Finite volume methods in the quantum sinh-Gordon theory

Abstract One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 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 model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). 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 ≪ 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 Ec dependence, which disappears according only to a very slow power law in Ec. 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 naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.Abstract One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 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 model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). 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 ≪ 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 Ec dependence, which disappears according only to a very slow power law in Ec. 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 naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

arXiv:2007.00154Journal of High Energy PhysicsJan 5, 202122 citations

The lattice quantum Sine-Gordon model

Anatoli G. Izergin, V. E. Korepin

Letters in Mathematical PhysicsMay 1, 198178 citations

Finite temperature expectation values of local fields in the sinh-Gordon model

Lukyanov, S

Sklyanin's method of separation of variables is employed in a calculation of finite temperature expectation values. An essential element of the approach is Baxter's $Q$ -function. We propose its explicit form corresponding to the ground state of the sinh-Gordon theory. With the method of separation of variables we calculate the finite temperature expectation values of the exponential fields to one-loop order of the semi-classical expansion.Sklyanin's method of separation of variables is employed in a calculation of finite temperature expectation values. An essential element of the approach is Baxter's $Q$-function. We propose its explicit form corresponding to the ground state of the sinh-Gordon theory. With the method of separation of variables we calculate the finite temperature expectation values of the exponential fields to one-loop order of the semi-classical expansion.

May 2, 200045 citations

Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model

Andrei Bytsko, Jörg Teschner

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.

arXiv:hep-th/0602093Journal of Physics A Mathematical and GeneralSep 27, 2006167 citations

Structure of the vacuum in the quantum sine-Gordon model

N. M. Bogoliubov, V. E. Korepin, Anatoli G. Izergin

Physics Letters BSep 1, 19858 citations

Sklyanin's method of separation of variables is employed in a calculation of finite temperature expectation values. An essential element of the approach is Baxter's

Q

-function. We propose its explicit form corresponding to the ground state of the sinh-Gordon theory. With the method of separation of variables we calculate the finite temperature expectation values of the exponential fields to one-loop order of the semi-classical expansion.