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:1810.09779 · Physical review. B./Physical review. B

Algebraic localization from power-law couplings in disordered quantum wires

Thomas Botzung, Davide Vodola, Piero Naldesi, Markus Müller, Elisa Ercolessi, Guido Pupillo

Published Oct 22, 2019• 26 citations
arXivHTMLPDF

Abstract

We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $\ensuremath{\ell}$ as a power law $1/{\ensuremath{\ell}}^{\ensuremath{\alpha}}$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $\ensuremath{\alpha}$. The exponent of algebraic decay depends only on $\ensuremath{\alpha}$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular, and solid-state physics.

DOI
10.1103/physrevb.100.155136

Related papers

Study of strongly correlated one-dimensional systems with long-range interactions

Thomas Botzung

During this Ph.D, we studied one-dimensional systems with long-range couplings. In the first part, we demonstrate that power-law couplings lead to an algebraic decay of correlations at long distances in disordered quantum wires. In the second chapter, we analyzed an extended Hubbard model where particles interact via a finite-range potential that induces frustration and new exotic phases. In the third chapter, we demonstrated that restoring energy extensivity has an influence on the low-energy properties of quantum model in the thermodynamic limit. Finally, we provide preliminary results on the modification of Anderson localization due to the coupling to a cavity mode.

•Jan 1, 2020•3 citations

Correlation Decay in Fermionic Lattice Systems with Power-Law Interactions at Nonzero Temperature

Senaida Hernández-Santana, Christian Gogolin, J. I. Cirac +1 more

We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.

arXiv:1702.00371•Physical Review Letters•Sep 13, 2017•25 citations

Many-body localization transition with power-law interactions: Statistics of eigenstates

Konstantin S. Tikhonov, A. D. Mirlin

We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as $1/{r}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ satisfying $d\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2d$, where $d$ is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where $W$ is the disorder strength and $L$ the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder $W$ with the system size $L$ and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.

arXiv:1803.04285•Physical review. B./Physical review. B•Jun 22, 2018•98 citations

Self-consistent theory of many-body localisation in a quantum spin chain with long-range interactions

Sthitadhi Roy, David E. Logan

Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.

arXiv:1903.04851•SciPost Physics•Oct 3, 2019•45 citations

Duality in Power-Law Localization in Disordered One-Dimensional Systems

Xiaolong Deng, V. E. Kravtsov, Georgy V. Shlyapnikov +1 more

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.

arXiv:1706.04088•Physical Review Letters•Mar 16, 2018•83 citations

Self-consistent theory of many-body localisation in a quantum spin chain with long-range interactions

Sthitadhi Roy, David E. Logan

Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.

•Oct 1, 2019•41 citations
We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance \ensuremathℓ\ensuremath{\ell}\ensuremathℓ as a power law 1/\ensuremathℓ\ensuremathα1/{\ensuremath{\ell}}^{\ensuremath{\alpha}}1/\ensuremathℓ\ensuremathα. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents \ensuremathα\ensuremath{\alpha}\ensuremathα. The exponent of algebraic decay depends only on \ensuremathα\ensuremath{\alpha}\ensuremathα, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular, and solid-state physics.
During this Ph.D, we studied one-dimensional systems with long-range couplings. In the first part, we demonstrate that power-law couplings lead to an algebraic decay of correlations at long distances in disordered quantum wires. In the second chapter, we analyzed an extended Hubbard model where particles interact via a finite-range potential that induces frustration and new exotic phases. In the third chapter, we demonstrated that restoring energy extensivity has an influence on the low-energy properties of quantum model in the thermodynamic limit. Finally, we provide preliminary results on the modification of Anderson localization due to the coupling to a cavity mode.
We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.
We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as $1/{r}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ satisfying $d\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2d$, where ddd is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where WWW is the disorder strength and LLL the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder WWW with the system size LLL and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.
Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.
The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.
Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.