While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an $SU(2)$ chiral field theory, and provide a semiclassically solvable definition of Abelian $2+1$ chiral topological orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.
While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an SU(2) chiral field theory, and provide a semiclassically solvable definition of Abelian 2+1 chiral topological orders. This leads to an exactly solvable definition of chiral U(1) SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.
A Lattice Chiral Boson Theory in 1+1d
Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a 1+1d chiral field theory as the edge of a 2+1d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the 2+1d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a 1+1d spacetime lattice which carries an anomalous chiral U(1) symmetry with zero chiral central charge. The lattice theory with anomalous chiral U(1) symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.
A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry G in 1+1D spacetime is proposed. More precisely, we ask whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry G defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry G? In particular, we show that the 3L-5R-4L-0R U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.
A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry G in 1+1D spacetime is proposed. More precisely, we ask ``whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing on-site symmetry G defined on a 1D spatial lattice with continuous time, such that its low-energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry G?'' In particular, we propose that the chiral fermion theory with chiral U(1) ${3}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{0}_{R}$ symmetry, with two left-moving fermions of charge 3 and 4, and two right-moving fermions of charge 5 and 0 at IR low energy, can emerge from a 1D UV spatial lattice with a chiral U(1) symmetry, if we include properly designed multi-fermion interactions with intermediate strength (i.e., the dimensionless coupling constant is naturally order 1). In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with on-site symmetry by using a quantum Hamiltonian with continuous time, but without suffering from Nielsen-Ninomiya theorem's fermion doubling, if we include properly designed interactions between matter fields. We propose how to design such interactions by looking for extra anomaly-free symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror decoupling. Since a lattice on-site internal symmetry can be gauged, we thus can further define a nonperturbative regularization of any anomaly-free U(1) chiral gauge theory in 1+1D. As an additional remark, we show a topological nonperturbative proof of the equivalence relation between the G-symmetric 't Hooft anomaly cancellation conditions and the G-symmetry-preserving gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multiple U(1) symmetries. We expect that our result holds universally regardless of spatial Hamiltonian or spacetime Lagrangian/path integral formulation of quantum theory. Numerical tests on our proposal are demanding tasks but highly desirable for future work.
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.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
We introduce the \emph{flavoured lattice Schwinger model}, a (1+1)-dimensional U(1) lattice gauge theory in which the fermion doubling problem is resolved by staggering a Z2 flavour degree of freedom rather than staggering chirality. Unlike the standard approaches, this construction preserves an exact axial U(1) symmetry at finite lattice spacing. We derive the continuum limit, showing that the model reduces to the \emph{two-flavour} massless Schwinger model, with flavours α∈{0,1} sharing one dynamical U(1) gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge QGA whose continuum non-conservation ⟨dQGA/dt⟩=−(2g/π)∫dx⟨E(x)⟩ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the χ flavour exposes a hidden UL(2)×UR(2) chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level-1SU(2) Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped (2+1)D Bernevig--Hughes--Zhang topological insulator and gauging the bulk in a constant background field factorises the boundary theory into \emph{two decoupled} single-flavour Schwinger models, one on each edge, identifying the lattice factor of 2 as one quantum of Schwinger anomaly per edge.