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

TL;DR

This work develops a 1D lattice formulation of a helical Luttinger liquid using tangent fermions to preserve time-reversal symmetry and avoid fermion doubling, enabling tensor-network simulations of interacting edge states without invoking a 2D bulk. By discretizing the momentum with a tangent dispersion and mapping to a local generalized eigenproblem, the authors include forward and two-particle backscattering, implemented via a fixed-bond-dimension MPO/DMRG framework. For forward scattering, the lattice results reproduce the bosonization-predicted power-law decays with a Luttinger parameter $K$ determined by forward-scattering couplings, validating the method. When backscattering is present, the numerics show a finite gap and spontaneous time-reversal symmetry breaking near half-filling for $K<1/2$, with degenerate ground states exchanged by $\mathcal{T}$ and saturating mass correlators, demonstrating nonperturbative interaction effects in a lattice realization of the helical liquid.

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.

Figures (5)

• Figure 1: Absolute value of the propagator $\bar{C}_\uparrow$ and transverse spin correlator $\bar{R}_x$, defined in Eq. \ref{['correlators']}, calculated in the tensor network of $L= 60$ sites (MPS bond dimension $\chi= 8192$). The data points are computed from the tangent discretization of the Luttinger Hamiltonian, for free fermions and for a repulsive intra-band and inter-band interaction. Only forward scattering is included in this figure ($t_{\rm U1}=0=t_{\rm U2})$. The curves are the analytical results in the continuum from bosonization theory. Panels a) and b) are for a half-filled band. Panel c) shows $\bar{R}_x$ away from half filling. The corresponding data for $\bar{C}_\uparrow$ is indistinguishable from panel a), so it is not included in the figure. For $\kappa_{\rm inter}=0$ the data is independent of $\kappa_{\rm intra}$, hence blue and orange data overlaps.
• Figure 2: Absolute value of the propagator $\bar{C}_\uparrow$ calculated in the tangent fermion tensor network for different $L$ (MPS bond dimension $\chi= 2048$ for $L=20$, $\chi=4096$ for $L=40,60,80$). For all panels, the band is half-filled ($N_\uparrow=N_\downarrow=L/2$). The top row includes only inter-band forward scattering, for three different values of $\kappa_{\rm inter}=0.5,0.7,0.8$, corresponding to the values of the Luttinger parameter $K$ indicated in each panel. The lower row also includes two-particle backscattering ($t_{\rm U2}=0.4\,t_0$). The propagator is scaled by the power law \ref{['Cscaling']}, so that in a gapless liquid curves for different $L$ collapse onto a single curve. This expected power law scaling breaks down in panels e) and f), with a crossover to an exponential decay indicating the opening of an excitation gap by two-particle backscattering for sufficiently strong forward scattering ($K<0.5$).
• Figure 3: Same as Figs. \ref{['fig_gap']}e) and \ref{['fig_gap']}f), but away from half-filling. There is now no indication of a gap opening.
• Figure 4: Same as Figs. \ref{['fig_gap']}d), \ref{['fig_gap']}e), and \ref{['fig_gap']}f), but for $t_{\rm U2}=0$ and nonzero $t_{\rm U1}$, to show that one-particle backscattering alone does not open a gap.
• Figure 5: Top row: Same as Figs. \ref{['fig_gap']}d), \ref{['fig_gap']}e), and \ref{['fig_gap']}f), but for the transverse spin correlator $\bar{R}_y$ (no rescaling with $L$). The saturation of the correlator at a nonzero value in panels b), c) is a signature of spontaneous time-reversal symmetry breaking. The bottom row shows the same for negative $t_{\rm U2}$, when the saturated correlator is $\bar{R}_x$ instead of $\bar{R}_y$.