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arXiv:2312.03172 · Physical review. D/Physical review. D.

Entanglement in massive Schwinger model at finite temperature and density

Sebastian Grieninger, Kazuki Ikeda, Dmitri E. Kharzeev, Ismail Zahed A

Published Jan 23, 2024• 14 citations
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Abstract

We evaluate the entanglement entropy and entropic function of massive two dimensional quantum electrodynamics (Schwinger model) at finite temperature, density, and θ -angle. In the strong coupling regime, the entropic function is dominated by the boson mass for large spatial intervals, and reduces to the conformal field theory result for small spatial intervals. We also discuss the entanglement spectrum at finite temperature and a finite θ -angle. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.016023

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We evaluate the entanglement entropy and entropic function of massive two dimensional quantum electrodynamics (Schwinger model) at finite temperature, density, and θ -angle. In the strong coupling regime, the entropic function is dominated by the boson mass for large spatial intervals, and reduces to the conformal field theory result for small spatial intervals. We also discuss the entanglement spectrum at finite temperature and a finite θ -angle. Published by the American Physical Society 2024
We compute the entanglement entropy and the entanglement spectrum of the vacuum state in the massive Schwinger model at a finite θθθ angle. By computing two-point correlation functions we show that the correlation length diverges in the critical region. We clarify the physical origin of entanglement entropy enhancement at θ=πθ=πθ=π by relating it to the competition between distinct electric-flux vacuum branches. We show that the peak near θ=πθ=πθ=π persists across the range of masses studied and corresponds to the point of maximal competition between distinct vacuum branches with opposite electric-field orientation, where quantum fluctuations due to fermion pair creation are maximized. While this entropy enhancement is generic, a pronounced compression of the low-lying entanglement spectrum at θ=πθ=πθ=π distinguishes the near-critical case (m/g≃0.33m/g\simeq0.33m/g≃0.33). Using the Bisognano--Wichmann (BW) theorem, we construct a lattice BW entanglement Hamiltonian and compare it with the exact modular Hamiltonian obtained from the reduced density matrix. We observe agreement in the infrared sector, indicating that the entanglement Hamiltonian is well approximated by a spatially weighted Hamiltonian. Finally, we discuss possible applications to entanglement in topological insulators and quantum wires. In this work, we used the chirally rotated formulation of the Schwinger model Hamiltonian. We identify an additive vacuum-energy artifact induced by the discretization of the chirally rotated lattice and derive its exact free-lattice form to subtract it in the continuum limit.
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In this thesis, we study the entanglement entropy of a chain of coupled harmonic oscillators, which is used to model a one-dimensional bosonic scalar field. Initially, a model for a massless scalar field is discussed. When calculating the entanglement entropy from the reduced density matrix we run into a problem due to the presence of a zero mode, and we are only able to compute the entanglement entropy of one particular case. These difficulties are avoided when we consider a massive scalar field instead, which we accomplish by adding a mass term to the dispersion relation. We numerically study the entanglement entropy of the massive scalar field model as a function of the number of coordinates and the mass parameter and compare with analytical results. By fitting our numerical data we construct a single cross-over function, which describes the entanglement entropy as a function of both the number of coordinates and the mass parameter. The cross-over function agrees well with the numerical data, especially for small values of the mass parameter. Taking then the massless limit, the entanglement entropy of a massless scalar field can nevertheless be determined and reproduces the prediction of conformal field theory.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.
We find the analytic expression of tr ρ n (L) for a free massive boson field in 1+1 dimensions, where ρ(L) is the reduced density matrix corresponding to an interval of length L. This is given exactly (except for a non-universal factor) in terms of a finite sum of solutions of non-linear differential equations of the Painlevé V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the sine–Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the asymptotic limits of the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two-dimensional entropy functions.
Using quantum simulations on classical hardware, we study the phase diagram of the massive Schwinger model with a $\ensuremath{\theta}$ term at finite chemical potential $\ensuremath{\mu}$. We find that the quantum critical point in the phase diagram of the model can be detected through the entanglement entropy and entanglement spectrum. As a first step, we chart the phase diagram using conventional methods by computing the dependence of the charge and chiral condensates on the fermion mass mmm, coupling constant ggg, and the chemical potential $\ensuremath{\mu}$. At zero density, the Schwinger model possesses a quantum critical point at $\ensuremath{\theta}=\ensuremath{\pi}$ and $m/g\ensuremath{\simeq}0.33$. We find that the position of this quantum critical point depends on the chemical potential. Near this quantum critical point, we observe a sharp maximum in the entanglement entropy. Moreover, we find that the quantum critical point can be located from the entanglement spectrum by detecting the position of the gap closing point.