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Register a new userBranch Point Twist Field Form Factors in the sine-Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement
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In this paper we continue the programme initiated in Part I, that is the study of entanglement measures in the sine-Gordon model. In both parts, we have focussed on one specific technique, that is the well-known connection between branch point twist field correlators and measures of entanglement in 1+1D integrable quantum field theory. Our papers apply this technique for the first time to a non-diagonal theory with an involved particle spectrum, the sine-Gordon model. In this Part II we focus on a different entanglement measure, the symmetry resolved entanglement, and develop its associated twist field description, exploiting the underlying U(1) symmetry of the theory. In this context, conventional branch point twist fields are no longer the fields required, but instead we must work with one of their composite generalisations, which can be understood as the field resulting from the fusion of a standard branch point twist field and the sine-Gordon exponential field associated with U(1) symmetry. The resulting composite twist field has correlators which as usual admit a form factor expansion. In this paper we write the associated form factor equations and solve them for various examples in the breather sector by using the method of angular quantisation. We show that, in the attractive regime, this is the sector which provides the leading contribution to the symmetry resolved entropies, both Renyi and von Neumann. We compute the latter in the limit of a large region size and show that they satisfy the property of equipartition, that is the leading contribution to the symmetry resolved entanglement is independent of the symmetry sector.
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In this paper, we apply the form factor bootstrap approach to branch point twist fields in the $q$-state Potts model for $qleq 3$. For $q=3$ this is an integrable interacting quantum field theory with an internal discrete $mathbb{Z}_3$ symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for $qleq 3$ the standard Renyi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with $qleq 3$ and for the composite (or symmetry resolved) branch point twist field with $q=3$. In both cases, the form factor equations are solved for two particles and the solutions are carefully checked via the $Delta$-sum rule. Using our analytic predictions, we compute the leading finite-size corrections to the entanglement entropy and entanglement equipartition for a single interval in the ground state.
We introduce and study the properties of a periodic model interpolating between the sine-- and the sinh--Gordon theories in $1+1$ dimensions. This model shows the peculiarities, due to the preservation of the functional form of their potential across RG flows, of the two limiting cases: the sine-Gordon, not having conventional order/magnetization at finite temperature, but exhibiting Berezinskii-Kosterlitz-Thouless (BKT) transition; and the sinh-Gordon, not having a phase transition, but being integrable. The considered interpolation, which we term as {em sn-Gordon} model, is performed with potentials written in terms of Jacobi functions. The critical properties of the sn-Gordon theory are discussed by a renormalization-group approach. The critical points, except the sinh-Gordon one, are found to be of BKT type. Explicit expressions for the critical coupling as a function of the elliptic modulus are given.
We consider the problem of the decomposition of the Renyi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider $SU(2)_k$ as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size $L$ the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on $L$ but only on the dimension of the representation. Moreover, a $loglog L$ contribution to the Renyi entropies exhibits a universal form related to the underlying symmetry group of the model, i.e. the dimension of the Lie group.
One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the $b rightarrow 1/b$ self-duality of its $S$-matrix, of which there is no trace in its Lagrangian formulation. Here $b$ is the coupling appearing in the models eponymous hyperbolic cosine present in its Lagrangian, $cosh(bphi)$. In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for $b ll 1$ and intermediate values of $b$, but as the self-dual point $b=1$ is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff $E_c$ dependence, which disappears according only to a very slow power law in $E_c$. Standard renormalization group strategies -- whether they be numerical or analytic -- also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of $b=1$. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how `quantum mechanical vs `quantum field theoretic the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of $b$ as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase $b > 1$ of the Lagrangian formulation of model may be different from what is naively inferred from its $S$-matrix. In particular, we present an argument that the theory is massless for $b>1$.
We consider the canonical symplectic form for sine-Gordon evaluated explicitly on the solitons of the model. The integral over space in the form, which arises because the canonical argument uses the Lagrangian density, is done explicitly in terms of functions arising in the group doublecrossproduct formulation of the inverse scattering procedure, and we are left with a simple expression given by two boundary terms. The expression is then evaluated explicitly in terms of the changes in the positions and momenta of the solitons, and we find agreement with a result of Babelon and Bernard who have evaluated the form using a different argument, where it is diagonal in terms of `in or `out co-ordinates. Using the result, we also investigate the higher conserved charges within the inverse scattering framework, check that they Poisson commute and evaluate them on the soliton solutions.
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