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Advanced Quantum Technologies

Topological Metal‐Insulator Phase Transition in A Square Lattice Model

Li‐Shuang Liu, Kai‐Xin Hu, Zhi‐Xu Zhang, Ji Cao, Wen‐Xue Cui, Shou Zhang, Hong‐Fu Wang

Published Jul 9, 2025• 1 citation
Publisher

Abstract

Abstract A square lattice model with next‐nearest‐neighbor hopping and staggered fluxes is investigated, revealing distinct topological characteristics in different phases emerging at specific filling levels. Specifically, at quarter filling, the system exhibits metallic phases with Chern insulator (CI)‐like features and a topological insulator phase with polarization . At half filling, the metallic phases, CI phases, and trivial insulator phases are characterized by , , and , respectively. Notably, the half‐filled metallic phases with CI‐like features exhibit three distinct spatial distributions of boundary states, i.e., type‐I corner states confined to the corners, type‐I edge states localized along the edges but bypassing the corners, and type‐II corner states localized near the corners. The half‐filled CI phases support the existence of type‐II edge states and type‐II corner states. Moreover, a scheme is proposed to achieve coupling with arbitrary phase factors in a topolectrical circuit. The work deepens the understanding of topological states and enriches the study of phase transitions in systems with broken time‐reversal symmetry.

DOI
10.1002/qute.202500381

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Abstract A square lattice model with next‐nearest‐neighbor hopping and staggered fluxes is investigated, revealing distinct topological characteristics in different phases emerging at specific filling levels. Specifically, at quarter filling, the system exhibits metallic phases with Chern insulator (CI)‐like features and a topological insulator phase with polarization . At half filling, the metallic phases, CI phases, and trivial insulator phases are characterized by , , and , respectively. Notably, the half‐filled metallic phases with CI‐like features exhibit three distinct spatial distributions of boundary states, i.e., type‐I corner states confined to the corners, type‐I edge states localized along the edges but bypassing the corners, and type‐II corner states localized near the corners. The half‐filled CI phases support the existence of type‐II edge states and type‐II corner states. Moreover, a scheme is proposed to achieve coupling with arbitrary phase factors in a topolectrical circuit. The work deepens the understanding of topological states and enriches the study of phase transitions in systems with broken time‐reversal symmetry.
We construct a dimerized square-octagon lattice model with the staggered magnetic fluxes threading plaquettes and find two types of topological states at half filling, i.e., the Chern insulator (CI) and the zero-Chern-number topological (ZCNT) states. The CI can be characterized on the basis of the chiral edge states and nonzero Chern number. Different from the CI, this ZCNT state hosts robust edge states with zero Chern number. Its topological nature can be identified by the quantized bulk polarization. More intriguingly, corner states emerge in this ZCNT state, even though its edge state is gapless. In addition, we present a phase diagram by tuning the dimerized hopping potential and the staggered fluxes. Our findings provide a promising approach to search more topological phases without time-reversal symmetry.
We construct an islamic lattice by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-NN hoppings. This model supports abundant quantum phases for various values of filling fractions. At1/4filling, Chern insulator (CI) phases with Chern numbersC=±1, -2and a zero-Chern-number topological insulator (ZCNTI) phase exist. At3/8filling, several CI phases with Chern numbersC=±1, 3and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbersC=±1, 2and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at1/4and3/4fillings. Phase diagrams for filling fractions1/8,1/2,5/8and7/8are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number|C|=1emerge in this model and exactly flat band is found at 3/8 filling. We further investigateν=1/2fractional Chern insulate state when hard-core bosons fill into this TFB model.
Abstract We explore extensively topological quantum phase transitions (TQPTs) of the breathing kagomé lattice model in the presence of staggered fluxes. We obtain rich topological phases, including the Chern insulator (CI) and the second-order topological insulator (SOTI) phases, by tuning the dimerized hopping parameter t 1 ′ and the staggered-flux parameter ϕ . The CI phases can be identified on the basis of the chiral edge states and the non-zero Chern numbers. However, in sharp contrast to the CI phases, the SOTI phases are characterized by the robust corner states and the quantized polarizations. In addition, we explore the TQPTs considering the next-nearest-neighbor hopping parameter t 2 . We demonstrate the existence of two-dimensional SOTIs with broken time-reversal symmetry and reveal the TQPTs between the CIs and the SOTIs.
Models with topological flat bands (TFBs) have been proposed on which the fractional Chern insulators (FCIs) have been studied. To flatten the lowest energy band, some long-range hopping terms are considered in Chern insulator (CI) models. In this paper, we study a tight-binding model on a maple leaf lattice with staggered magnetic fluxes. When the nearest-neighbor hopping term with a suitable phase factor is considered, a TFB with a large flatness ratio emerges, suggesting that staggered magnetic fluxes can give rise to TFBs. By adding the next-nearest-neighbor (NNN) hopping term, there are two types of topological states at one-third filling, i.e., the CI and the higher-order topological insulator (HOTI) states. Simultaneously, we find the CI with a high Chern number can transition to the HOTI state by tuning the NNN hopping parameter. When we consider more distant-neighbor hopping terms, a TFB with Chern number two is found. We further investigate FCI states when hard-core bosons are taken into account in these TFB models.
We theoretically investigate a tight-binding model of fermions hopping on the square-octagon lattice which consists of a square lattice with plaquette corners themselves decorated by squares. Upon the inclusion of second-neighbor spin-orbit coupling or non-Abelian gauge fields, time-reversal symmetric topological Z2{Z}_{2}Z2​ band insulators are realized. Additional insulating and gapless phases are also realized via the non-Abelian gauge fields. Some of the phase transitions involve topological changes to the Fermi surface. The stability of the topological phases to various symmetry-breaking terms is investigated via the entanglement spectrum. Our results enlarge the number of known exactly solvable models of Z2{Z}_{2}Z2​ band insulators and are potentially relevant to the realization and identification of topological phases in both the solid-state and cold atomic gases.
By means of exact diagonalizations, the Bernevig-Hughes-Zhang model at quarter-filling in the limit of strong Hubbard on-site repulsion is investigated. We find that the noninteracting metallic state will be turned into a Chern insulator with saturated magnetization under strong correlations. That is, at such a metal-insulator transition, both the topological and the magnetic properties of the system are changed due to spontaneous breaking of time-reversal symmetry in the ground states. According to our findings, this topological phase transition seems to be of first order. Our results illustrate the interesting physics in topological Mott transitions and provide guidance to the search of more interaction-induced topological phases in similar systems.