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arXiv:1907.05373 · Physical review. B./Physical review. B

Torus spectroscopy of the Gross-Neveu-Yukawa quantum field theory: Free Dirac versus chiral Ising fixed point

Michael Schuler, Stephan Heßelmann, Seth Whitsitt, Thomas C. Lang, Stefan Weßel, Andreas Martin Läuchli

Published Mar 12, 2021• 16 citations
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Abstract

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 ${n}_{\text{D}}=4$ component Dirac spinors in $D=(2+1)$ 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.

DOI
10.1103/physrevb.103.125128

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•May 11, 2018•0 citations
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 \ensuremathπ\ensuremath{\pi}\ensuremathπ-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 \ensuremathϵ=4\ensuremath−D\ensuremath{\epsilon}=4\ensuremath{-}D\ensuremathϵ=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.
Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the [Formula: see text] field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the [Formula: see text] field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher [Formula: see text]-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art [Formula: see text]-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.
We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.
It is expected that the Gross-Neveu-Yukawa (GNY) chiral Ising transition of Dirac fermions coupled with a scalar field in (2+1)(2+1)(2+1) dimensions will be the first fermionic quantum critical point that various methods, such as conformal bootstrap, perturbative renormalization group, and quantum Monte Carlo (QMC) simulations, would yield converged critical exponents---serving the same role as the Ising and O(N)O(N)O(N) models in the textbooks of statistical and quantum physics. However, such an expectation has not been fully realized from the lattice QMC simulations due to the obstacles introduced by the UV finite-size effect. In this Letter, by means of the elective-momentum ultrasize (EMUS)-QMC method, we compute the critical exponents of the $O{(N/2)}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{2}$ GNY N=8N=8N=8 chiral Ising transition on a two-dimensional $\ensuremath{\pi}$-flux fermion lattice model between Dirac semimetal and quantum spin Hall insulator phases. With the matching of fermionic and bosonic momentum transfer and collective update in momentum space, our QMC results provide fully consistent exponents with those obtained from the bootstrap and perturbative approaches. In this way, the EMUS now live happily on the N=8N=8N=8 island and could explore the chiral Gross-Neveu-Yukawa archipelago with ease.
We perform large-scale quantum Monte Carlo simulations of the Hubbard model at half filling with a single Dirac cone close to the critical point, which separates a Dirac semimetal from an antiferromagnetically ordered phase where SU(2) spin rotational symmetry is spontaneously broken. We discuss the implementation of a single Dirac cone in the SLAC formulation for eight Dirac components and the influence of dynamically induced long-range superexchange interactions. The finite-size behavior of dimensionless ratios and the finite-size scaling properties of the Hubbard model with a single Dirac cone are shown to be superior compared to the honeycomb lattice. We extract the critical exponent believed to belong to the chiral Heisenberg Gross-Neveu-Yukawa universality class: the critical exponent ν = 1.02 ( 3 ) coincides for the two lattice types once honeycomb lattices of linear dimension L > 15 are considered. In contrast to the SLAC formulation, where the anomalous dimensions are estimated to be η ϕ = 0.73 ( 1 ) and η ψ = 0.09 ( 1 ) , they remain less stable on honeycomb lattices, but tend towards the estimates from the SLAC formulation.
We perform large-scale quantum Monte Carlo simulations of SLAC fermions on a two-dimensional square lattice at half filling with a single Dirac cone with N=2 spinor components and repulsive on-site interactions. Despite the presence of a sign problem, we accurately identify the critical interaction strength U_{c}=7.28±0.02 in units of the hopping amplitude, for a continuous quantum phase transition between a paramagnetic Dirac semimetal and a ferromagnetic insulator. Using finite-size scaling, we extract the critical exponents for the corresponding N=2 chiral Ising Gross-Neveu universality class: the inverse correlation length exponent ν^{-1}=1.19±0.03, the order parameter anomalous dimension η_{ϕ}=0.31±0.01, and the fermion anomalous dimension η_{ψ}=0.136±0.005.
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.