The ground state of the sine-Gordon soliton

At one loop, we provide an explicit formula for the ground state of the one- soliton sector in the sine-Gordon theory. The state is given in the basis of eigenstates of the field operator, or equivalently as a Schrodinger wave functional. The formula readily generalizes to other solitons in other models and as an example we also provide the ground state of the kink in the (1+1)-dimensional ϕ4 double well.


Background material
The prototypical strong-weak duality is that between the sine-Gordon model and the massive Thirring model [1]. The central role in this duality is played by the sine-Gordon soliton, which becomes the fundamental fermion in the massive Thirring model. The soliton is a solution of the classical equations of the motion of the classical sine-Gordon theory. In the quantum theory, it is so far understood only in a singular limit of a tree-level approximation [2]. But what lies between these two limits? In this note, we will find the ground state of the one-soliton sector at one loop.

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The classical equations of motion derived from (1.1) admit the soliton solution (1.4) It will be convenient to define a new Hamiltonian H K which is related to the original Hamiltonian by a similarity transformation where D f is the translation operator which shifts the field by the soliton solution. In classical field theory, H K describes oscillations about the soliton configuration. As we have normal-ordered the Hamiltonian in (1.1), the theory is finite and regularization is not necessary. However, had we adapted a more general regulator, which would be necessary in a theory with fermions or more dimensions, then it is essential that the full, regulated H be used in eq. (1.5).
Let |K be the soliton ground state and, motivated by [4,5], define O to be any operator such that If E is the minimum soliton energy, then Left multiplying by D −1 f one finds and so O|0 is the lowest eigenvector of H K . It was shown in ref. [6] that O can be chosen to be equal to the identity plus quantum corrections. 1 Let us make such a choice, so that (1.8) can be solved in perturbation theory. Let Q 0 be the classical soliton energy. Then where H I contains all interaction terms (1.10) As λ has dimensions of inverse action, the loop expansion is an expansion in the dimensionless combination λ . We set = 1 and so each loop brings a power of λ, begining with terms of order 1/λ which appear in Q 0 . The terms in H I , which are all at least O(λ 1/2 ), only contribute at two loops and beyond. In this paper we will work at one loop,

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where the only contribution to the Hamiltonian is the free Pöschl-Teller Hamiltonian with Hamiltonian density (1.11) whose classical equations of motion have constant frequency solutions g k (x) parameterized by k and a bound state solution g B (x) representing the soliton Goldstone mode with respective frequencies These have been normalized so that (1.14) As H K is free, the classical equations of motion are linear, real eigenvalue equations and so their solutions are orthogonal and they satisfy the reality conditions In the ground state sector it was convenient to decompose the field φ(x) into plane waves (1.2) to obtain the Heisenberg operators a p . Similarly, in the one-soliton sector it is convenient to decompose φ(x) into the constant frequency solutions 2 (1.12) Using the orthogonality relations (1.14) these can be inverted

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We will consistently use the variable and index k to denote this decomposition into Pöschl-Teller wave functions, while p and q will be reserved for the decomposition into plane waves (1.2). Note that, with this notation, φ k and φ p are quite different, they are related by where we have defined the inverse Fourier transforms After a tedious calculation one finds [6] where Q 1 is the one-loop soliton energy. Thus the lowest eigenstate (1.8) solves So finishes our review of ref. [6].

The sine-Gordon soliton state
We will work in the basis of states provided by the eigenvectors |F of the field operator φ(x). Here F is a real-valued function of the space coordinate x. The basis states are defined by the eigenvalue equation The basis |F is complete because it is the set of eigenvectors of the Hermitian operator φ(x). Any state |Ψ can be expanded in terms of the basis states F where DF is a measure on the space of functions F and Ψ[F ] is the complex-valued Schrodinger wave functional Ψ [7,8] evaluated on the function F . This is the analogue of a wave function in quantum mechanics Just as in quantum mechanics, states can be equivalently described by Dirac kets |Ψ or their matrix elements, the wave functionals Ψ[F ] and in fact both of these objects satisfy the same operator equations. Thus for simplicity we will work with the wave functionals and not the kets. Before solving eq. (1.23) for the soliton state, let us remind the reader of the solution of

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which describes the ground state of the free massive scalar theory. Working in terms of wave functionals, this is Let us try Using the commutator in (1.3) one finds and so establishing that Ψ 0 is the ground state wave functional of the vacuum sector.
Similarly, for the soliton wave functional corresponding to O|0 try Note that this is not equal to Ψ 0 in (2.7) because φ q is the transform of φ with respect to plane waves while φ k is the decomposition with respect to Pöschl-Teller functions. As Ψ O is independent of φ 0 and π 0 commutes with φ k , trivially In other words, this state is translation-invariant and so it has zero momentum. Following the argument above and so Ψ O is annihilated by b −k and solves (1.23). Therefore we have shown that According to the definition (1.6), to obtain the kink state, we need to left multiply this result by (2.14) Using the fundamental property of the translation operator JHEP07(2020)099 and the orthogonality relations (1.14) one finds the action of D f on φ 0 and φ k Thus we can compute the action of D f on any function of φ 0 and the set of φ k , it simply translates each argument. The soliton ground state is then This is our main result.

The double well kink state
We expect the above construction to apply to stationary classical solutions in a range of quantum field theories. In this section we will show how trivially it is extended to the φ 4 double well in 1+1 dimensions, first treated at one loop in ref. [9]. This theory is described by the Hamiltonian and has a classical kink solution Again we define a kink Hamiltonian by the similarity transform (1.5), where f (x) is now given by eq. (3.2). At one loop the corresponding Hamiltonian density is where β = m/2. Like (1.11) in the case of the sine-Gordon theory, this is a reflectionless Pöschl-Teller Hamiltonian. However, now it is at level 2 instead of level 1, due to the factor of 3. The difference in level in the potential means that in addition to the Goldstone mode g B (x), H K has an additional, odd, classical bound state g BO (x). Overall the eigenfunctions are The frequencies, orthogonality relations and reality conditions are as in the sine-Gordon case except now we also have

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Let us decompose the field and its conjugate Using the orthogonality relations (1.14) and (1.15) together with these can be inverted Note that φ BO and π BO are anti-Hermitian which is the reason for the wrong sign in eq. (3.9). We adopt a star instead of a dagger for Hermitian conjugation of φ BO for later convenience, because the field φ in a Schrodinger wave functional is interpreted as a function and not an operator. However strictly speaking a dagger should be used at this step, as φ BO is an operator, to be replaced by a star only at the end of the calculation when we write the wave functional. The interpretation of φ as a function, instead of an operator, in the wave functional is possible because π does not appear, and so φ commutes with everything. In quantum mechanics the analogous statement is that x is an operator, but when the wave function is presented x may be interpreted as simply a coordinate. After some calculation one finds [10] Thus the lowest eigenstate solves This is identical to the sine-Gordon case except for the b BO condition, which states that the quantum harmonic oscillator corresponding to this oscillation mode of the kink is in its ground state. To solve this condition note that

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and so Thus the Schrodinger wave functional corresponding to O|0 is As in the sine-Gordon case, to obtain the kink ground state we need to multiply by The wrong-sign canonical commutation relations in (3.9) lead to and so the kink ground state Schrodinger wave functional is 4 What next?
We are motivated by a desire to understand monopole condensation in N = 2 super Yang-Mills. Solitons may only condense in the deep nonperturbative regime, because within the validity of perturbation theory the tree level contribution to their energy is dominant, and this is positive in the theories of interest. Thus only nonperturbatively can the energy be reduced by a soliton condensate. To go beyond pertubation theory, we feel that we must first understand perturbation theory. In this paper we have found the ground state at one loop. We found that the state is independent of φ 0 . This is reasonable, as the Hamilonian is translation invariant and any breaking of translation invariance would lead to a Goldstone boson, which is forbidden in 1+1 dimensions. However φ 0 is not precisely the same as the location of the center of the soliton. They agree only for small perturbations. Already at two loops, the theory is sensitive to perturbations beyond this linear approximation. Therefore the translation symmetry, which is an exact symmetry of the ground state, is nonlinearly realized in terms of the fields. Moreover, there will be states with arbitrarily small momenta in the x direction, leading to a continuous spectrum. In the presence of a continuous spectrum, the usual perturbation theory is inapplicable as it involves denominators with differences in energies, but these differences are arbitrarily small. In general this break down of perturbation theory can lead to a host of interesting effects [11,12]. However, as in the current case the continuous spectrum comes from a symmetry of the ground state, we expect that it will be possible to simply impose the translation symmetry by hand and so render the spectrum discrete, after which perturbation theory may be applied. At one-loop, the translation mode φ 0 did not couple to other modes, indeed it did not even appear in the Hamiltonian, and so this was not necessary. However at two loops it will be necessary. One possible approach to this problem is that pioneered in ref. [13], but it may not be the most economical at two loops. where, for example in the case of the sine-Gordon theory, Ψ 0 is given in (2.7) and Ψ O in (2.10). Both are infinite-dimensional Gaussian functions. In both cases, the Hessian matrix has the same eigenvalues, the set of all negative numbers. In the case of the sine-Gordon theory, even the weight function, whose normalization is fixed by the 1/2 in the exponent, is identical for both Ψ 0 and Ψ O . In the case of the φ 4 theory the densities of the eigenvalues are not the same as a result of the odd ground state. One might have expected such a problem for the even ground state in both theories, but Ψ O is independent of φ 0 and so it does not contribute directly to the wave functionals, although it contributes indirectly by making the ψ k fail to span the space of functions.

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As the action of O on all states except for Ψ 0 is arbitrary, there are many solutions to (A.1). The simplest is to first rotate the eigenvectors φ p of the Hessian of Ψ 0 to the eigenvectors φ k of the Hessian of ψ O , and then use a squeeze operator to adjust the eigenvalues. If the rotation takes φ p to φ k(p) for some function k(p), then the squeeze operator must squeeze the Hessian eigenvalue from ω p to ω k(p) .
In the rest of this appendix we will concentrate on the sine-Gordon case. As the eigenvalues of the two Hessians are identical, with the same weights, we may choose in which case the squeeze operator is not necessary. With this choice, the operator O is simply a rotation which takes each φ p to φ k where k = p. Using the inverse of eq. (1.20) this condition can be written Formallyg p (q) is a matrix and its logarithm N pq can be defined by a power series. Then this equation is solved by Note that the δ term ing is proportional to the identity matrix and so only contributes a constant to N . This solution for O is independent of λ.

B Completeness
The basis of functions g is complete. There are two ways to see this. First of all, by the classical equations of motion, it is the set of all eigenvectors of the real operator ] is the potential term in the Hamiltonian density. This is shown, for the general Pochl-Teller potential which covers both of our examples, in ref. [14]. Briefly reviewing the argument there, one makes the substitution y = cosh 2 (mx), or y = cosh 2 (βx) (B.1) in the classical equations of motion (respectively) for the two theories and then divides by y or y 3/2 respectively. After this, one arrives at the hypergeometric equation. Its even and odd solutions are given by hypergeometric functions multiplied by monomials and parametrized by a single complex number k. Our g(x) is equal to the even function plus i times the odd function. When k is real and positive, one obtains the distorted plane waves g k (x). However there are normalizable functions g k (x) for some discrete values of k off the real axis. These are the bound states, and there is precisely one for the sine-Gordon model and two for the kink as these correspond to Poschl-Teller potentials at level 1 and 2 respectively. There is also a more concrete albeit numerical argument. Assuming completeness of the g(x), one can decompose φ(x) as in (1.17). The condition that φ(x) and π(y) satisfy the canonical algebra implied (1.19). But (1.19) in turn can be used to determine the algebra satisfied by φ(x) and π(y), which is the canonical algebra, as had been assumed, if and only if the completeness relation is satisfied in the case of the sine-Gordon model or in the case of φ 4 . In general, a sum of the norm squared of all bound states appears on the left. This does not prove that the completeness relation holds, only that it is equivalent to the validity of the decomposition. However we have checked the completeness relation numerically. Defining we integrate x 2 x 1 dyI(x, y).

(B.5)
If (1.14) or (B.3) is satisfied, then, as L tends to ∞, this integral approaches 1 if y ∈ [x 1 , x 2 ] and otherwise it approaches 0. In figure 1 we show that numerically this seems to be the realized. respectively. In black we plot the entire integral. One sees that while I(x, y) integrates to about unity for all large values of L, as L increases this function has a support closer to 3, as expected as I(x, y) approaches a δ(3 − x). Bottom: same as the right panel but for the φ 4 theory at β = 1.
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