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arXiv:1912.10045 · Physical Review A

Scaling functions for eigenstate entanglement crossovers in harmonic lattices

Thomas Barthel, Qiang Miao

Published Aug 16, 2021• 6 citations
arXivHTMLPDF

Abstract

For quantum matter, eigenstate entanglement entropies obey an area law or log-area law at low energies and small subsystem sizes and cross over to volume laws for high energies and large subsystems. This transition is captured by crossover functions, which assume a universal scaling form in quantum critical regimes. We demonstrate this for the harmonic lattice model, which describes quantized lattice vibrations and is a regularization for free scalar field theories, modeling, e.g., spin-0 bosonic particles. In one dimension, the ground-state entanglement obeys a log-area law. For dimensions $d\ensuremath{\ge}2$, it displays area laws, even at criticality. The distribution of excited-state entanglement entropies is found to be sharply peaked around subsystem entropies of corresponding thermodynamic ensembles in accordance with the eigenstate thermalization hypothesis. Numerically, we determine crossover scaling functions for the quantum critical regime of the model and do a large-deviation analysis. We show how infrared singularities of the system can be handled and how to access the thermodynamic limit using a perturbative trick for the covariance matrix. Eigenstates for quasi-free bosonic systems are not Gaussian. We resolve this problem by considering appropriate squeezed states instead. For these, entanglement entropies can be evaluated efficiently.

DOI
10.1103/physreva.104.022414

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For quantum matter, eigenstate entanglement entropies obey an area law or log-area law at low energies and small subsystem sizes and cross over to volume laws for high energies and large subsystems. This transition is captured by crossover functions, which assume a universal scaling form in quantum critical regimes. We demonstrate this for the harmonic lattice model, which describes quantized lattice vibrations and is a regularization for free scalar field theories, modeling, e.g., spin-0 bosonic particles. In one dimension, the ground-state entanglement obeys a log-area law. For dimensions d\ensuremath≥2d\ensuremath{\ge}2d\ensuremath≥2, it displays area laws, even at criticality. The distribution of excited-state entanglement entropies is found to be sharply peaked around subsystem entropies of corresponding thermodynamic ensembles in accordance with the eigenstate thermalization hypothesis. Numerically, we determine crossover scaling functions for the quantum critical regime of the model and do a large-deviation analysis. We show how infrared singularities of the system can be handled and how to access the thermodynamic limit using a perturbative trick for the covariance matrix. Eigenstates for quasi-free bosonic systems are not Gaussian. We resolve this problem by considering appropriate squeezed states instead. For these, entanglement entropies can be evaluated efficiently.
For the typical quantum many-body systems that obey the eigenstate thermalization hypothesis (ETH), we argue that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function. The ETH implies that the crossover functions can be deduced from subsystem entropies of thermal ensembles and have universal properties. These functions capture the full crossover from the ground-state entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For critical one-dimensional systems, a universal scaling function follows from conformal field theory and can be adapted for nonlinear dispersions. We use it to also deduce the crossover scaling function for Fermi liquids in d>1 dimensions. The analytical results are complemented by numerics for large noninteracting systems of fermions in d≤3 dimensions and have also been confirmed for bosonic systems and nonintegrable spin chains.
With increasing subsystem size and energy, bipartite entanglement entropies of energy eigenstates cross over from the groundstate scaling to a volume law. In previous work, we pointed out that, when strong or weak eigenstate thermalization (ETH) applies, the entanglement entropies of all or, respectively, almost all eigenstates follow a single crossover function. The crossover functions are determined by the subsystem entropy of thermal states and assume universal scaling forms in quantum-critical regimes. This was demonstrated by field-theoretical arguments and the analysis of large systems of non-interacting fermions and bosons. Here, we substantiate such scaling properties for integrable and non-integrable interacting spin-1/2 chains at criticality using exact diagonalization. In particular, we analyze XXZ and transverse-field Ising models with and without next-nearest-neighbor interactions. Indeed, the crossover of thermal subsystem entropies can be described by a universal scaling function following from conformal field theory. Furthermore, we analyze the validity of ETH for entanglement in these models. Even for the relatively small system sizes that can be simulated, the distributions of eigenstate entanglement entropies are sharply peaked around the subsystem entropies of the corresponding thermal ensembles.
A bstract The entanglement entropy of a free scalar field in its ground state is dominated by an area law term. It is noteworthy, however, that the study of entanglement in scalar field theory has not advanced far beyond the ground state. In this paper, we extend the study of entanglement of harmonic systems, which include free scalar field theory as a continuum limit, to the case of the most general Gaussian states, namely the squeezed states. We find the eigenstates and the spectrum of the reduced density matrix and we calculate the entanglement entropy. We show that our method is equivalent to the correlation matrix method. Finally, we apply our method to free scalar field theory in 1+1 dimensions and show that, for very squeezed states, the entanglement entropy is dominated by a volume term, unlike the ground-state case. Even though the state of the system is time-dependent in a non-trivial manner, this volume term is time-independent. We expect this behaviour to hold in higher dimensions as well, as it emerges in a large-squeezing expansion of the entanglement entropy for a general harmonic system.
Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit); i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.
The scaling of entanglement entropy with subsystem size fails to distinguish between the gapped and the gapless ground states of a scalar field theory in d>1d>1d>1 dimensions. We show that the scaling of the angular momentum resolved entanglement entropy ${S}_{\ensuremath{\ell}}$ with the subsystem radius RRR can clearly distinguish between these states. For a massless theory with momentum cutoff $\mathrm{\ensuremath{\Lambda}}, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[\mathrm{\ensuremath{\Lambda}}R/\ensuremath{\ell}]$ for $\mathrm{\ensuremath{\Lambda}}R\ensuremath{\gg}\ensuremath{\ell}$, while ${S}_{\ensuremath{\ell}}\ensuremath{\sim}{R}^{0}$ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector ${k}_{F}, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[{k}_{F}R]$ for ${k}_{F}R\ensuremath{\gg}\ensuremath{\ell}$. We show how this leads to an ``area-log'' scaling of total entanglement entropy of fermions, while the extra factor of $\ensuremath{\ell}$ leads to a leading area law even for massless bosons. At finite temperatures, we find that there is a crossover in the scaling of ${S}_{\ensuremath{\ell}}$ from the T=0T=0T=0 logarithmic scaling to a high TTT linear scaling ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\pi}TR/3$. The logarithmic scaling exists for larger subsystem sizes for larger values of $\ensuremath{\ell}$. We provide estimates of temperatures and subsystem sizes where this critical scaling can be seen in experiments on ultracold atoms.
Signature of Criticality in Angular Momentum Resolved Entanglement of Scalar Fields in d>1d>1d>1
The scaling of entanglement entropy with subsystem size fails to distinguish between gapped and gapless ground state of a scalar field theory in d>1d>1d>1 dimensions. We show that the scaling of angular momentum resolved entanglement entropy SℓS_\ellSℓ​ with the subsystem radius RRR can clearly distinguish between these states. For a massless theory with momentum cut-off ΛΛΛ, Sℓ∼ln⁡[ΛR/ℓ]S_\ell \sim \ln [ΛR/\ell]Sℓ​∼ln[ΛR/ℓ] for ΛR≫ℓΛR \gg \ellΛR≫ℓ, while Sℓ∼R0S_\ell \sim R^0Sℓ​∼R0 for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector kFk_FkF​, Sℓ∼ln⁡[kFR]S_\ell \sim \ln [k_F R]Sℓ​∼ln[kF​R] for kFR≫ℓk_F R \gg \ellkF​R≫ℓ. We show how this leads to an ``area-log'' scaling of total entanglement entropy of Fermions, while the extra factor of ℓ\ellℓ leads to a leading area law even for massless Bosons.