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arXiv:2308.01964

Signature of Criticality in Angular Momentum Resolved Entanglement of Scalar Fields in $d>1$

Mrinal Kanti Sarkar, Saranyo Moitra, Rajdeep Sensarma

Published Aug 3, 2023• 0 citations
arXivHTMLPDF

Abstract

The scaling of entanglement entropy with subsystem size fails to distinguish between gapped and gapless ground state of a scalar field theory in $d>1$ dimensions. We show that the scaling of angular momentum resolved entanglement entropy $S_\ell$ with the subsystem radius $R$ can clearly distinguish between these states. For a massless theory with momentum cut-off $Λ$, $S_\ell \sim \ln [ΛR/\ell]$ for $ΛR \gg \ell$, while $S_\ell \sim R^0$ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector $k_F$, $S_\ell \sim \ln [k_F R]$ for $k_F R \gg \ell$. We show how this leads to an ``area-log'' scaling of total entanglement entropy of Fermions, while the extra factor of $\ell$ leads to a leading area law even for massless Bosons.

DOI
10.48550/arxiv.2308.01964

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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 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.
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.
Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law SX=Θ(LXdb)S_X=Θ\left(L_X^{d_{\mathrm b}}\right)SX​=Θ(LXdb​​) under a uniform gap and satisfies LXdb≲SX≲LXdbln⁡LXL_X^{d_{\mathrm b}}\lesssim S_X\lesssim L_X^{d_{\mathrm b}}\ln L_XLXdb​​≲SX​≲LXdb​​lnLX​ when the gap closes no faster than algebraically. Here LXL_XLX​ is the subsystem linear size and dbd_{\mathrm b}db​ the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmixS_X^{\mathrm{mix}}SXmix​ with max⁡{SX,zXln⁡2}⩽SXmix⩽SX+zXln⁡2\max\left\{S_X,z_X\ln2\right\}\leqslant S_X^{\mathrm{mix}}\leqslant S_X+z_X\ln2max{SX​,zX​ln2}⩽SXmix​⩽SX​+zX​ln2, where zXz_XzX​ is the local zero-mode weight; an extensive zXz_XzX​ thus makes SXmixS_X^{\mathrm{mix}}SXmix​ follow the bulk dimension.
Abstract We report on the calculation of the symmetry resolved entanglement entropies in two-dimensional many-body systems of free bosons and fermions by dimensional reduction . When the subsystem is translational invariant in a transverse direction, this strategy allows us to reduce the initial two-dimensional problem into decoupled one-dimensional ones in a mixed space-momentum representation. While the idea straightforwardly applies to any dimension d , here we focus on the case d = 2 and derive explicit expressions for two lattice models possessing a U (1) symmetry, i.e., free non-relativistic massless fermions and free complex (massive and massless) bosons. Although our focus is on symmetry resolved entropies, some results for the total entanglement are also new. Our derivation gives a transparent understanding of the well known different behaviours between massless bosons and fermions in d ⩾ 2: massless fermions presents logarithmic violation of the area which instead strictly hold for bosons, even massless. This is true both for the total and the symmetry resolved entropies. Interestingly, we find that the equipartition of entanglement into different symmetry sectors holds also in two dimensions at leading order in subsystem size; we identify for both systems the first term breaking it. All our findings are quantitatively tested against exact numerical calculations in lattice models for both bosons and fermions.
Abstract In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem’s reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher–Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement (‘flux-resolved entanglement’) for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.
A codimension-one critical surface in momentum space can be either a familiar Fermi surface, which separates occupied states from empty ones in the noninteracting fermion case, or a novel Bose surface, where gapless bosonic excitations are anchored. The presence of such surfaces gives rise to logarithmic violation of entanglement entropy area law. When they are convex, we show that the shape of these critical surfaces can be determined by inspecting the leading logarithmic term of real-space entanglement entropy. The fundamental difference between a Fermi surface and a Bose surface is revealed by the fact that the logarithmic terms in entanglement entropies differ by a factor of 2: SlogBose=2SlogFermi{S}_{\mathrm{log}}^{\mathrm{Bose}}=2{S}_{\mathrm{log}}^{\mathrm{Fermi}}SlogBose​=2SlogFermi​, even when they have identical geometry. Our method has remarkable similarity with determining Fermi surface shape using quantum oscillation. We also discuss possible probes of concave critical surfaces in momentum space.