Helical Luttinger liquid on a space-time lattice
Abstract: 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 \textit{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.
- An overview of methods to avoid fermion doubling in lattice gauge theory can be found in chapter 4 of David Tong’s lecture notes: https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html.
- We set the branch cut of the logarithm along the negative real axis, so lneikasuperscript𝑒𝑖𝑘𝑎\ln e^{ika}roman_ln italic_e start_POSTSUPERSCRIPT italic_i italic_k italic_a end_POSTSUPERSCRIPT is a sawtooth that jumps at ka=π+2nπ𝑘𝑎𝜋2𝑛𝜋ka=\pi+2n\piitalic_k italic_a = italic_π + 2 italic_n italic_π.
- The continuum limit Cσ(k)=−12sign(σk)subscript𝐶𝜎𝑘12sign𝜎𝑘C_{\sigma}(k)=-\tfrac{1}{2}\operatorname{sign}(\sigma k)italic_C start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_k ) = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_sign ( italic_σ italic_k ) differs from the zero-temperature Fermi function θ(−σk)𝜃𝜎𝑘\theta(-\sigma k)italic_θ ( - italic_σ italic_k ) by a 1/2121/21 / 2 offset. This offset corresponds to a delta function δ(x−x′)𝛿𝑥superscript𝑥′\delta(x-x^{\prime})italic_δ ( italic_x - italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) contribution to the propagator (9), which is lost in the discretization.
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