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arXiv:1605.02477 · Physical review. B./Physical review. B

Entanglement entropies and fermion signs of critical metals

Nikolaos K. Kaplis, Frank Krüger, Jan Zaanen

Published Apr 3, 2017• 6 citations
arXivHTMLPDF

Abstract

The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron systems. Only recently has it been suggested that fermion signs are fundamental for the universal behavior of critical metallic systems and crucially enhance their degree of quantum entanglement. In this work we explore potential connections between emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis is based on a wave-function Ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants. Such wave functions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces, a representation of the fermion sign structure in many-particle configurations space, show fractal behavior up to a length scale $\ensuremath{\xi}$ that diverges at a critical backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on $\ensuremath{\xi}$, the number of fermions and the exponent of the backflow. For the same wave functions we numerically calculate the second R\'enyi entanglement entropy ${S}_{2}$. Our results show a crossover from volume scaling, ${S}_{2}\ensuremath{\sim}{\ensuremath{\ell}}^{\ensuremath{\theta}}$ $(\ensuremath{\theta}=2$ in $d=2$ dimensions), to the characteristic Fermi-liquid behavior ${S}_{2}\ensuremath{\sim}\ensuremath{\ell}ln\ensuremath{\ell}$ on scales larger than $\ensuremath{\xi}$. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.

DOI
10.1103/physrevb.95.155102

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The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron systems. Only recently has it been suggested that fermion signs are fundamental for the universal behavior of critical metallic systems and crucially enhance their degree of quantum entanglement. In this work we explore potential connections between emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis is based on a wave-function Ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants. Such wave functions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces, a representation of the fermion sign structure in many-particle configurations space, show fractal behavior up to a length scale \ensuremathξ\ensuremath{\xi}\ensuremathξ that diverges at a critical backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on \ensuremathξ\ensuremath{\xi}\ensuremathξ, the number of fermions and the exponent of the backflow. For the same wave functions we numerically calculate the second R'enyi entanglement entropy S2{S}_{2}S2​. Our results show a crossover from volume scaling, S2\ensuremath∼\ensuremathℓ\ensuremathθ{S}_{2}\ensuremath{\sim}{\ensuremath{\ell}}^{\ensuremath{\theta}}S2​\ensuremath∼\ensuremathℓ\ensuremathθ (\ensuremathθ=2(\ensuremath{\theta}=2(\ensuremathθ=2 in d=2d=2d=2 dimensions), to the characteristic Fermi-liquid behavior S2\ensuremath∼\ensuremathℓln\ensuremathℓ{S}_{2}\ensuremath{\sim}\ensuremath{\ell}ln\ensuremath{\ell}S2​\ensuremath∼\ensuremathℓln\ensuremathℓ on scales larger than \ensuremathξ\ensuremath{\xi}\ensuremathξ. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.
The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron systems. Only recently, it has been suggested that fermion signs are fundamental for the universal behavior of critical metallic systems and crucially enhance their degree of quantum entanglement. In this work we explore potential connections between emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis is based on a wavefunction ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants. Such wavefunctions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces -- a representation of the fermion sign structure in many-particle configurations space -- show fractal behavior up to a length scale ξ\xiξ that diverges at a critical backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on ξ\xiξ, the number of fermions and the exponent of the backflow. For the same wavefunctions we numerically calculate the second R\'enyi entanglement entropy S2S_2S2​. Our results show a cross-over from volume scaling, S2∼ℓθS_2\sim \ell^\thetaS2​∼ℓθ (θ=2\theta=2θ=2 in d=2d=2d=2 dimensions), to the characteristic Fermi-liquid behavior S2∼ℓln⁡ℓS_2\sim \ell\ln \ellS2​∼ℓlnℓ on scales larger than ξ\xiξ. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.
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.
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the \log{L} log L scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log L L log L scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.
Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy S(2) on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, S(2) can be calculated on relatively large systems (>324 spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with S(2) enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.
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