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 $k$ -leg Bethe lattice $(d=1)$ and $d=2$ rectangular lattices with a subsystem of ${L}^{d}$ sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$ entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by $m$ 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.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 $k$-leg Bethe lattice $(d=1)$ and $d=2$ rectangular lattices with a subsystem of ${L}^{d}$ sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$ entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by $m$ 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.

arXiv:1402.5437Physical Review AMay 8, 20144 citations

Calculation of Entanglement Entropy in Fermion Lattices

Michelle Storms, Rajiv R. P. Singh

Bulletin of the American Physical SocietyOct 4, 20130 citations

Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

Alexander Elgart, Leonid Andreevich Pastur, Mariya V Shcherbina

arXiv:1601.00294Journal of Statistical PhysicsNov 7, 201629 citations

Explicit Hamiltonians inducing volume law for entanglement entropy in fermionic lattices

Giacomo Gori, Simone Paganelli, Auditya Sharma +2 more

We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets $A$ and $\overline{A}$ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on $A$ and $\overline{A}$ . We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy ${S}_{A}$ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where ${S}_{A}(L)\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ being intermediate between $\ensuremath{\beta}=0$ (area law) and $\ensuremath{\beta}=1$ (BP state inducing volume law). For this model, the dispersion relation has a ``zigzag'' structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\ensuremath{\beta}$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets $A$ and $\overline{A}$ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on $A$ and $\overline{A}$. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy ${S}_{A}$ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where ${S}_{A}(L)\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ being intermediate between $\ensuremath{\beta}=0$ (area law) and $\ensuremath{\beta}=1$ (BP state inducing volume law). For this model, the dispersion relation has a ``zigzag'' structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\ensuremath{\beta}$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.

arXiv:1405.3616Physical Review BJun 17, 201548 citations

Entanglement entropy of a massive fermion on a torus

Christopher P. Herzog, Tatsuma Nishioka

arXiv:1301.0336Journal of High Energy PhysicsMar 1, 201372 citations

Area Law Scaling for the Entropy of Disordered Quasifree Fermions

Leonid Andreevich Pastur, Victor V. Slavin

We study theoretically and numerically the entanglement entropy of the $d$ -dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩$ of the $d$ dimension cube $\mathrm{\ensuremath{\Lambda}}$ of side length $l$ admits the area law scaling $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩\ensuremath{\sim}{l}^{(d\ensuremath{-}1)},l\ensuremath{\gg}1$, even in the gapless case, thereby manifesting the area law in the mean for our model. For $d=1$ and $l\ensuremath{\gg}1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if $l\ensuremath{\gg}1$.We study theoretically and numerically the entanglement entropy of the $d$-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩$ of the $d$ dimension cube $\mathrm{\ensuremath{\Lambda}}$ of side length $l$ admits the area law scaling $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩\ensuremath{\sim}{l}^{(d\ensuremath{-}1)},l\ensuremath{\gg}1$, even in the gapless case, thereby manifesting the area law in the mean for our model. For $d=1$ and $l\ensuremath{\gg}1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if $l\ensuremath{\gg}1$.

arXiv:1408.2570Physical Review LettersOct 9, 201454 citations

k

-leg Bethe lattice

(d=1)

and

d=2

rectangular lattices with a subsystem of

{L}^{d}

sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the

O(L)

entropy is an expression of the subdominant

O(L)

entropy of the bulk entropy-area law. For

d=2

(square) lattices connected by

m

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.

A

and

\overline{A}

is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on

A

and

\overline{A}

. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy

{S}_{A}

between the area and the volume law, building an explicit example of a one-dimensional free fermion model where ${S}_{A}(L)\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ being intermediate between $\ensuremath{\beta}=0$ (area law) and $\ensuremath{\beta}=1$ (BP state inducing volume law). For this model, the dispersion relation has a ``zigzag'' structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\ensuremath{\beta}$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.

We study theoretically and numerically the entanglement entropy of the

d

-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩$ of the

d

dimension cube $\mathrm{\ensuremath{\Lambda}}$ of side length

l

admits the area law scaling $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩\ensuremath{\sim}{l}^{(d\ensuremath{-}1)},l\ensuremath{\gg}1$, even in the gapless case, thereby manifesting the area law in the mean for our model. For

d=1

and $l\ensuremath{\gg}1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if $l\ensuremath{\gg}1$.