Understanding critical phenomena is of central importance to condensed-matter and high-energy physics. Such an understanding is reflected in our ability to sort observables based on their degeneracy, symmetries, and power-law decays. Here, we study such critical properties of the Ising-Higgs gauge theory in ( 2 + 1 ) D along the self-dual line which have recently been a subject of debate. Using machine learning techniques, we determine the low-energy operator content of the associated field theory. Our approach enables us to largely refute the existence of an emergent current operator and with it the standing conjecture that this transition is of the X Y * universality class. We contrast these results with the ones obtained for the ( 2 + 1 ) D Ashkin-Teller transverse field Ising model where we find the expected current operator. Our numerical technique extends the recently proposed real-space mutual information allowing us to extract subleading nonlinear operators. This allows a controlled and computationally scalable approach to target the conformal field theory spectrum. and discern universality classes beyond ( 1 + 1 ) D from Monte Carlo data. Published by the American Physical Society 2024
Understanding critical phenomena is of central importance to condensed-matter and high-energy physics. Such an understanding is reflected in our ability to sort observables based on their degeneracy, symmetries, and power-law decays. Here, we study such critical properties of the Ising-Higgs gauge theory in ( 2 + 1 ) D along the self-dual line which have recently been a subject of debate. Using machine learning techniques, we determine the low-energy operator content of the associated field theory. Our approach enables us to largely refute the existence of an emergent current operator and with it the standing conjecture that this transition is of the X Y * universality class. We contrast these results with the ones obtained for the ( 2 + 1 ) D Ashkin-Teller transverse field Ising model where we find the expected current operator. Our numerical technique extends the recently proposed real-space mutual information allowing us to extract subleading nonlinear operators. This allows a controlled and computationally scalable approach to target the conformal field theory spectrum. and discern universality classes beyond ( 1 + 1 ) D from Monte Carlo data. Published by the American Physical Society 2024
We critically discuss the results reported in arXiv:2311.17994v1 by L. Oppenheim, M. Koch-Janusz, S. Gazit, and Z. Ringel, on the multicritical behavior of the three-dimensional Ising-Gauge model at the multicritical point. We argue that their results do not contradict the theoretical scenario put forward in ``Multicritical point of the three-dimensional Z2 gauge Higgs model'', Phys. Rev. B 105, 165138 (2022), arXiv:2112.01824, that predicted a multicritical behavior controlled by the stable XY fixed point of an effective three-dimensional Z2⊕Z2 Landau-Ginzburg-Wilson Φ4 field theory. Actually, their results, as well as all numerical results reported so far in the literature, are consistent with a multicritical XY scenario.
The deconfined phase of (2+1)-dimensional Zk gauge theory exhibits topological order, with e and m anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either e or m drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges e and m anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level k to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for k=2 disappears generically for sufficiently large k; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order 1/k2 in the large-k limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.
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.
The presence of gauge symmetry in 1+1 dimensions is known to be redundant, since it does not imply the existence of dynamical gauge bosons. As a consequence, in the continuum, the Abelian-Higgs model (i.e., the theory of bosonic matter interacting with photons) just possesses a single phase, as the higher-dimensional Higgs and Coulomb phases are connected via nonperturbative effects. However, recent research published in Phys. Rev. Lett. 128, 090601 (2022) has revealed an unexpected phase transition when the system is discretized on the lattice. This transition is described by a conformal field theory with a central charge of c=3/2. In this paper, we aim to characterize the two components of this c=3/2 theory---namely the free Majorana fermionic and bosonic parts---through equilibrium and out-of-equilibrium spectral analyses.
At a quantum critical point, the low-energy physics of a quantum spin chain is described by conformal field theory (CFT). Given the Hamiltonian of a critical spin chain, we propose a variational method to build an approximate lattice representation ϕ_{α} of the corresponding primary CFT operators ϕ_{α}^{CFT}. We then show how to numerically compute the operator product expansion coefficients C_{αβγ}^{CFT} governing the fusion of two primary fields. In this way, we complete the implementation of Cardy's program, outlined in the 1980s, which aims to extract the universality class of a phase transition, as encoded in the so-called conformal data of the underlying CFT, starting from a microscopic description. Our approach, demonstrated here for the critical quantum Ising model, only requires a generic (i.e., in general, nonintegrable) critical lattice Hamiltonian as its input.
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.