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arXiv:1606.03385 · Physical review. D/Physical review. D.

Hamiltonian simulation of the Schwinger model at finite temperature

Boye Buyens, Frank Verstraete, Karel Van Acoleyen

Published Oct 21, 2016• 86 citations
arXivHTMLPDF

Abstract

Using matrix product operators, the Schwinger model is simulated in thermal equilibrium. The variational manifold of gauge-invariant matrix product operators is constructed to represent Gibbs states. As a first application, the chiral condensate in thermal equilibrium is computed, and agreement with earlier studies is found. Furthermore, as a new application, the Schwinger model is probed with a fractional charged static quark-antiquark pair separated infinitely far from each other. A critical temperature beyond which the string tension is exponentially suppressed is found and is in qualitative agreement with analytical studies in the strong coupling limit. Finally, the $CT$ symmetry breaking is investigated, and our results strongly suggest that the symmetry is restored at any nonzero temperature.

DOI
10.1103/physrevd.94.085018

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Using matrix product operators, the Schwinger model is simulated in thermal equilibrium. The variational manifold of gauge-invariant matrix product operators is constructed to represent Gibbs states. As a first application, the chiral condensate in thermal equilibrium is computed, and agreement with earlier studies is found. Furthermore, as a new application, the Schwinger model is probed with a fractional charged static quark-antiquark pair separated infinitely far from each other. A critical temperature beyond which the string tension is exponentially suppressed is found and is in qualitative agreement with analytical studies in the strong coupling limit. Finally, the CTCTCT symmetry breaking is investigated, and our results strongly suggest that the symmetry is restored at any nonzero temperature.
We demonstrate the suitability of tensor network techniques for describing the thermal evolution of lattice gauge theories. As a benchmark case, we have studied the temperature dependence of the chiral condensate in the Schwinger model, using matrix product operators to approximate the thermal equilibrium states for finite system sizes with nonzero lattice spacings. We show how these techniques allow for reliable extrapolations in bond dimension, step width, system size and lattice spacing, and for a systematic estimation and control of all error sources involved in the calculation. The reached values of the lattice spacing are small enough to capture the most challenging region of high temperatures and the final results are consistent with the analytical prediction by Sachs and Wipf over a broad temperature range.
Confinement of quarks due to the strong interaction and the deconfinement at high temperatures and high densities are a basic paradigm for understanding the nuclear matter. Their simulation, however, is very challenging for classical computers due to the sign problem of solving equilibrium states of finite-temperature quantum chromodynamical systems at finite density. In this paper, we propose a variational approach, using the lattice Schwinger model, to simulate the confinement or deconfinement by investigating the string tension. We adopt an ansatz that the string tension can be evaluated without referring to quantum protocols for measuring the entropy in the free energy. Results of numeral simulation show that the string tension decreases both along the increasing of the temperature and the chemical potential, which can be an analog of the phase diagram of QCD. Through numerical simulations on the classical computer, we demonstrate the potential of exploiting near-term quantum computers for investigating the phase diagram of finite-temperature and finite-density nuclear matters.
Tensor network (TN) methods, in particular the matrix product states (MPS) ansatz, have proven to be a useful tool in analyzing the properties of lattice gauge theories. They allow for a very good precision, much better than standard Monte Carlo (MC) techniques for the models that have been studied so far, due to the possibility of reaching much smaller lattice spacings. The real reason for the interest in the TN approach, however, is its ability, shown so far in several condensed matter models, to deal with theories which exhibit the notorious sign problem in MC simulations. This makes it prospective for dealing with the nonzero chemical potential in QCD and other lattice gauge theories, as well as with real-time simulations. In this paper, using matrix product operators, we extend our analysis of the Schwinger model at zero temperature to show the feasibility of this approach also at finite temperature. This is an important step on the way to deal with the sign problem of QCD. We analyze in detail the chiral symmetry breaking in the massless and massive cases and show that the method works very well and gives good control over a broad range of temperatures, essentially from zero to infinite temperature.
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The entanglement asymmetry has emerged in recent years as a practical quantity to study phases of matter. We present the first study of entanglement asymmetry in gauge theories by considering the chiral anomaly of the analytically solvable massless Schwinger model at both zero and finite temperatures. At zero temperature, we find the asymmetry exhibits logarithmic growth with system size. At finite temperature, we show that it is parametrically more sensitive to chiral symmetry-breaking than the corresponding local order parameter: while the chiral condensate decays exponentially, the asymmetry decreases only logarithmically. This establishes the entanglement asymmetry as a promising tool to probe (finite-temperature) phase transitions in gauge theories.
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