is α = a.
Therefore, the parameter of the conformal symmetry breaking is the distance between the rest point of the harmonic oscillator and the singularity of the potential 1/r 2 expressed in units of r 0 . At vanishing α the restored conformal symmetry is the well-known hidden SL(2) symmetry [7,8,9] of quantum theories with potential r 2 + 1/r 2 . The SL (2) conformal symmetry without the oscillatorial term was introduced in [10] (see [1] and [11] for its formulation in the topological setting).
5 The shape of the potential. Are there instantons?
We follow the method explained by Coleman in [5]. The potential (4. In quantum mechanics with more than one degree of freedom one is often concerned with the stability of the Euclidean solutions, even when they are energetically possible for connecting two vacua. We should point out that in the present case the potential We give the explicit expression of the two maxima M ± , both in polar (r ± , θ ± ) and in Cartesian (x ± , y ± ) coordinates. It is convenient to introduce the angle ϕ, In terms of ϕ we can write, in polar coordinates, r + = αr 0 cos ϕ, θ + = −ϕ, In Cartesian coordinates we have x + = αr 0 cos 2 ϕ, y + = − 1 2 αr 0 sin(2ϕ) = − r 0 α , Due to the discrete T symmetry of the potential, the following relations are found: as well as In order to further clarify the structure of the theory we compute the extremal points of the potential (4.7), obtained by solving the coupled system of equations At the extrema, we check the sign of the second-order derivatives to determine local minima, maxima and saddle points. We present the complete set of solutions in the different ranges of the order parameter α: i) at α = 0 we recover the O(2) rotational invariant theory. The case is analogous to the one investigated in [1], with the important difference that the harmonic term allows for a discrete spectrum with a normalizable vacuum. The unbroken O(2) symmetry implies a S 1 circle of degenerate maxima. It is given by the points at distance r 0 from the origin and −V (r 0 , θ) < 0; ii) in the 0 < α < √ 2 range two extremal points are found (we call them M 0a and M 0b ).
Their polar coordinates (which do not depend on α) are iv) in the range α > √ 2 we have four extremal points, the two maxima M ± given by Eq.

The fermionic part of the action
The fermionic part of the I susy γ (g, ω) action (3.6) is By taking into account Eq. (3.7) for the operator Q one has We call this expression the "classical fermionic action"; the fermions Ψ x , Ψ y , Ψ x , Ψ y are time-dependent anticommuting Grassmann variables.
The fermionic zero modes are defined as the solutions of the following equation, By setting Ψ = Ψ x + iΨ y one can also writė In the above equation the functions r(t) and θ(t) solve the self-duality equations (3.4).
By redefining Ψ → Ψ = exp(−iθ)Ψ, one gets the equatioṅ By expressingθ in terms of the second equation of (3.4), one haṡ so that one gets the following zero mode equatioṅ This first order differential equation indicates that, most likely, fermionic zero modes exist in the background of the r(t), θ(t) solutions of the self-duality equations (3.4).

The supersymmetric Hamiltonian
The passage from the supersymmetric Lagrangian entering (3.6) to the classical Hamiltonian formulation is done following the standard prescription, by performing a Legendre transformation for both bosonic and fermionic coordinates. One introduces the conjugate momenta p i to the coordinates q i , while the fermionic Grassmann fields Ψ i turn out to be conjugate to the fermionic fields Ψ i . The Poisson brackets are Z 2 -graded. The non-vanishing Poisson brackets are The classical Hamiltonian H is N = 2 supersymmetric. It corresponds to the Morse The supersymmetric charges Q, Q are given by A quick computation shows that the classical fermionic part of the Hamiltonian is A convenient presentation of the quantum fermionic operators Ψ i , Ψ i is via the 4 × 4 matrices ξ ±i , through the positions where ξ ±i are given by The matrices ξ ±i generate the Cl(2, 2) Clifford algebra. Indeed, by setting for µ = 1, 2, 3, 4, we end up with the basic relations, for their anticommutators, where η µν is the flat metric with diagonal entries (1, 1, −1, −1).
The hamiltonian H and the supercharges Q, Q can be expressed as 4 × 4 differential matrix operators. In polar coordinates the hamiltonian is where V 0 (r, θ) coincides with the potential in Eq. (4.2), namely V 0 (r, θ) = 1 2 while The supercharge Q is explicitly given by We note that the Hamiltonian (7.11) cannot be diagonalized. Indeed, a rotation, acting on fermions alone, which diagonalizes the potential V 1 (r, θ), is θ-dependent, so that it produces non-diagonal terms when applied to the Laplacian. By realizing that the lower two-dimensional block in V 1 (r, θ) can be expressed as − g r 2 R −θ CR θ , where R θ , C have been introduced in (2.8), a simpler expression for the quantum dynamics is given by the hamiltonian H, unitarily equivalent to H via the transformation One has where the 4 × 4 matrix V 1 (r, θ) is decomposed in 2 × 2 blocks according to .

(7.20)
σ 3 is the diagonal Pauli matrix, while C = σ 1 is the conjugation matrix introduced in (2.8). The non-diagonal terms correspond to a first-order differential operator.

Conclusions
We introduced a deformation of the conformally invariant topological quantum model [1] of a particle moving on the punctured plane. The modified potential depends on a length scale r 0 and on a dimensionless parameter α ≥ 0. It explicitly breaks the rotational and the conformal invariance, but it preseves a discrete Z 2 simmetry given by the idempotent map r/r 0 → r 0 /r and θ → −θ − π/4 . The role of α is that of an order parameter with a critical value α c = √ 2. Below that value (for 0 < α ≤ √ 2) the ground state is unique.
Above that value (α > √ 2) the ground state is doubly degenerate.
Two key features distinguish this potential with respect to the well-known Higgs potential λ(| q| 2 − m λ ) 2 in the plane. The first one is the presence of an angular dependence (since the model is not rotationally invariant). The second relevant feature is that, above the critical value, the extremal points are the two ground states and two saddle points (therefore, the theory does not possess local maxima).
We proved (with techniques similar as those applied in [5], [12]) that, in the Eu- The BRSTQFT (see [6]) gauge-fixing method induces a N = 2 supersymmetric quantum mechanics that we constructed both in the path-integral (the fermionic timedependent fields are Grassmann coordinates) and in the operatorial (the fermionic degrees of freedom are realized by Clifford matrices) frameworks.
In the operatorial (and real time) formulation the Hamiltonian is a 4 × 4 hermitian differential operator. By construction the Hamiltonian, for any α ≥ 0, has a well-defined, discrete and bounded from below, spectrum. The Hamiltonian is non-diagonalizable and this is another indication of the richness and the non-triviality of the model.
We leave for a future paper the computation of the spectrum (numerical, if not analytical) of this quantum theory and of its further properties.
As mentioned in the Introduction the construction of this model with such a critical phenomenon was found by searching for a potential that can possibly simulate the constraints of a mechanical device that acts as a series of locks, allowing the propagation of signals from one side to the other of a chain, reproducing the behaviour of some of the machines described in [3].