Entanglement and Subsystem Particle Numbers in Free Fermion Systems

We study the relationship between bipartite entanglement, subsystem particle number and topology in a half-filled free fermion system. It is proposed that the spin-projected particle numbers can distinguish the quantum spin Hall state from other states, and can be used to establish a new topological index for the system. Furthermore, we apply the new topological invariant to a disordered system and show that a topological phase transition occurs when the disorder strength is increased beyond a critical value. It is also shown that the subsystem particle number fluctuation displays behavior very similar to that of the entanglement entropy. This provides a lower-bound estimation for the entanglement entropy, which can be utilized to obtain an estimate of the entanglement entropy experimentally.We study the relationship between bipartite entanglement, subsystem particle number and topology in a half-filled free fermion system. It is proposed that the spin-projected particle numbers can distinguish the quantum spin Hall state from other states, and can be used to establish a new topological index for the system. Furthermore, we apply the new topological invariant to a disordered system and show that a topological phase transition occurs when the disorder strength is increased beyond a critical value. It is also shown that the subsystem particle number fluctuation displays behavior very similar to that of the entanglement entropy. This provides a lower-bound estimation for the entanglement entropy, which can be utilized to obtain an estimate of the entanglement entropy experimentally.

arXiv:1111.0791Journal of Physics Condensed MatterFeb 19, 20143 citations

Bounds on the entanglement entropy by the number entropy in non-interacting fermionic systems

Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Jesko Sirker +1 more

Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.

arXiv:2003.03112SciPost PhysicsJun 3, 202070 citations

Full counting statistics and entanglement in a disordered free fermion system

Michael Bantegui, Joseph A. Burg, Gregory Levine

Bulletin of the American Physical SocietyFeb 28, 20120 citations

Entropic entanglement criteria for Fermion systems

Claudia Zander, Angel Ricardo Plastino, M. Casas +1 more

arXiv:1103.0569The European Physical Journal DJan 1, 201221 citations

Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture

Dimitri Gioev, Israel Klich

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 $S$ .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 $S$.

arXiv:quant-ph/0504151Physical Review LettersMar 14, 2006513 citations

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, $S\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega}){L}^{d\ensuremath{-}1}\mathrm{log}L$ 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

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