A study of the quantum Sinh-Gordon model with relativistic continuous matrix product states

TL;DR

This paper applies relativistic continuous matrix product states to study the ground states of the quantum Sine-Gordon and Sinh-Gordon models, demonstrating high accuracy in certain regimes and revealing potential limitations at strong coupling.

Contribution

It introduces the use of RCMPS for the quantum SG and ShG models, providing insights into their ground states without UV regulators and highlighting the method's strengths and limitations.

Findings

RCMPS accurately approximates the SG ground state up to the free Fermion point

RCMPS fits the ShG ground state well up to a certain coupling, then deviates

The results suggest possible changes in the ShG ground state structure at strong coupling

Abstract

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 $\beta=\sqrt{4\pi}$ in equal-time quantization convention, or $b=1/\sqrt{2}$ in CFT convention], where the ground energy collapses to $-\infty$, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime $\beta \approx 1$ is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately $b=1/\sqrt{2}$, 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.