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.
<!--HTML-->The Sinh-Gordon model is a 1+1 dimensional quantum field theory with a potential cosh(b phi) that is quite peculiar. It is at the same time exactly solvable (for many observables) and not well understood. I will present the results of a variational exploration of its strong coupling regime with a recent generalization of continuous matrix product states. The advantage of this method is that it does not require introducing a cutoff, UV or IR, is fully non-perturbative, typically converges fast for Hamiltonians with polynomial interactions, and gives rigorous energy upper bounds. Its application to the Sinh-Gordon model is only partly successful: observables can be computed accurately up to fairly large values of the coupling, but the ultra strong coupling regime remains difficult to access without extrapolations. As a result, the behavior near the self-dual point is not yet fully settled. I will show the basics of relativistic continuous matrix product states, explain how they typically work for phi^4 like theories or even the easier Sine-Gordon model and finally discuss my attempts at taming Sinh-Gordon.
I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling β=4π in equal-time quantization convention, or b=1/2 in CFT convention], where the ground energy collapses to −∞, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime β≈1 is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately b=1/2, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.
We compare the spectrum of the elliptic Ruijsenaars–Schneider model with the finite-size spectrum of the sine-Gordon model, highlighting both their similarities and differences. Our analysis focuses on the two-particle sector in the center-of-mass frame. At the free point, we carry out an analytic comparison, while at generic couplings we employ non-perturbative numerical calculations based on the truncated Hilbert space method adapted to difference operators. To benchmark this numerical approach, we first study the trigonometric limit, where analytic results are available. We then compare the Bethe–Yang momentum quantization conditions, derived from infinite-volume scattering phases, with the exact finite-volume solutions. We find that these conditions hold exactly in the rational and trigonometric cases, but acquire finite-size corrections in the hyperbolic and elliptic cases, however, these corrections are different in quantum field theories and in many body systems.
A bstract We present efficient algorithms for obtaining the Hamiltonian in Lightcone Conformal Truncation (LCT) for a 2d scalar field with a generic potential. We apply this method to the sine-Gordon and sinh-Gordon models in 1 +1 d , and find precise agreement with integrability results when the scaling dimension ∆ of the deforming cosine/cosinh potential is in the range ∆ ≤ 1. The agreement provides additional evidence for a recent conjecture for how to compute the effective lightcone Hamiltonian in this class of models. In addition, to high precision, we provide the first direct confirmation for the conjectured self-duality of the sinh-Gordon model (∆ < 0), which relates ∆ ↔ 4/∆. As the dimension approaches the upper limit ∆ = 1 from below, we show analytically that the Hamiltonian matrix elements exactly reproduce those of a free Majorana fermion, demonstrating how bosonization is manifested in the LCT basis. We comment on the possible extension of the approach to ∆ > 1.
The repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model (S-matrix, Lagrangian, and operators) can be put in correspondence with those of the former. We use this mapping, together with the thermodynamical Bethe ansatz equations and the exact form factors of the sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperatures. The computation of one-point correlators is thoroughly detailed and when possible compared with known results in the literature.
We propose herein an extension of truncated spectrum methodologies, a nonperturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, H , into two parts, one part, H1 that is “kept” for the numerical computations, and one part, H2 , that is discarded or “truncated.” Even though H2 is discarded, truncated spectrum methodologies will often try to incorporate the effects of H2 in some effective way. In these terms, we propose to keep the dimension of H1 small. We pair this choice of H1 with a Krylov subspace iterative approach able to take into account the effects of H2 . This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the next-to-leading order approach of Elias-Miró [NLO renormalization in the Hamiltonian truncation, ]. To demonstrate this approach, we focus on the ( 1+1d )-dimensional ϕ4 model and compute the bulk energy and mass gaps in both the Z2 -broken and unbroken sectors. We estimate the critical ϕ4 coupling in the broken phase to be gc=0.2645±0.002 . Published by the American Physical Society 2024