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arXiv:1607.05279 · Physical review. B./Physical review. B

Spectrum of conformal gauge theories on a torus

Alex Thomson, Subir Sachdev A

Published May 18, 2017• 13 citations
arXivHTMLPDF

Abstract

Many model quantum spin systems have been proposed to realize critical points or phases described by 2+1 dimensional conformal gauge theories. On a torus of size $L$ and modular parameter $\ensuremath{\tau}$, the energy levels of such gauge theories equal $(1/L)$ times universal functions of $\ensuremath{\tau}$. We compute the universal spectrum of ${\mathrm{QED}}_{3}$, a U(1) gauge theory with ${N}_{f}$ two-component massless Dirac fermions, in the large-${N}_{f}$ limit. We also allow for a Chern-Simons term at level $k$, and show how the topological $k$-fold ground state degeneracy in the absence of fermions transforms into the universal spectrum in the presence of fermions; these computations are performed at fixed ${N}_{f}/k$ in the large-${N}_{f}$ limit.

DOI
10.1103/physrevb.95.205128

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Many model quantum spin systems have been proposed to realize critical points or phases described by 2+1 dimensional conformal gauge theories. On a torus of size LLL and modular parameter \ensuremathτ\ensuremath{\tau}\ensuremathτ, the energy levels of such gauge theories equal (1/L)(1/L)(1/L) times universal functions of \ensuremathτ\ensuremath{\tau}\ensuremathτ. We compute the universal spectrum of QED3{\mathrm{QED}}_{3}QED3​, a U(1) gauge theory with Nf{N}_{f}Nf​ two-component massless Dirac fermions, in the large-Nf{N}_{f}Nf​ limit. We also allow for a Chern-Simons term at level kkk, and show how the topological kkk-fold ground state degeneracy in the absence of fermions transforms into the universal spectrum in the presence of fermions; these computations are performed at fixed Nf/k{N}_{f}/kNf​/k in the large-Nf{N}_{f}Nf​ limit.
Abstract Two dimensional conformal field theories have been extensively studied in the past. When considered on the torus, they are strongly constrained by modular invariance. However, introducing relevant deformations or chemical potentials pushes these theories away from criticality, where many of their aspects are still poorly understood. In this note we make a step towards filling this gap, by analyzing the theory of a Dirac fermion on the torus, deformed by a mass term and a chemical potential for the particle number symmetry. The theory breaks conformal and Lorentz invariance, and we study its spectrum and partition function. We also focus on two limits that are interesting on their own right: a massless relativistic fermion with nonzero chemical potential (a simple model for CFTs at finite density), and nonrelativistic Schrodinger fermions (of relevance in condensed matter systems). Taking inspiration from recent developments in massive modular forms, we obtain a representation of the torus free energy based on Fourier-transforming over a twisted boundary condition. This dual representation fullfills many properties analogous to modular invariance in CFTs. In particular, we use this result to derive Cardy-like formulas for the high energy density of states of these theories.
We establish the universal torus low-energy spectra at the free Dirac fixed point and at the strongly coupled chiral Ising fixed point and their subtle crossover behavior in the Gross-Neuveu-Yukawa field theory with nD=4{n}_{\text{D}}=4nD​=4 component Dirac spinors in D=(2+1)D=(2+1)D=(2+1) dimensions. These fixed points and the field theories are directly relevant for the long-wavelength physics of certain interacting Dirac systems, such as repulsive spinless fermions on the honeycomb lattice or $\ensuremath{\pi}$-flux square lattice. The torus energy spectrum has been shown previously to serve as a characteristic fingerprint of relativistic fixed points and is a powerful tool to discriminate quantum critical behavior in numerical simulations. Here, we use a combination of exact diagonalization and quantum Monte Carlo simulations of strongly interacting fermionic lattice models, to compute the critical torus energy spectrum on finite-size clusters with periodic boundaries and extrapolate them to the thermodynamic limit. Additionally, we compute the torus energy spectrum analytically using the perturbative expansion in $\ensuremath{\epsilon}=4\ensuremath{-}D$, which is in good agreement with the numerical results, thereby validating the presence of the chiral Ising fixed point in the lattice models at hand. We show that the strong interaction between the spinor field and the scalar order-parameter field strongly influences the critical torus energy spectrum and we observe prominent multiplicity features related to an emergent symmetry predicted from the quantum field theory. Building on these results we are able to address the subtle crossover physics of the low-energy spectrum flowing from the chiral Ising fixed point to the Dirac fixed point, and analyze earlier flawed attempts to extract Fermi velocity renormalizations from the low-energy spectrum.
This dissertation is devoted to the study of universal observables in quantum critical systems, most of which have a non-local character. The overarching goal of this work is to compute new universal observables of critical points to aid theoretical understanding of these systems.\nWe first consider the finite-size energy spectrum of quantum critical points on the torus. We compute the spectrum for the Wilson-Fisher conformal field theory in the epsilon=3−d\\epsilon = 3 - depsilon=3−d expansion, where the energy spectrum maps onto a strongly-coupled problem in quantum mechanics. We also compute the energy spectrum to leading order in 1/N1/N1/N, and we compare the two expansions. We then study the torus spectra associated with a class of confinement transitions in states with mathbbZ2\\mathbb{Z}_2mathbbZ2​ topological order. After introducing these universality classes, we show that the critical torus spectrum can be used to detect nontrivial effects like spontaneous symmetry breaking and emergent gauge degrees of freedom. We compare our analytic results with numerical simulations where available, demonstrating the utility of the torus spectrum as a useful characterization of the universality class of a quantum critical point.\nWe then present a computation of the von Neumann entanglement entropy of the Wilson-Fisher and Gross-Neveu conformal field theories in the large NNN limit. We obtain an exact mapping to the von Neumann entanglement entropy of a free quantum field theory, allowing an exact determination of the entanglement entropy in a number of cases. \nWe also study a critical point displaying impurity-driven critical behavior in a boson superfluid-insulator phase transition. The presence of an impurity drives the system to a new interacting universality class, which has critical exponents associated with the scaling dimensions of the impurity degree of freedom. We present the universal quantum field theory of this transition and compute the critical exponents and finite-temperature compressibility at the critical point using the epsilon\\epsilonepsilon expansion.
The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for (2+1)D systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyze the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a Z_{2} topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.
We consider relativistic U(1) gauge theories in 2+12+12+1 dimensions, with Nb{N}_{b}Nb​ species of complex bosons and Nf{N}_{f}Nf​ species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a conformal field theory (CFT). At large Nb{N}_{b}Nb​ and Nf{N}_{f}Nf​, but for arbitrary values of the ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​, we present computations of various critical exponents and universal amplitudes for these CFTs. We make contact with the different spin liquids, charge liquids, and deconfined critical points of quantum magnets that these field theories describe. We compute physical observables that may be measured in experiments or numerical simulations of insulating and doped quantum magnets.
The low-energy spin excitations in the flux phase of the (2+1)-dimensional large-nnn SU(n)\mathrm{SU}(n)SU(n)Heisenberg antiferromagnet are relativistic fermions coupled to a U(1) gauge field. These excitations are studied in the physical n=2n=2n=2 case with the use of a 1n\frac{1}{n}n1​ expansion. A Chern-Simons term is included to remove gauge configurations forbidden in the n=2n=2n=2 theory.