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arXiv:1603.03042 · Physical Review Letters

Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories

Michael Schuler, Seth Whitsitt, Louis-Paul Henry, Subir Sachdev A, Andreas Martin Läuchli

Published Nov 16, 2016• 74 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.117.210401

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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.
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.
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.
The authors study the finite-size energy spectrum of the O(NNN) symmetric Wilson-Fisher conformal field theory (CFT) on the spatial torus. They first perform an analytic calculation of the spectrum using the ϵϵϵ expansion with a careful treatment of the zero-momentum modes. The authors also perform exact diagonalization on several lattice Hamiltonians whose critical points are known to be described by the Wilson-Fisher CFT. Comparison of the analytic and numerical data demonstrates that the energy spectrum is a useful diagnostic for determining the universality class of a critical point. These results have applications in determining the critical behavior of more complicated many-body systems.
Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.
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 $\ensuremath{\tau}$, the energy levels of such gauge theories equal (1/L)(1/L)(1/L) times universal functions of $\ensuremath{\tau}$. 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.
Conformal field theories (CFTs) feature prominently in high-energy physics, statistical mechanics, and condensed matter. For example, CFTs govern emergent universal properties of systems tuned to quantum phase transitions, including their entanglement, correlations, and low-energy excitation spectra. Much of the rich structure predicted by CFTs nevertheless remains unobserved in experiment. Here we directly observe the energy excitation spectra of emergent CFTs at quantum phase transitions -- recovering universal energy ratios characteristic of the underlying field theories. Specifically, we develop and implement a modulation technique to resolve a Rydberg chain's finite-size spectra, variably tuned to quantum phase transitions described by either Ising or tricritical Ising CFTs. We also employ local control to distinguish parities of excitations under reflection and, in the tricritical Ising chain, to induce transitions between distinct CFT spectra associated with changing boundary conditions. By utilizing a variant of the modulation technique, we furthermore study the dynamical structure factor of the critical system, which is closely related to the correlation of an underlying Ising conformal field. Our work not only probes the emergence of CFT features in a quantum simulator, but also provides a technique for diagnosing a priori unknown universality classes in future experiments.