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arXiv:2211.02960 · Physical review. B./Physical review. B

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

Zhenjiu Wang, Fakher F. Assaad, Maksim Ulybyshev

Published Jul 5, 2023• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.108.045105

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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.
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.
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.
Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling J_K J K at fixed doping x. At large positive J_K J K , we confirm the expected conventional Luttinger liquid phase with 2k_F=\frac{1+x}{2} 2 k F = 1 + x 2 (in units of 2\pi 2 π ), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the J_K ≤ 0 J K ≤ 0 side, our simulation finds the existence of a fractional Luttinger liquid (LL \star ⋆ ) phase with 2k_F=\frac{x}{2} 2 k F = x 2 , accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL \star ⋆ ) phase in higher dimensions. The LL \star ⋆ phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive J_K J K . Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of J_K J K , the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035 J 0.035 J ) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z=+ z = + . The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.
We demonstrate the equivalence between two distinct Luttinger liquid impurity problems. The first concerns a one-dimensional topological superconductor coupled at one end to the ends of two single channel Luttinger liquids. The second concerns a point contact in the quantum spin Hall effect, where four helical Luttinger liquids meet at a point. Both problems have been studied previously and exhibit several stable phases depending on the Luttinger parameter K, that can be characterized in terms of simple conformally invariant boundary conditions describing perfect normal (or Andreev) transmission or reflection. In addition, both problems exhibit critical points that are described by "intermediate" fixed points similar to those found in earlier studies of an impurity in a Luttinger liquid with spin. Though these two models have different symmetries and numbers of modes, we show they are equivalent and are related by a duality transformation, and we show that the non-trivial intermediate critical points are the same. In the non-interacting limit, K=1, the duality involves two distinct free fermion representations that are related by a non-local transformation that derives from the triality of SO(8). Using the explicit translation between the two theories, we translate results from one problem to the other and vice versa. This allows us to make new predictions about the topological superconductor-Luttinger liquid junction, including predictions about the global behavior of the critical conductance G*(K), as well predictions for the critical exponents and universal crossover scaling functions. In this paper we introduce both models from scratch, using a common notation that facilitates their comparison, and we discuss in detail the dualities that relate them, along with their free fermion limits. We close with a discussion of open problems and future directions.
Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.
Understanding interactions in quantum many-body systems remains one of the most profound and difficult challenges in condensed matter physics. While free particle systems can be solved through single-particle techniques, the introduction of interactions turns the problem essentially many-body and quickly intractable. The Hilbert space grows exponentially, and traditional approaches become useless. To deal with this, one typically turns to numerical methods by discretizing the system on a lattice. This approach generally works well for many 1D problems, including those described by Schrödinger equation. However, when it comes to massless Dirac fermions with protected chirality, the situation becomes more complicated. Any naive discretization inevitably runs into the fermion doubling problem, formalized by the Nielsen–Ninomiya theorem, which forbids a straightforward lattice realization of a single chiral mode without either breaking symmetry or introducing unphysical degrees of freedom. My thesis is motivated by this problem: the need to describe 1D interacting systems, and the difficulty of doing so numerically due to these fundamental obstacles. The main focus of this work is to develop and explore lattice-based numerical methods for strongly correlated chiral fermions in one dimension, with the Luttinger liquid as a central case study.