Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules

TL;DR

This work develops a non-perturbative, lattice-based framework to regularize anomaly-free 1+1D chiral matter with onsite symmetry. By embedding the theory on a 2+1D Chern-Simons bulk and engineering symmetry-preserving gapping terms on a boundary, the authors prove a rigorous, non-perturbative equivalence between ’t Hooft anomaly matching conditions and boundary fully gapping rules, using the 3$_L$-5$_R$-4$_L$-0$_R$ model as a concrete test. They demonstrate a concrete mapping from a continuum chiral theory to a finite, onsite lattice Hamiltonian with intermediate-strength interactions that gap the mirror edge while preserving the chiral edge, and provide both non-perturbative and perturbative arguments for the boundary gapping mechanism. The approach offers a general construction principle for non-perturbative anomaly-free chiral matter from SPT bulk-edge data and clarifies the distinctions and potential advantages over Eichten-Preskill and Chen-Giedt-Poppitz schemes, particularly in the role of onsite symmetry and carefully chosen interaction terms. The results have implications for robust lattice realizations of chiral gauge theories and for understanding anomaly constraints via topological boundary conditions.

Abstract

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, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" In particular, we propose that the 3$_L$-5$_R$-4$_L$-0$_R$ 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 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 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. We show a topological non-perturbative proof of the equivalence between $G$-symmetric 't Hooft anomaly cancellation conditions and $G$-symmetric gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multi-U(1) symmetry. We expect our result holds universally regardless of spatial Hamiltonian or Lagrangian/spacetime path integral formulation. Numerical tests are demanding tasks but highly desirable for future work.

Figures (10)

• Figure 1: We construct a UV (ultraviolet high-energy) lattice model in Eq. (\ref{['H3540']}) and (\ref{['eq:H-int-all']}), whose energy scale $\Lambda_{3,\text{UV}} \simeq 1/a$. In contrast, the lattice QCD community usually employs a direct lattice regularization of a continuum field theory at another energy scale $\Lambda_{2,\text{UV}}$. In this work, we do not explore the UV lattice regularization of a field theory model. However, we consider the models marked with the (✓) mark: The UV continuum field theory, including both the fermionic model (Eq. (\ref{['Lf3-5-4-0']}) and (\ref{['eq:fermionize-all']})) and the bosonic model (Eq. (\ref{['b3540']}) and (\ref{['eq:bosonize-all']})) at another energy scale $\Lambda_{1,\text{UV}}$. The UV continuum field theory does not have to be renormalizable in the renormalization group (RG) sense; however, we provide a deeper UV completion of this UV continuum field theory by the UV Hamiltonian model at $\Lambda_{3,\text{UV}}$. In this work, we set the $\Lambda_{3,\text{UV}} \gtrsim \Lambda_{2,\text{UV}} \gtrsim\Lambda_{1,\text{UV}}$. Since the energy scale $\Lambda_{3,\text{UV}} \gtrsim \Lambda_{2,\text{UV}} \gtrsim\Lambda_{1,\text{UV}}$ is set about the same, the RG flow analysis can be controlled along the way. This includes the controlled RG flow $\dasharrow$. Here we do not study anything along with the flows of two dotted arrows ($\cdots$), since we do not attempt from the UV lattice field theory model (which is a conventional model of the lattice QCD community). We can also analyze along with the two dashed-dotted arrows (-.-.-.): We find that the RG flows to a completely gapped phase for the mirror sector, which is known in QFT and condensed matter literature.h95Kapustin:2010hkWang:2012amLevin:2013gaa The boldface $\longleftrightarrow$ arrow is based on the standard bosonization/fermionization method in 1+1D.
• Figure 2: 3$_L$-5$_R$-4$_L$-0$_R$ chiral fermion model: (a) The fermions carry U(1) charge $3$,$5$,$4$,$0$ for $\psi_{L,3},$$\psi_{R,5},$$\psi_{L,4},$$\psi_{R,0}$ on the edge A, and also for its mirror partners $\tilde{\psi}_{R,3},$$\tilde{\psi}_{L,5},$$\tilde{\psi}_{R,4},$$\tilde{\psi}_{L,0}$ on the edge B. We focus on the model with a periodic boundary condition along $x$, and a finite-size length along $y$, effectively as, (b) on a spatial cylinder. (c) The ladder model on a cylinder with the $t$ hopping term along black links, the $t'$ hopping term along brown links. The shadow on the edge B indicates the gapping terms with $G_1,G_2$ couplings in Eq.(\ref{['H3540']}) are imposed.
• Figure 3: Chiral $\pi$-flux square lattice: (a) A unit cell is indicated as the shaded darker region, containing two sublattice as a black dot $a$ and a white dot $b$. The lattice Hamiltonian has hopping constants, $t_1 e^{\mathrm{i}\pi/4}$ along the black arrow direction, $t_2$ along dashed brown links, $-t_2$ along dotted brown links. (b) Put the lattice on a cylinder. (c) The ladder: the lattice on a cylinder with a square lattice width. The chirality of edge state is along the direction of blue arrows.
• Figure 4: Two nearly-flat energy bands $\mathop{\mathrm{E}}_{\pm}$ in Brillouin zone for the kinetic hopping terms of our model Eq.(\ref{['H3540']}).
• Figure 5: The energy spectrum $\mathop{\mathrm{E}}(k_x)$ and the density matrix $\langle f^\dagger f \rangle$ of the chiral $\pi$-flux model on a cylinder: (a) On a 10-sites width ($9a_y$-width) cylinder: The blue curves are edge states spectrum. The black curves are for states extending in the bulk. The chemical potential at zero energy fills eigenstates in solid curves, and leaves eigenstates in dashed curves unfilled. (b) On the ladder, a 2-sites width ($1a_y$-width) cylinder: the same as the (a)'s convention. (c) The density $\langle f^\dagger f \rangle$ of the edge eigenstates (the solid blue curve in (b)) on the ladder lattice. The dotted blue curve shows the total density sums to 1, the darker purple curve shows $\langle f_{\mathop{\mathrm{A}}}^\dagger f_{\mathop{\mathrm{A}}} \rangle$ on the left edge A, and the lighter purple curve shows $\langle f_{\mathop{\mathrm{B}}}^\dagger f_{\mathop{\mathrm{B}}} \rangle$ on the right edge B. The dotted darker (or lighter) purple curve shows density $\langle f_{\mathop{\mathrm{A}},a}^\dagger f_{\mathop{\mathrm{A}},a} \rangle$ (or $\langle f_{\mathop{\mathrm{B}},a}^\dagger f_{\mathop{\mathrm{B}},a} \rangle$) on sublattice $a$, while the dashed darker (or lighter) purple curve shows density $\langle f_{\mathop{\mathrm{A}},b}^\dagger f_{\mathop{\mathrm{A}},b} \rangle$ (or $\langle f_{\mathop{\mathrm{B}},b}^\dagger f_{\mathop{\mathrm{B}},b} \rangle$) on sublattice $b$. This edge eigenstate has the left edge A density with majority quantum number $k_x<0$, and has the right edge B density with majority quantum number $k_x>0$. Densities on two sublattice $a,b$ are equally distributed as we desire. Note: Here we do not use the domain wall fermion approach with a large extra dimension, and we do not require a 1D domain wall in an infinite large 2D lattice system. We cannot over-emphasize that our 1D spatial lattice model (effectively 1D ladder, or a 2D cylinder with a finite width along $y$, here we focus on the quadratic free part of Hamiltonian in Eq. (\ref{['H3540']})) with a finite Hilbert space can already effectively simulate the relativistic 1+1D Weyl fermion doubling theory at low energy.