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:2601.09563 · Annalen der Physik

Lattice fermion simulation of spontaneous time-reversal symmetry breaking in a helical Luttinger liquid

V. A. Zakharov, J. Sánchez Fernán, C. W. J. Beenakker

cond-mat.str-elquant-phSubmitted Jan 14, 2026• Announced Jan 15, 2026• 1 citation
v1 · Jan 14, 2026v2 · May 11, 2026
arXivHTMLPDF

Abstract

We extend a recently developed "tangent fermion" method to discretize the Hamiltonian of a helical Luttinger liquid on a one-dimensional lattice, including two-particle backscattering processes that may open a gap in the spectrum. The fermion-doubling obstruction of the sine dispersion is avoided by working with a tangent dispersion, preserving the time-reversal symmetry of the Hamiltonian. The numerical results from a tensor network calculation on a finite lattice confirm the expectation from infinite-system analytics, that a gapped phase with spontaneously broken time-reversal symmetry emerges when the Fermi level is tuned to the Dirac point and the Luttinger parameter crosses a critical value.

Journal reference
Annalen Physik 538, e00004 (2026)
DOI
10.1002/andp.202600004
Comments
7 pages, 5 figures

Related papers

Helical Luttinger Liquid on a Space-Time Lattice

V. A. Zakharov, J. Tworzydło, C. W. J. Beenakker +1 more

The Luttinger model is a paradigm for the breakdown due to interactions of the Fermi liquid description of one-dimensional massless Dirac fermions. Attempts to discretize the model on a one-dimensional lattice have failed to reproduce the established bosonization results because of the fermion-doubling obstruction: a local and symmetry-preserving discretization of the Hamiltonian introduces a spurious second species of low-energy excitations, while a nonlocal discretization opens a single-particle gap at the Dirac point. Here, we show how to work around this obstruction by discretizing both space and time to obtain a local Lagrangian for a helical Luttinger liquid with Hubbard interaction. The approach enables quantum Monte Carlo simulations that preserve the topological protection of an unpaired Dirac cone.

arXiv:2401.10828•Physical Review Letters•Sep 9, 2024•6 citations

Lattice fermion simulation of spontaneous time-reversal symmetry breaking in a helical Luttinger liquid

Zakharov, Vladimir A., J. Sánchez Fernán, C. W. J. Beenakker

This repository contains Python code based on TeNPy for constructing a finite-size MPO of a lattice-regularized helical Luttinger liquid using tangent fermion method. It includes DMRG simulations and routines to compute and compare fermionic propagators and transverse spin correlators with bosonization predictions.

•Jan 9, 2026•0 citations

Validity of SLAC fermions for the (1+1) -dimensional helical Luttinger liquid

Zhenjiu Wang, Fakher F. Assaad, Maksim Ulybyshev

The Nielson-Ninomiya theorem states that a chirally invariant free fermion lattice action, which is local, translation invariant, and real, necessarily has fermion doubling. The SLAC approach gives up on locality, and long-range hopping leads to a linear dispersion with singularity at the zone boundary. We introduce a SLAC Hamiltonian formulation that is expected to realize a U(1) helical Luttinger liquid in a naive continuum limit. We argue that nonlocality and concomitant singularity at the zone edge have important implications. Large momentum transfers yield spurious features already in the noninteracting case. Upon switching on interactions, nonlocality invalidates the Mermin-Wagner theorem and allows for long-ranged magnetic ordering. In fact, in the strong-coupling limit the model maps onto an XXZ-spin chain with $1/{r}^{2}$ exchange. Here, both spin-wave and DMRG calculations support long-ranged order. While the long-ranged order opens a single-particle gap at the Dirac point, the singularity at the zone boundary persists for any finite value of the interaction strength such that the ground state remains metallic. Hence, the SLAC Hamiltonian does not flow to the 1D helical Luttinger liquid fixed point. Aside from DMRG simulations, we have used auxiliary field quantum Monte Carlo simulations to arrive at the above conclusions.

arXiv:2211.02960•Physical review. B./Physical review. B•Jul 5, 2023•10 citations

Luttinger liquid tensor network: Sine versus tangent dispersion of massless Dirac fermions

V. A. Zakharov, Stefano Polla, A. Donís Vela +4 more

To apply the powerful many-body techniques of tensor networks to massless Dirac fermions, one wants to discretize the p · σ Hamiltonian and construct a matrix-product-operator (MPO) representation. We compare two alternative discretization schemes, one with a sine dispersion, the other with a tangent dispersion, applied to a one-dimensional Luttinger liquid with Hubbard interaction. Both types of lattice fermions allow for an exact MPO representation of low bond dimension, so they are efficiently computable, but only the tangent dispersion gives a power law decay of the propagator in agreement with the continuum limit: The sine dispersion is gapped by the interactions, evidenced by an exponentially decaying propagator. Our construction of a tensor network with an unpaired Dirac cone works around the fermion-doubling obstruction by exploiting the fact that the Hamiltonian of tangent fermions permits a generalized eigenproblem. Published by the American Physical Society 2024

arXiv:2407.06713•Physical Review Research•Oct 23, 2024•5 citations

Time-reversal symmetry breaking at the edge states of a three-dimensional topological band insulator

Cenke Xu

We study the time-reversal $(T)$ symmetry breaking of two-dimensional helical Fermi liquid, with application to the edge states of three-dimensional (3D) topological band insulators with only one two-component Dirac fermion at finite chemical potential, as well as other systems with spin-orbit coupling. The $T$-breaking Ising order parameter is not overdamped and the theory is different from the ordinary Hertz-Millis theory for order parameters at zero momentum. We argue that the $T$-breaking phase transition is a 3D Ising transition, and the quasiparticles are well defined in the quantum critical regime.

arXiv:0908.2147•Physical Review B•Jan 26, 2010•17 citations

Quantum-critical electrodynamics of Luttinger fermions

Santanu Dey, Joseph Maciejko

We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative U(1) quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations and can thus also be viewed as a U(1) spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.

arXiv:2204.05319•Physical review. B./Physical review. B•Jul 20, 2022•3 citations
We extend a recently developed "tangent fermion" method to discretize the Hamiltonian of a helical Luttinger liquid on a one-dimensional lattice, including two-particle backscattering processes that may open a gap in the spectrum. The fermion-doubling obstruction of the sine dispersion is avoided by working with a tangent dispersion, preserving the time-reversal symmetry of the Hamiltonian. The numerical results from a tensor network calculation on a finite lattice confirm the expectation from infinite-system analytics, that a gapped phase with spontaneously broken time-reversal symmetry emerges when the Fermi level is tuned to the Dirac point and the Luttinger parameter crosses a critical value.
The Luttinger model is a paradigm for the breakdown due to interactions of the Fermi liquid description of one-dimensional massless Dirac fermions. Attempts to discretize the model on a one-dimensional lattice have failed to reproduce the established bosonization results because of the fermion-doubling obstruction: a local and symmetry-preserving discretization of the Hamiltonian introduces a spurious second species of low-energy excitations, while a nonlocal discretization opens a single-particle gap at the Dirac point. Here, we show how to work around this obstruction by discretizing both space and time to obtain a local Lagrangian for a helical Luttinger liquid with Hubbard interaction. The approach enables quantum Monte Carlo simulations that preserve the topological protection of an unpaired Dirac cone.
The Nielson-Ninomiya theorem states that a chirally invariant free fermion lattice action, which is local, translation invariant, and real, necessarily has fermion doubling. The SLAC approach gives up on locality, and long-range hopping leads to a linear dispersion with singularity at the zone boundary. We introduce a SLAC Hamiltonian formulation that is expected to realize a U(1) helical Luttinger liquid in a naive continuum limit. We argue that nonlocality and concomitant singularity at the zone edge have important implications. Large momentum transfers yield spurious features already in the noninteracting case. Upon switching on interactions, nonlocality invalidates the Mermin-Wagner theorem and allows for long-ranged magnetic ordering. In fact, in the strong-coupling limit the model maps onto an XXZ-spin chain with 1/r21/{r}^{2}1/r2 exchange. Here, both spin-wave and DMRG calculations support long-ranged order. While the long-ranged order opens a single-particle gap at the Dirac point, the singularity at the zone boundary persists for any finite value of the interaction strength such that the ground state remains metallic. Hence, the SLAC Hamiltonian does not flow to the 1D helical Luttinger liquid fixed point. Aside from DMRG simulations, we have used auxiliary field quantum Monte Carlo simulations to arrive at the above conclusions.
To apply the powerful many-body techniques of tensor networks to massless Dirac fermions, one wants to discretize the p · σ Hamiltonian and construct a matrix-product-operator (MPO) representation. We compare two alternative discretization schemes, one with a sine dispersion, the other with a tangent dispersion, applied to a one-dimensional Luttinger liquid with Hubbard interaction. Both types of lattice fermions allow for an exact MPO representation of low bond dimension, so they are efficiently computable, but only the tangent dispersion gives a power law decay of the propagator in agreement with the continuum limit: The sine dispersion is gapped by the interactions, evidenced by an exponentially decaying propagator. Our construction of a tensor network with an unpaired Dirac cone works around the fermion-doubling obstruction by exploiting the fact that the Hamiltonian of tangent fermions permits a generalized eigenproblem. Published by the American Physical Society 2024
We study the time-reversal (T)(T)(T) symmetry breaking of two-dimensional helical Fermi liquid, with application to the edge states of three-dimensional (3D) topological band insulators with only one two-component Dirac fermion at finite chemical potential, as well as other systems with spin-orbit coupling. The TTT-breaking Ising order parameter is not overdamped and the theory is different from the ordinary Hertz-Millis theory for order parameters at zero momentum. We argue that the TTT-breaking phase transition is a 3D Ising transition, and the quasiparticles are well defined in the quantum critical regime.
We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative U(1) quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations and can thus also be viewed as a U(1) spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.