Quantum simulation of the phase transition of the massive Thirring model

TL;DR

This work assesses the feasibility of quantum minimally entangled typical thermal states (QMETTS) to simulate the chiral phase transition and dual topological transition in the massive Thirring model at finite temperature. By mapping the model to a lattice with staggered fermions and using Jordan-Wigner transformations, the authors compare QMETTS results against exact diagonalization for observables such as the chiral condensate $\langle \bar{\psi}\psi\rangle$ and the fermion number $\langle \bar{\psi}\gamma^0\psi\rangle$, across varying temperature and coupling. The study demonstrates that QMETTS can closely reproduce known phase structures, including symmetry breaking at low temperature or weak coupling and integer topological sectors, while systematically analyzing model- and method-related errors (finite-volume, discretization, truncation of $A$, Trotterization, and statistical fluctuations). The results support the potential of QMETTS (and QITE-based approaches) to tackle finite-temperature fermionic systems and topological transitions, offering benchmarks for future quantum-simulator hardware and sign-problem-free quantum methods in related field theories.

Abstract

Recent advancements in quantum computing technology have enabled the study of fermionic systems at finite temperature via quantum simulations. This presents a novel approach to investigating the chiral phase transition in such systems. Among these, the quantum minimally entangled typical thermal states~(QMETTS) algorithm has recently attracted considerable interest. The massive Thirring model, which exhibits a variety of phenomena at low temperatures, includes both a chiral phase transition and a topologically non-trivial ground state. It therefore raises the intriguing question of whether its phase transition can be studied using a quantum simulation approach. In this study, the chiral phase transition of the massive Thirring model and its dual topological phase transition are studied using the QMETTS algorithm. Numerical results are obtained on a classical computer simulating circuit-based quantum computations. The results show that QMETTS is able to accurately reproduce the phase transition and thermodynamic properties of the massive Thirring model.

Figures (10)

• Figure 1: Chiral condensation $\langle \bar{\psi}\psi\rangle$ (the left panels) and fermion number (topological charge) $\langle \bar{\psi}\gamma^0\psi\rangle$ (the right panels) as functions of $T$ and $g^2$ calculated by using exact diagonalization. The panels in the first row correspond to $H_E$, and the ones in the second row correspond to $H_M$, respectively. Chiral symmetry breaks at low temperatures or weak coupling strengths. At low temperatures, the topological charge takes integer values, indicating a topological structure.
• Figure 2: $\langle \bar{\psi}\psi\rangle$ (the left panels) and $\langle \bar{\psi}\gamma^0\psi\rangle$ (the right panels) as functions of $T$ and $g^2$ calculated by using QMETTS. The panels in the first row correspond to $H_E$, and the ones in the second row correspond to $H_M$, respectively. For clarity, statistical uncertainties are omitted , with their analysis deferred to the following section. Comparison with Fig. \ref{['fig:exactdiagonal']} demonstrates that QMETTS generally captures the evolution of both the chiral condensation and topological charge with varying temperature and interaction strength.
• Figure 3: At $a^{-1}\beta=1$ ($T=100\;{\rm MeV}$) and different $g^2$, the average chiral condensate $\langle \bar{\psi}\psi\rangle / N$ and topological charge $\langle \bar{\psi}\gamma _0\psi\rangle / N$ as functions of $1/V$ for lattice sizes $N=4$, $6$, $8$, $10$, and $12$, along with corresponding linear extrapolations. The upper panels are the results of $H_M$, the bottom panels are results of $H_E$, respectively. It can be observed that finite-volume effects are negligible at higher temperatures.
• Figure 4: Same as Fig. \ref{['fig:fv-beta1']} but for $a^{-1}\beta=10$ ($T=10\;{\rm MeV}$). The effect of finite volume is significant at smaller $g^2$, especially for the case of $H_M$ (the upper panels). However, the regime with $1/V \to 0$ can be effectively captured through linear extrapolations.
• Figure 5: .At $T=100\;{\rm MeV}$ and different $g^2$, the average chiral condensate $\langle \bar{\psi}\psi\rangle / N$ and topological charge $\langle \bar{\psi}\gamma _0\psi\rangle / N$ as functions of $a$ for lattice sizes $N=4$, $6$, $8$, $10$, and $12$, along with corresponding linear extrapolations. The upper panels are the results of $H_M$, the bottom panels are results of $H_E$, respectively. The convergence behavior of observables is not purely linear, therefore the linear extrapolation is performed with the data points of $N=8,10,12$.