Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2302.06743 · Physical Review A

Entanglement entropy in ground states of long-range fermionic systems

Debarghya Chakraborty, Nikolaos Angelinos

Published Oct 11, 2024• 3 citations
arXivHTMLPDF

Abstract

We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law $\ensuremath{\alpha}$ is varied. We ask if a common ${\ensuremath{\alpha}}_{c}$ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as $\ensuremath{\alpha}$ approaches zero. These general features are expected to persist on turning on interactions.

DOI
10.1103/physreva.110.042408

Related papers

Entanglement transitions and multifractality in monitored free-fermions with random long-range hopping

Bo Fan, Antonio M. García-García

We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent $α$. For $α\lesssim 1$, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size $L$ is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small $α\lesssim 1/2$. For $α> 3/2$, the EE is in the area-law phase, namely, no scaling with $L$, for any monitoring strength. For $1 < α\lesssim 3/2$, we identify an $α$-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.

quant-pharXiv:2607.23871•Jul 28, 2026•0 citations

Entanglement Entropy of Free Fermions on Random Fractal Lattices

Arkadiusz Kosior, Jia Wang

Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability $p$, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.

quant-pharXiv:2607.05611•Jul 8, 2026•0 citations

Logarithmic, fractal and volume-law entanglement in a Kitaev chain with long-range hopping and pairing

Andrea Solfanelli, Stefano Ruffo, Sauro Succi A +1 more

A bstract Thanks to their prominent collective character, long-range interactions promote information spreading and generate forms of entanglement scaling, which cannot be observed in traditional systems with local interactions. In this work, we study the asymptotic behavior of the entanglement entropy for Kitaev chains with long-range hopping and pairing couplings decaying with a power law of the distance. We provide a fully-fledged analytical and numerical characterization of the asymptotic growth of the ground state entanglement in the large subsystem size limit, finding that the truly non-local nature of the model leads to an extremely rich phenomenology. Most significantly, in the strong long-range regime, we discovered that the system ground state may have a logarithmic, fractal, or volume-law entanglement scaling, depending on the value of the chemical potential and on the strength of the power law decay.

arXiv:2301.13231•Journal of High Energy Physics•May 8, 2023•14 citations

Entanglement-limited linear response in fermionic systems

Hadi Cheraghi, Ali G. Moghaddam, Teemu Ojanen

We propose a general connection between entanglement-entropy scaling laws and the linear response functions of particle-conserving fermionic systems in their ground state. Specifically, we show that the response to perturbations coupled to the particle number within a finite region exhibits the same size scaling as the entanglement entropy of that region. We explicitly verify this scaling in free-fermion systems that display area-law, volume-law, and critical forms of entanglement. The resulting entanglement-governed scaling of response functions leads to unexpected physical consequences. For instance, contrary to conventional expectations, the energy absorption rate and particle-number fluctuations in gapped systems scale with the boundary of the perturbed region rather than with its volume. Our work thus establishes a direct link between linear-response properties and many-body entanglement.

quant-pharXiv:2511.19415•Nov 25, 2025•0 citations

Entanglement entropy of free fermions in timelike slices

Bowei Liu, Hao Chen, Biao Lian

We define the entanglement entropy of free fermion quantum states in an arbitrary space-time slice of a discrete set of points and particularly investigate timelike (causal) slices. For one-dimensional lattice free fermions with an energy bandwidth ${E}_{0}$, we calculate the time-direction entanglement entropy ${S}_{A}$ in a time-direction slice of a set of times ${t}_{n}=n\ensuremath{\tau}$ $(1\ensuremath{\le}n\ensuremath{\le}K)$ spanning a time length $t$ on the same site. For zero-temperature ground states, we find that ${S}_{A}$ shows volume law when $\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\tau}}_{0}=2\ensuremath{\pi}/{E}_{0}$; in contrast, ${S}_{A}\ensuremath{\sim}\frac{1}{3}lnt$ when $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$, and ${S}_{A}\ensuremath{\sim}\frac{1}{6}lnt$ when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite-temperature thermal states, the mutual information also saturates when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$. For noneigenstates, volume law in $t$ and signatures of the Lieb-Robinson bound velocity can be observed in ${S}_{A}$. For generic space-time slices with one point per site, the zero-temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.

arXiv:2210.03134•Physical review. B./Physical review. B•Oct 17, 2024•15 citations

Scaling laws of entanglement entropy in fractal dimensions

Zheyu Cheng, Kailin Song, Jian-Wei Liu +5 more

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 $S_X=Θ\left(L_X^{d_{\mathrm b}}\right)$ under a uniform gap and satisfies $L_X^{d_{\mathrm b}}\lesssim S_X\lesssim L_X^{d_{\mathrm b}}\ln L_X$ when the gap closes no faster than algebraically. Here $L_X$ is the subsystem linear size and $d_{\mathrm b}$ the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy $S_X^{\mathrm{mix}}$ with $\max\left\{S_X,z_X\ln2\right\}\leqslant S_X^{\mathrm{mix}}\leqslant S_X+z_X\ln2$, where $z_X$ is the local zero-mode weight; an extensive $z_X$ thus makes $S_X^{\mathrm{mix}}$ follow the bulk dimension.

quant-pharXiv:2610.41761•Nov 2, 2026
We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law \ensuremathα\ensuremath{\alpha}\ensuremathα is varied. We ask if a common \ensuremathαc{\ensuremath{\alpha}}_{c}\ensuremathαc​ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as \ensuremathα\ensuremath{\alpha}\ensuremathα approaches zero. These general features are expected to persist on turning on interactions.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent ααα. For α≲1α\lesssim 1α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
A bstract Thanks to their prominent collective character, long-range interactions promote information spreading and generate forms of entanglement scaling, which cannot be observed in traditional systems with local interactions. In this work, we study the asymptotic behavior of the entanglement entropy for Kitaev chains with long-range hopping and pairing couplings decaying with a power law of the distance. We provide a fully-fledged analytical and numerical characterization of the asymptotic growth of the ground state entanglement in the large subsystem size limit, finding that the truly non-local nature of the model leads to an extremely rich phenomenology. Most significantly, in the strong long-range regime, we discovered that the system ground state may have a logarithmic, fractal, or volume-law entanglement scaling, depending on the value of the chemical potential and on the strength of the power law decay.
We propose a general connection between entanglement-entropy scaling laws and the linear response functions of particle-conserving fermionic systems in their ground state. Specifically, we show that the response to perturbations coupled to the particle number within a finite region exhibits the same size scaling as the entanglement entropy of that region. We explicitly verify this scaling in free-fermion systems that display area-law, volume-law, and critical forms of entanglement. The resulting entanglement-governed scaling of response functions leads to unexpected physical consequences. For instance, contrary to conventional expectations, the energy absorption rate and particle-number fluctuations in gapped systems scale with the boundary of the perturbed region rather than with its volume. Our work thus establishes a direct link between linear-response properties and many-body entanglement.
We define the entanglement entropy of free fermion quantum states in an arbitrary space-time slice of a discrete set of points and particularly investigate timelike (causal) slices. For one-dimensional lattice free fermions with an energy bandwidth E0{E}_{0}E0​, we calculate the time-direction entanglement entropy SA{S}_{A}SA​ in a time-direction slice of a set of times ${t}_{n}=n\ensuremath{\tau}$ $(1\ensuremath{\le}n\ensuremath{\le}K)$ spanning a time length ttt on the same site. For zero-temperature ground states, we find that SA{S}_{A}SA​ shows volume law when $\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\tau}}_{0}=2\ensuremath{\pi}/{E}_{0}$; in contrast, ${S}_{A}\ensuremath{\sim}\frac{1}{3}lnt$ when $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$, and ${S}_{A}\ensuremath{\sim}\frac{1}{6}lnt$ when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite-temperature thermal states, the mutual information also saturates when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$. For noneigenstates, volume law in ttt and signatures of the Lieb-Robinson bound velocity can be observed in SA{S}_{A}SA​. For generic space-time slices with one point per site, the zero-temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.
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.