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:2502.04866 · Journal of Physics Condensed Matter

Crossover from Wannier-Stark localization to charge density waves for interacting spinless fermions in one dimension

N. Aucar Boidi, Amnon Aharony A, Ora Entin-wohlman A, K. Hallberg, César R. Proetto

Published Sep 12, 2025• 0 citations
arXivHTMLPDF

Abstract

We study spinless fermions on a finite chain with nearest-neighbor repulsion and in the presence of a Wannier-Stark (WS) linearly varying electric field potential. In the absence of the interaction, the eigenstates are localized for system's sizes larger than the localization length. We present several analytical expressions for the localization length, which in all cases is proportional to the inverse of the electric field. Using the density matrix renormalization group numerical technique, we observe that the ground state exhibits a decrease of the occupation on the chain sites from the 'bulk', with occupation 1, to the vacuum, with occupation 0. The width of this intermediate 'edge' region is also inversely proportional to the electric field, increasing linearly with the strength of the nearest-neighbor repulsion. For strong interactions, the occupations in the intermediate region exhibit a charge density wave. We also present the local density of states for sites in the intermediate 'domain wall' region. For the non-interacting case, the spectrum shows an increasing energy-localized structure as the field is increased, which is a consequence of the uniform energy distribution of the localized states (WS ladder). This structure survives for small interactions, and it smears out in the strongly interacting limit. Experimental variations of the slope of the potential (the electric field) on cold atom chains may test these predictions.

DOI
10.1088/1361-648x/ae0672

Related papers

Coexistence of insulating phases in confined fermionic chains with a Wannier-Stark potential

N. Aucar Boidi, K. Hallberg, Amnon Aharony A +1 more

We study fermions on a finite chain, interacting repulsively when residing on the same and on nearest-neighbor sites, and subjected to a Wannier-Stark linearly varying potential. Using the density matrix renormalization-group numerical technique to solve this generalized extended Hubbard model, the ground state exhibits a staircase of (quasi) plateaus in the average local site density along the chain, decreasing from being doubly filled to empty as the potential increases. These ``plateaus'' represent locked-in commensurate phases of charge density waves together with band and Mott insulators. These phases are separated by incompressible regions with incommensurate fillings. These results differ from the many-body localization proposed for this model earlier. It is suggested that experimental variations of the slope of the potential and the range of the repulsive interactions will produce such a coexistence of phases which have been individually expected theoretically and observed experimentally for uniform systems.

arXiv:2310.00291•Physical review. B./Physical review. B•Jan 12, 2024•2 citations

Stark Many-Body Localization

Maximilian Schulz, C. Hooley, Roderich Moessner +1 more

We consider spinless fermions on a finite one-dimensional lattice, interacting via nearest-neighbor repulsion and subject to a strong electric field. In the noninteracting case, due to Wannier-Stark localization, the single-particle wave functions are exponentially localized even though the model has no quenched disorder. We show that this system remains localized in the presence of interactions and exhibits physics analogous to models of conventional many-body localization (MBL). In particular, the entanglement entropy grows logarithmically with time after a quench, albeit with a slightly different functional form from the MBL case, and the level statistics of the many-body energy spectrum are Poissonian. We moreover predict that a quench experiment starting from a charge-density wave state would show results similar to those of Schreiber et al. [Science 349, 842 (2015)SCIEAS0036-807510.1126/science.aaa7432].

arXiv:1808.01250•Physical Review Letters•Jan 30, 2019•253 citations

Topological order versus many-body localization in periodically modulated spin chains

Takahiro Orito, Yoshihito Kuno, Ikuo Ichinose

In this paper, we study periodically modulated $s=1/2$ spin chain in a linear gradient potential (LP) that is generated by an external magnetic field. In the absence of the LP, the system has topological states that exhibit a magnetization plateau for a uniform external magnetic field. These topological states have a finite integer Chern number and their stability is clarified by an equivalent spinless fermion system derived by a Jordan-Wigner transformation. We show that the LP, which is nothing but a constant electric field in the spinless fermion system, destabilizes the topological states, because it induces localization called Wannier-Stark (WS) localization. We clarify the phase diagram in the presence of the LP and on-site diagonal disorder. To this end, we carefully study edge excitations under the open boundary condition, which are a hallmark of the topological order. We find a very interesting phenomenon indicating existence of a quasiedge modes that take the place of the genuine edge modes in certain parameter regions. This is a precursor of the WS localization realized in topological states. Finally, we investigate many-body localization induced by a sufficiently strong LP or disorder. To this end, we study the energy-level statistics for whole energy levels, and we find unexpected extended-state regimes located in intermediate potential-gradient and weak on-site disorder regimes. We verify this phenomenon by calculating variance of the entanglement entropy. The present system is closely related to quantum Hall state in two dimensions, and therefore our findings can be observed not only in experiments on ultracold atomic gases but also quantum Hall physics.

arXiv:1909.13457•Physical review. B./Physical review. B•Dec 3, 2019•9 citations

Interacting spinless fermions in a diamond chain

Alexandre Lopes, R. G. Dias

We study spinless fermions in a flux threaded AB${}_{2}$ chain taking into account nearest-neighbor Coulomb interactions. The exact diagonalization of the spinless AB${}_{2}$ chain is presented in the limiting cases of infinite or zero nearest-neighbor Coulomb repulsion for any filling. Without interactions, the AB${}_{2}$ chain has a flat band even in the presence of magnetic flux. We show that the respective localized states can be written in the most compact form as standing waves in one or two consecutive plaquettes. We show that this result is easily generalized to other frustrated lattices such as the Lieb lattice. A restricted Hartree-Fock study of the $V/t$ versus filling phase diagram of the AB${}_{2}$ chain has also been carried out. The validity of the mean-field approach is discussed taking into account the exact results in the case of infinite repulsion. The ground-state energy as a function of filling and interaction $V$ is determined using the mean-field approach and exactly for infinite or zero $V$. In the strong coupling limit, two kinds of localized states occur: one-particle localized states due to geometry and two-particle localized states due to interaction and geometry. These localized fermions create open-boundary regions for itinerant carriers. At filling $\ensuremath{\rho}=2/9$ and in order to avoid the existence of itinerant fermions with positive kinetic energy, phase separation occurs between a high-density phase ($\ensuremath{\rho}=2/3$) and a low-density phase ($\ensuremath{\rho}=2/9$) leading to a metal-insulator transition. The ground-state energy reflects such phase separation by becoming linear on filling above $2/9$. We argue that for filling near or larger than 2/9, the spectrum of the t-V AB${}_{2}$ chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with different fillings, boundary conditions, and LL velocities.

arXiv:1110.2224•Physical Review B•Aug 24, 2011•40 citations

Many-body localization of spinless fermions with attractive interactions in one dimension

Sheng-Hsuan Lin, Björn Sbierski, Florian Dorfner +2 more

We study the finite-energy density phase diagram of spinless fermions with attractive interactions in one dimension in the presence of uncorrelated diagonal disorder. Unlike the case of repulsive interactions, a delocalized Luttinger-liquid phase persists at weak disorder in the ground state, which is a well-known result. We revisit the ground-state phase diagram and show that the recently introduced occupation-spectrum discontinuity computed from the eigenspectrum of the one-particle density matrix is noticeably smaller in the Luttinger liquid compared to the localized regions. Moreover, we use the functional renormalization group scheme to study the finite-size dependence of the conductance, which also resolves the existence of the Luttinger liquid and is computationally cheap. Our main results concern the finite-energy density case. Using exact diagonalization and by computing various established measures of the many-body localization-delocalization transition, we argue that the zero-temperature Luttinger liquid smoothly evolves into a finite-energy density ergodic phase without any intermediate phase transition.

arXiv:1707.06759•SciPost Physics•Jan 18, 2018•23 citations

Local density of states of the one-dimensional spinless fermion model

E Jeckelmann

We investigate the local density of states of the one-dimensional half-filled spinless fermion model with nearest-neighbour hopping t > 0 and interaction V in its Luttinger liquid phase -2t < V ≤ 2t. The bulk density of states and the local density of states in open chains are calculated over the full band width ∼4t with an energy resolution ≤0.08t using the dynamical density-matrix renormalization group (DDMRG) method. We also perform DDMRG simulations with a resolution of 0.01t around the Fermi energy to reveal the power-law behaviour D(ϵ) ∼ |ϵ - ϵ(F)|(α) predicted by the Luttinger liquid theory for bulk and boundary density of states. The exponents α are determined using a finite-size scaling analysis of DDMRG data for lattices with up to 3200 sites. The results agree with the exact exponents given by the Luttinger liquid theory combined with the Bethe Ansatz solution. The crossover from boundary to bulk density of states is analysed. We have found that boundary effects can be seen in the local density of states at all energies, even far away from the chain edges.

arXiv:1111.6545•Journal of Physics Condensed Matter•Dec 5, 2012•15 citations
We study spinless fermions on a finite chain with nearest-neighbor repulsion and in the presence of a Wannier-Stark (WS) linearly varying electric field potential. In the absence of the interaction, the eigenstates are localized for system's sizes larger than the localization length. We present several analytical expressions for the localization length, which in all cases is proportional to the inverse of the electric field. Using the density matrix renormalization group numerical technique, we observe that the ground state exhibits a decrease of the occupation on the chain sites from the 'bulk', with occupation 1, to the vacuum, with occupation 0. The width of this intermediate 'edge' region is also inversely proportional to the electric field, increasing linearly with the strength of the nearest-neighbor repulsion. For strong interactions, the occupations in the intermediate region exhibit a charge density wave. We also present the local density of states for sites in the intermediate 'domain wall' region. For the non-interacting case, the spectrum shows an increasing energy-localized structure as the field is increased, which is a consequence of the uniform energy distribution of the localized states (WS ladder). This structure survives for small interactions, and it smears out in the strongly interacting limit. Experimental variations of the slope of the potential (the electric field) on cold atom chains may test these predictions.
We study fermions on a finite chain, interacting repulsively when residing on the same and on nearest-neighbor sites, and subjected to a Wannier-Stark linearly varying potential. Using the density matrix renormalization-group numerical technique to solve this generalized extended Hubbard model, the ground state exhibits a staircase of (quasi) plateaus in the average local site density along the chain, decreasing from being doubly filled to empty as the potential increases. These ``plateaus'' represent locked-in commensurate phases of charge density waves together with band and Mott insulators. These phases are separated by incompressible regions with incommensurate fillings. These results differ from the many-body localization proposed for this model earlier. It is suggested that experimental variations of the slope of the potential and the range of the repulsive interactions will produce such a coexistence of phases which have been individually expected theoretically and observed experimentally for uniform systems.
We consider spinless fermions on a finite one-dimensional lattice, interacting via nearest-neighbor repulsion and subject to a strong electric field. In the noninteracting case, due to Wannier-Stark localization, the single-particle wave functions are exponentially localized even though the model has no quenched disorder. We show that this system remains localized in the presence of interactions and exhibits physics analogous to models of conventional many-body localization (MBL). In particular, the entanglement entropy grows logarithmically with time after a quench, albeit with a slightly different functional form from the MBL case, and the level statistics of the many-body energy spectrum are Poissonian. We moreover predict that a quench experiment starting from a charge-density wave state would show results similar to those of Schreiber et al. [Science 349, 842 (2015)SCIEAS0036-807510.1126/science.aaa7432].
In this paper, we study periodically modulated s=1/2s=1/2s=1/2 spin chain in a linear gradient potential (LP) that is generated by an external magnetic field. In the absence of the LP, the system has topological states that exhibit a magnetization plateau for a uniform external magnetic field. These topological states have a finite integer Chern number and their stability is clarified by an equivalent spinless fermion system derived by a Jordan-Wigner transformation. We show that the LP, which is nothing but a constant electric field in the spinless fermion system, destabilizes the topological states, because it induces localization called Wannier-Stark (WS) localization. We clarify the phase diagram in the presence of the LP and on-site diagonal disorder. To this end, we carefully study edge excitations under the open boundary condition, which are a hallmark of the topological order. We find a very interesting phenomenon indicating existence of a quasiedge modes that take the place of the genuine edge modes in certain parameter regions. This is a precursor of the WS localization realized in topological states. Finally, we investigate many-body localization induced by a sufficiently strong LP or disorder. To this end, we study the energy-level statistics for whole energy levels, and we find unexpected extended-state regimes located in intermediate potential-gradient and weak on-site disorder regimes. We verify this phenomenon by calculating variance of the entanglement entropy. The present system is closely related to quantum Hall state in two dimensions, and therefore our findings can be observed not only in experiments on ultracold atomic gases but also quantum Hall physics.
We study spinless fermions in a flux threaded AB2{}_{2}2​ chain taking into account nearest-neighbor Coulomb interactions. The exact diagonalization of the spinless AB2{}_{2}2​ chain is presented in the limiting cases of infinite or zero nearest-neighbor Coulomb repulsion for any filling. Without interactions, the AB2{}_{2}2​ chain has a flat band even in the presence of magnetic flux. We show that the respective localized states can be written in the most compact form as standing waves in one or two consecutive plaquettes. We show that this result is easily generalized to other frustrated lattices such as the Lieb lattice. A restricted Hartree-Fock study of the V/tV/tV/t versus filling phase diagram of the AB2{}_{2}2​ chain has also been carried out. The validity of the mean-field approach is discussed taking into account the exact results in the case of infinite repulsion. The ground-state energy as a function of filling and interaction VVV is determined using the mean-field approach and exactly for infinite or zero VVV. In the strong coupling limit, two kinds of localized states occur: one-particle localized states due to geometry and two-particle localized states due to interaction and geometry. These localized fermions create open-boundary regions for itinerant carriers. At filling $\ensuremath{\rho}=2/9$ and in order to avoid the existence of itinerant fermions with positive kinetic energy, phase separation occurs between a high-density phase ($\ensuremath{\rho}=2/3$) and a low-density phase ($\ensuremath{\rho}=2/9$) leading to a metal-insulator transition. The ground-state energy reflects such phase separation by becoming linear on filling above 2/92/92/9. We argue that for filling near or larger than 2/9, the spectrum of the t-V AB2{}_{2}2​ chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with different fillings, boundary conditions, and LL velocities.
We study the finite-energy density phase diagram of spinless fermions with attractive interactions in one dimension in the presence of uncorrelated diagonal disorder. Unlike the case of repulsive interactions, a delocalized Luttinger-liquid phase persists at weak disorder in the ground state, which is a well-known result. We revisit the ground-state phase diagram and show that the recently introduced occupation-spectrum discontinuity computed from the eigenspectrum of the one-particle density matrix is noticeably smaller in the Luttinger liquid compared to the localized regions. Moreover, we use the functional renormalization group scheme to study the finite-size dependence of the conductance, which also resolves the existence of the Luttinger liquid and is computationally cheap. Our main results concern the finite-energy density case. Using exact diagonalization and by computing various established measures of the many-body localization-delocalization transition, we argue that the zero-temperature Luttinger liquid smoothly evolves into a finite-energy density ergodic phase without any intermediate phase transition.
We investigate the local density of states of the one-dimensional half-filled spinless fermion model with nearest-neighbour hopping t > 0 and interaction V in its Luttinger liquid phase -2t < V ≤ 2t. The bulk density of states and the local density of states in open chains are calculated over the full band width ∼4t with an energy resolution ≤0.08t using the dynamical density-matrix renormalization group (DDMRG) method. We also perform DDMRG simulations with a resolution of 0.01t around the Fermi energy to reveal the power-law behaviour D(ϵ) ∼ |ϵ - ϵ(F)|(α) predicted by the Luttinger liquid theory for bulk and boundary density of states. The exponents α are determined using a finite-size scaling analysis of DDMRG data for lattices with up to 3200 sites. The results agree with the exact exponents given by the Luttinger liquid theory combined with the Bethe Ansatz solution. The crossover from boundary to bulk density of states is analysed. We have found that boundary effects can be seen in the local density of states at all energies, even far away from the chain edges.