© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1305.0278

Searching for Topological Degeneracy in the Hubbard Model with Quantum Monte Carlo

Bryan K. Clark

Published May 1, 2013• 1 citation

Abstract

Z2 spin liquids have topological order. One manifestation of this is that a Z2 spin liquid on a torus exhibits a four-fold degeneracy. Recent numerical evidence has argued for the existence of a spin liquid ground state in the Hubbard model on a honeycomb lattice near U approximately 4. The evidence for this claim involves the presence of a gapped state that lacks any identifiable order. This argument relies on being able to distinguish small order from no order which is notoriously difficult. In this paper we demonstrate an approach which uses quantum Monte Carlo to search for one of the key features that positively identify the topological spin liquid: the topological degeneracy. For any finite system, this topological degeneracy is split where the splitting decays exponentially with system size. We search for low lying states in the energy spectrum that could be identified as these topologically degenerate states. We show that, for system sizes up to N=162 sites, there is no evidence for these states being significantly below the first excited state giving evidence against the existence of a topological phase. We discuss the possible options for the Hubbard model on the honeycomb lattice in the absence of such degeneracy.

DOI: 10.48550/arxiv.1305.0278

Related papers

Absence of a Spin Liquid Phase in the Hubbard Model on the Honeycomb Lattice

Sandro Sorella, Yuichi Otsuka, Seiji Yunoki

A spin liquid is a novel quantum state of matter with no conventional order parameter where a finite charge gap exists even though the band theory would predict metallic behavior. Finding a stable spin liquid in two or higher spatial dimensions is one of the most challenging and debated issues in condensed matter physics. Very recently, it has been reported that a model of graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin liquid ground state in a wide region of the phase diagram, between a semi-metal (SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically exact quantum Monte Carlo simulations, we extend the previous study to much larger clusters (containing up to 2592 sites), and find, if any, a very weak evidence of this spin liquid region. Instead, our calculations strongly indicate a direct and continuous quantum phase transition between SM and AFMI.

arXiv:1207.1783•Scientific Reports•Dec 18, 2012•327 citations

Gapped Topological Spin-Orbital Liquid on the Honeycomb Lattice

Masahiko G. Yamada

We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z_{4} topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π-flux variational state. By exploiting full SU(4) symmetry and keeping up to 12 800 SU(4) multiplets, corresponding to more than 1×10^{6} U(1) states, we achieve unprecedented accuracy for two-dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.

arXiv:2601.06549•Physical Review Letters•Sep 10, 2026•0 citations

Quantum Monte Carlo study of the half-filled Hubbard model on the honeycomb lattice

Yuichi Otsuka, Seiji Yunoki, Sandro Sorella

The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.

Journal of Physics Conference Series•Aug 12, 2013•2 citations

Obtaining topological degenerate ground states by the density matrix renormalization group

Yin-Chen He, D. N. Sheng, Yan Chen

We develop the density matrix renormalization group approach to systematically identify the topological order of the quantum spin liquid (QSL) through adiabatically obtaining different topological degenerate sectors of the QSL on an infinite cylinder. As an application, we study the anisotropic kagome Heisenberg model known for hosting a ${Z}_{2}$ QSL, however no numerical simulations have been able to access all four sectors before. We obtain the complete set of four topological degenerate ground states distinguished by the presence or absence of the spinon and vison quasiparticle line, which fully characterizes the topological nature of the quantum phase. We have also studied the kagome Heisenberg model, which has recently attracted a lot of attention. We find two topological sectors accurately and also estimate various properties of the other topological sectors, where the larger correlation length is found indicating the possible proximity to another phase.

arXiv:1309.5669•Physical Review B•Feb 11, 2014•46 citations

Quantum Entanglement and Detection of Topological Order in Numerics

Tarun Grover

"Topological ordered" phases such as gapped quantum spin-liquids and fractional quantum Hall states possess ground state degeneracy on a torus. We show that the topological nature of this degeneracy has interesting consequences for the entanglement structure of the degenerate ground states. This leads to a simple method for detecting topological order, which is ready-made for numerical schemes such as density matrix renormalization group. The method is valid for phases that do or do not possess edge states alike. We demonstrate it by calculating the entanglement spectrum and the topological entanglement entropy for a chiral spin-liquid state on a 18 site system.

arXiv:1112.2215•Dec 9, 2011•3 citations

Absence of topological degeneracy in the Hubbard model on honeycomb lattice

Tao Li

It is shown that the unique sign structure of the ground state of the Hubbard model on honeycomb lattice, which is shown to be insensitive to the trapped Z 2 gauge flux when the system is defined on a torus, may cause the absence of topological degeneracy on this bipartite system. Examples of variational Mott insulating state on the honeycomb lattice are given to illustrate the close relation between the sign structure of the ground state and the (absence of) topological degeneracy.

arXiv:1006.2222•Europhysics Letters (EPL)•Feb 1, 2011•7 citations

Z2 spin liquids have topological order. One manifestation of this is that a Z2 spin liquid on a torus exhibits a four-fold degeneracy. Recent numerical evidence has argued for the existence of a spin liquid ground state in the Hubbard model on a honeycomb lattice near U approximately 4. The evidence for this claim involves the presence of a gapped state that lacks any identifiable order. This argument relies on being able to distinguish small order from no order which is notoriously difficult. In this paper we demonstrate an approach which uses quantum Monte Carlo to search for one of the key features that positively identify the topological spin liquid: the topological degeneracy. For any finite system, this topological degeneracy is split where the splitting decays exponentially with system size. We search for low lying states in the energy spectrum that could be identified as these topologically degenerate states. We show that, for system sizes up to N=162 sites, there is no evidence for these states being significantly below the first excited state giving evidence against the existence of a topological phase. We discuss the possible options for the Hubbard model on the honeycomb lattice in the absence of such degeneracy.

A spin liquid is a novel quantum state of matter with no conventional order parameter where a finite charge gap exists even though the band theory would predict metallic behavior. Finding a stable spin liquid in two or higher spatial dimensions is one of the most challenging and debated issues in condensed matter physics. Very recently, it has been reported that a model of graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin liquid ground state in a wide region of the phase diagram, between a semi-metal (SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically exact quantum Monte Carlo simulations, we extend the previous study to much larger clusters (containing up to 2592 sites), and find, if any, a very weak evidence of this spin liquid region. Instead, our calculations strongly indicate a direct and continuous quantum phase transition between SM and AFMI.

We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z_{4} topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π-flux variational state. By exploiting full SU(4) symmetry and keeping up to 12 800 SU(4) multiplets, corresponding to more than 1×10^{6} U(1) states, we achieve unprecedented accuracy for two-dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.

The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.

We develop the density matrix renormalization group approach to systematically identify the topological order of the quantum spin liquid (QSL) through adiabatically obtaining different topological degenerate sectors of the QSL on an infinite cylinder. As an application, we study the anisotropic kagome Heisenberg model known for hosting a

{Z}_{2}

QSL, however no numerical simulations have been able to access all four sectors before. We obtain the complete set of four topological degenerate ground states distinguished by the presence or absence of the spinon and vison quasiparticle line, which fully characterizes the topological nature of the quantum phase. We have also studied the kagome Heisenberg model, which has recently attracted a lot of attention. We find two topological sectors accurately and also estimate various properties of the other topological sectors, where the larger correlation length is found indicating the possible proximity to another phase.

"Topological ordered" phases such as gapped quantum spin-liquids and fractional quantum Hall states possess ground state degeneracy on a torus. We show that the topological nature of this degeneracy has interesting consequences for the entanglement structure of the degenerate ground states. This leads to a simple method for detecting topological order, which is ready-made for numerical schemes such as density matrix renormalization group. The method is valid for phases that do or do not possess edge states alike. We demonstrate it by calculating the entanglement spectrum and the topological entanglement entropy for a chiral spin-liquid state on a 18 site system.