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arXiv:2010.00787 · SciPost Physics

Entanglement gap, corners, and symmetry breaking

Vincenzo Alba

Published Mar 8, 2021• 6 citations
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Abstract

We investigate the finite-size scaling of the lowest entanglement gap \delta\xi δ ξ in the ordered phase of the two-dimensional quantum spherical model (QSM). The entanglement gap decays as \delta\xi=\Omega/\sqrt{L\ln(L)} δ ξ = Ω / L ln ( L ) . This is in contrast with the purely logarithmic behaviour as \delta\xi=\pi^2/\ln(L) δ ξ = π 2 / ln ( L ) at the critical point. The faster decay in the ordered phase reflects the presence of magnetic order. We analytically determine the constant \Omega Ω , which depends on the low-energy part of the model dispersion and on the geometry of the bipartition. In particular, we are able to compute the corner contribution to \Omega Ω , at least for the case of a square corner.

DOI
10.21468/scipostphys.10.3.056

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We investigate the finite-size scaling of the lowest entanglement gap \delta\xi δ ξ in the ordered phase of the two-dimensional quantum spherical model (QSM). The entanglement gap decays as \delta\xi=\Omega/\sqrt{L\ln(L)} δ ξ = Ω / L ln ( L ) . This is in contrast with the purely logarithmic behaviour as \delta\xi=\pi^2/\ln(L) δ ξ = π 2 / ln ( L ) at the critical point. The faster decay in the ordered phase reflects the presence of magnetic order. We analytically determine the constant \Omega Ω , which depends on the low-energy part of the model dispersion and on the geometry of the bipartition. In particular, we are able to compute the corner contribution to \Omega Ω , at least for the case of a square corner.
The study of entanglement spectra is a powerful tool to detect or elucidate universal behavior in quantum many-body systems. We investigate the scaling of the entanglement (or Schmidt) gap δξ , i.e., the lowest-laying gap of the entanglement spectrum, at a two-dimensional quantum critical point. We focus on the paradigmatic quantum spherical model, which exhibits a second-order transition and is mappable to free bosons with an additional external constraint. We analytically show that the Schmidt gap vanishes at the critical point, although only logarithmically. For a system on a torus and the half-system bipartition, the entanglement gap vanishes as π2/ ln(L), with L the linear system size. The entanglement gap is nonzero in the paramagnetic phase and exhibits a faster decay in the ordered phase. The rescaled gap δξ ln(L) exhibits a crossing for different system sizes at the transition, although logarithmic corrections prevent a precise verification of the finite-size scaling. Interestingly, the change of the entanglement gap across the phase diagram is reflected in the zero-mode eigenvector of the spin-spin correlator. At the transition quantum fluctuations give rise to a nontrivial structure of the eigenvector, whereas in the ordered phase it is flat. We also show that the vanishing of the entanglement gap at criticality can be qualitatively but not quantitatively captured by neglecting the structure of the zero-mode eigenvector.
Abstract We investigate the finite-size scaling of the entanglement gap in the one-dimensional long-range quantum spherical model (QSM). We focus on the weak long-range QSM, for which the thermodynamic limit is well-defined. This model exhibits a continuous phase transition, separating a paramagnetic from a ferromagnet phase. The universality class of the transition depends on the long-range exponent α. We show that in the thermodynamic limit the entanglement gap is finite in the paramagnetic phase, and it vanishes in the ferromagnetic phase. In the ferromagnetic phase the entanglement gap is understood in terms of standard magnetic correlation functions. The half-system entanglement gap decays as δ ξ ≃ C α L − ( 1 / 2 − α / 4 ) , where the constant C α depends on the low-energy properties of the model and L is the system size. This reflects that the lower part of the dispersion is affected by the long range physics. Finally, multiplicative logarithmic corrections are absent in the scaling of the entanglement gap, in contrast with the higher-dimensional case.
We study the Rényi entanglement entropy (EE) of the two-dimensional J-Q model, the emblematic quantum spin model of deconfined criticality at the phase transition between antiferromagnetic and valence-bond-solid ground states. State-of-the-art quantum Monte Carlo calculations of the EE reveal critical corner contributions that scale logarithmically with the system size, with a coefficient in remarkable agreement with the form expected from a large-N conformal field theory with SO(N=5) symmetry. However, details of the bipartition of the lattice are crucial in order to observe this behavior. If the subsystem for the reduced density matrix does not properly accommodate valence-bond fluctuations, logarithmic contributions appear even for cornerless bipartitions. We here use a 45° tilted cut on the square lattice. Beyond supporting an SO(5) deconfined quantum critical point, our results for both the regular and tilted cuts demonstrate important microscopic aspects of the EE that are not captured by conformal field theory.
The deconfined quantum critical point (DQCP) in square lattice S=1/2 quantum antiferromagnets has been extensively studied with a large body of evidence pointing to a weakly first-order transition scenario. Recent studies, which focused on entanglement at this nearly continuous DQCP in square lattice J-Q models, have observed conflicting bipartite entanglement entropy (EE) scaling behavior. One bipartition choice gave scaling coefficients in remarkable agreement with predictions from the unitary CFT corresponding to the putative DQCP. While another equally natural choice gave scaling coefficients in complete violation of unitary CFT that may be attributed to lack of scale invariance at the known weakly first-order behavior of the model. This motivates the exploration of DQCP behavior via entanglement measures in lattice models with distinct crystalline symmetries. Here we study a S=1/2 honeycomb model that hosts a nearly continuous transition between Néel and valence-bond-solid ground states relevant to probing DQCP. Using large-scale quantum Monte Carlo simulations, we compute the Rényi EE for a variety of bipartitions and test the CFT based description of the DQCP on the honeycomb lattice. For smooth bipartitions, we find no evidence of logarithmic corrections, in accordance with CFT, thereby essentially ruling out contributions from Goldstone modes. For subsystems with corners, CFT predicts universal logarithmic contributions, which we extract for corners with 60 and 120 degree angles and find close agreement with an emergent SO(5) CFT. While we observe scaling consistent with a critical system in the majority of cases, we also demonstrate an intriguing counterexample of the hexagon subsystem that exhibits a subtle period three oscillation. This results in three separate finite-size series, where the sign of the logarithmic term apparently changes depending on the series.
We perform a quantum Monte Carlo simulation of the resonating valence bond wavefunction on a two-dimensional square lattice with periodic boundary conditions. Using two replicas of the system, we calculate the second Renyi entropy on a spatial bipartition with a square geometry. Through a finite-size scaling analysis, we extract the logarithmic correction to the area law due to the presence of the sharp corners in the entangling surface. We find that the coefficient of this logarithm is positive with a value of 0.073 for a single 90∘90^{\circ}90∘ corner.
In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.