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arXiv:quant-ph/0504151 · Physical Review Letters

Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture

Dimitri Gioev, Israel Klich

Published Mar 14, 2006• 513 citations
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Abstract

We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling ${L}^{d\ensuremath{-}1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension $d$, $S\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega}){L}^{d\ensuremath{-}1}\mathrm{log}L$ as the size of a subsystem $L\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\partial}\ensuremath{\Gamma}$ is the Fermi surface and $\ensuremath{\partial}\ensuremath{\Omega}$ is the boundary of the region in real space. The expression for the constant $c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega})$ is based on a conjecture due to Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimate on the entropy $S$.

DOI
10.1103/physrevlett.96.100503

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We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling Ld\ensuremath−1{L}^{d\ensuremath{-}1}Ld\ensuremath−1 for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension ddd, S\ensuremath∼c(\ensuremath∂\ensuremathΓ,\ensuremath∂\ensuremathΩ)Ld\ensuremath−1logLS\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega}){L}^{d\ensuremath{-}1}\mathrm{log}LS\ensuremath∼c(\ensuremath∂\ensuremathΓ,\ensuremath∂\ensuremathΩ)Ld\ensuremath−1logL as the size of a subsystem L\ensuremath→\ensuremath∞L\ensuremath{\rightarrow}\ensuremath{\infty}L\ensuremath→\ensuremath∞, where \ensuremath∂\ensuremathΓ\ensuremath{\partial}\ensuremath{\Gamma}\ensuremath∂\ensuremathΓ is the Fermi surface and \ensuremath∂\ensuremathΩ\ensuremath{\partial}\ensuremath{\Omega}\ensuremath∂\ensuremathΩ is the boundary of the region in real space. The expression for the constant c(\ensuremath∂\ensuremathΓ,\ensuremath∂\ensuremathΩ)c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega})c(\ensuremath∂\ensuremathΓ,\ensuremath∂\ensuremathΩ) is based on a conjecture due to Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimate on the entropy SSS.
We study the entanglement entropy of connected bipartitions in free-fermion gases of N particles in arbitrary dimension d . We show that the von Neumann and Rényi entanglement entropies grow asymptotically as N ( d −1)/ d ln N , with a prefactor that is analytically computed using the Widom conjecture both for periodic and open boundary conditions. The logarithmic correction to the power-law behavior is related to the area-law violation in lattice free fermions. These asymptotic large- N behaviors are checked against exact numerical calculations for N -particle systems.
Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size L in d spatial dimensions is S∼L(d-1)logL, a result that should be contrasted with the usual boundary law S∼L(d-1). This term depends only on the geometry of the Fermi surface and on the boundary of the region in question. I give an intuitive account of this anomalous scaling based on a low energy description of the Fermi surface as a collection of one-dimensional gapless modes. Using this picture, I predict a violation of the boundary law in a number of other strongly correlated systems.
Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\ensuremath{\sim}{L}^{d\ensuremath{-}1}logL$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wave function of the $\ensuremath{\nu}=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wave function on various finite-size geometries, we argue that the prefactor of the leading LlogLL\phantom{\rule{0.16em}{0ex}}logLLlogL term is equivalent to that in the analogous free fermion wave function. In contrast to the recent results of Shao et al. [Phys. Rev. Lett. 114, 206402 (2015)], we thus conclude that the ``Widom formula'' holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wave function sign structure and $S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$ entanglement scaling in such highly entangled gapless phases.
We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space L2(Rd)L^2(\R^d)L2(Rd). As already observed by Gioev and Klich, this implies that the bi-partite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.
The scaling of entanglement entropy is computationally studied in several ($1\ensuremath{\le}d\ensuremath{\le}2$)-dimensional free-fermion systems that are connected by one or more point contacts (PCs). For both the kkk-leg Bethe lattice (d=1)(d=1)(d=1) and d=2d=2d=2 rectangular lattices with a subsystem of Ld{L}^{d}Ld sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the O(L)O(L)O(L) entropy is an expression of the subdominant O(L)O(L)O(L) entropy of the bulk entropy-area law. For d=2d=2d=2 (square) lattices connected by mmm PCs, the area law is found to be $S\ensuremath{\sim}a{L}^{d\ensuremath{-}1}+bm\phantom{\rule{0.16em}{0ex}}logL$ and is thus consistent with the anomalous area law for free fermions ($S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$) as $m\ensuremath{\rightarrow}L$. For the Bethe lattice, the relevance of this result to density-matrix renormalization-group schemes for interacting fermions is discussed.