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Physical review. B./Physical review. B

Signature of criticality in angular momentum resolved entanglement of scalar fields in d > 1

Mrinal Kanti Sarkar, Saranyo Moitra, Rajdeep Sensarma

Published Aug 13, 2024• 2 citations
Publisher

Abstract

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 $d>1$ dimensions. We show that the scaling of the angular momentum resolved entanglement entropy ${S}_{\ensuremath{\ell}}$ with the subsystem radius $R$ 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 $T=0$ logarithmic scaling to a high $T$ 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.

DOI
10.1103/physrevb.110.075128

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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\ensuremathℓ{S}_{\ensuremath{\ell}}S\ensuremathℓ​ with the subsystem radius RRR can clearly distinguish between these states. For a massless theory with momentum cutoff \ensuremathΛ,S\ensuremathℓ\ensuremath∼ln[\ensuremathΛR/\ensuremathℓ]\mathrm{\ensuremath{\Lambda}}, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[\mathrm{\ensuremath{\Lambda}}R/\ensuremath{\ell}]\ensuremathΛ,S\ensuremathℓ​\ensuremath∼ln[\ensuremathΛR/\ensuremathℓ] for \ensuremathΛR\ensuremath≫\ensuremathℓ\mathrm{\ensuremath{\Lambda}}R\ensuremath{\gg}\ensuremath{\ell}\ensuremathΛR\ensuremath≫\ensuremathℓ, while S\ensuremathℓ\ensuremath∼R0{S}_{\ensuremath{\ell}}\ensuremath{\sim}{R}^{0}S\ensuremathℓ​\ensuremath∼R0 for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector kF,S\ensuremathℓ\ensuremath∼ln[kFR]{k}_{F}, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[{k}_{F}R]kF​,S\ensuremathℓ​\ensuremath∼ln[kF​R] for kFR\ensuremath≫\ensuremathℓ{k}_{F}R\ensuremath{\gg}\ensuremath{\ell}kF​R\ensuremath≫\ensuremathℓ. We show how this leads to an ``area-log'' scaling of total entanglement entropy of fermions, while the extra factor of \ensuremathℓ\ensuremath{\ell}\ensuremathℓ 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\ensuremathℓ{S}_{\ensuremath{\ell}}S\ensuremathℓ​ from the T=0T=0T=0 logarithmic scaling to a high TTT linear scaling S\ensuremathℓ\ensuremath∼\ensuremathπTR/3{S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\pi}TR/3S\ensuremathℓ​\ensuremath∼\ensuremathπTR/3. The logarithmic scaling exists for larger subsystem sizes for larger values of \ensuremathℓ\ensuremath{\ell}\ensuremathℓ. 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.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
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.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to L=96L=96L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation FA=∑kνk(1−νk)F_A=\sum_kν_k(1-ν_k)FA​=∑k​νk​(1−νk​) and a participation-style effective number of entangling modes, MA=exp⁡[−∑kwkln⁡wk]\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]MA​=exp[−∑k​wk​lnwk​], with wk∝νk(1−νk)w_k\proptoν_k(1-ν_k)wk​∝νk​(1−νk​). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M\mathcal{M}M is nearly a deterministic function of FFF: a pooled curve explains 99.7\% of its variance across all rates, so M\mathcal{M}M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2\sim2∼2 smaller than that of the entropy or FFF under strong monitoring. Subsystem-scaling comparisons at L=48L=48L=48--969696 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, FFF remains the natural experimentally motivated companion to the entropy, whereas M\mathcal{M}M is best viewed as a variance-reduced numerical summary of essentially the same information.
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.
We study the scaling behavior of the eigenvalues of correlation matrices, which characterize the entanglement of a subsystem with its complement part of a total pure state. A most distinguishing feature of entanglement entropy is its logarithmic dependence on the subsystem size for the groundstate of one-dimensional critical systems. Despite its robust universal character and relevance to a wide range of topics, a thorough understanding of this result requires sophisticated mathematical physics techniques or conformal field theory. The aim of our work is to shed light on this from the underlying eigenvalue distribution perspective. The central object is the correlation matrix, which takes the form of Toeplitz or block-Toeplitz matrix. We develop a circulant matrix approximation in the large matrix dimension limit, thus allowing for the individual eigenvalues behavior to be analysed analytically. We find that for both free lattice fermions and transverse field Ising chain, eigenvalues in the bulk of the eigenspectrum scales as 1/LA1/L_A1/LA​ with the subsystem size. Together with the extensivity of the entropy function, it explains the robust log⁡2LA\log_2 L_Alog2​LA​ scaling of entanglement at criticality. Perturbing from the entanglement-free limit of the Ising chain, we find a smooth crossover behavior to `non-critical' scaling that is characterized by a very slow logarithmic dependence rendering it seemingly a constant entanglement value expected of non-critical systems.