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arXiv:2305.00996 · Physical review. D/Physical review. D.

Detecting the critical point through entanglement in the Schwinger model

Kazuki Ikeda, Dmitri E. Kharzeev, René Meyer, Shuzhe Shi

Published Nov 9, 2023• 18 citations
arXivHTMLPDF

Abstract

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 $m$, coupling constant $g$, 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.

DOI
10.1103/physrevd.108.l091501

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Using quantum simulations on classical hardware, we study the phase diagram of the massive Schwinger model with a \ensuremathθ\ensuremath{\theta}\ensuremathθ term at finite chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ. 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 \ensuremathμ\ensuremath{\mu}\ensuremathμ. At zero density, the Schwinger model possesses a quantum critical point at \ensuremathθ=\ensuremathπ\ensuremath{\theta}=\ensuremath{\pi}\ensuremathθ=\ensuremathπ and m/g\ensuremath≃0.33m/g\ensuremath{\simeq}0.33m/g\ensuremath≃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.
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.
Abstract The Schwinger model, 1D quantum electrodynamics, has CP symmetry at θ = π due to the topological nature of the θ term. At zero temperature, it is known that as the fermion mass increases, the system undergoes a second-order phase transition to the CP broken phase, which belongs to the same universality class as the quantum Ising chain. In this paper, we obtain the phase diagram near the quantum critical point (QCP) in the temperature and fermion mass plane using first-principle Monte Carlo simulations, while avoiding the sign problem by using the lattice formulation of the bosonized Schwinger model. Specifically, we perform a detailed investigation of the correlation function of the electric field near the QCP and find that its asymptotic behavior can be described by the universal scaling function of the quantum Ising chain. This finding indicates the existence of three regions near the QCP, each characterized by a specific asymptotic form of the correlation length, and demonstrates that the CP symmetry is restored at any nonzero temperature, entirely analogous to the quantum Ising chain. The range of the scaling behavior is also examined and found to be particularly wide.
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
Abstract As pointed out by Coleman, physical quantities in the Schwinger model depend on a parameter θ that determines the background electric field. There is a phase transition for θ = π only. We develop a momentum space formalism on a lattice and use it to perform a quantum computation of the critical point of this phase transition on the NISQ device IMB Q Lima. After error mitigation, our results give strong indication of the existence of a critical point at m / e ≃ 0.32, where m is the bare fermion mass and e is the coupling strength, in good agreement with the classical numerical result m / e ≃ 0.3335.
We perform a numerical analysis of the massive Schwinger model in the presence of a background electric field. Using the Density Matrix Renormalization Group approach, we efficiently compute the spectrum of the Schwinger model on a staggered lattice with up to 3000 qubits. As a result, we achieve a precise computation of the critical mass of the massive Schwinger model to five digits using four different “criticality criteria,” observing perfect agreement among them Additionally, we discuss the effect of a four-fermion operator deformation of the Schwinger model and compute the critical mass for various values of the deformation parameter.
We consider the fermionic (logarithmic) negativity between two fermionic modes in the Schwinger model. Recent results pointed out that fermionic systems can exhibit stronger entanglement than bosonic systems, exhibiting a negativity that decays only algebraically. The Schwinger model is described by fermionic excitations at short distances, while its asymptotic spectrum is the one of a bosonic theory. We show that the two-mode negativity detects this confining, fermion-to-boson transition, shifting from an algebraic decay to an exponential decay at distances of the order of the de Broglie wavelength of the first excited state. We derive analytical expressions in the massless Schwinger model and confront them with tensor network simulations. We also perform tensor network simulations in the massive model, which is not solvable analytically, and close to the Ising quantum critical point of the Schwinger model, where we show that the negativity behaves as its bosonic counterpart. Published by the American Physical Society 2024