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arXiv:2210.15998 · Journal of High Energy Physics

Spatially random disorder in unitary fermion system in (4 − ϵ)-dimensions and effective action at finite temperature

Rajesh K. Gupta, Meenu

Published Jul 3, 2023• 1 citation
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Abstract

Abstract Non-relativistic conformal field theory is significant to understand various aspects of an ultra-cold system. In this paper, we study a non-relativistic system of two-component fermions interacting with a complex boson with Yukawa-like interactions near d = 4-spatial dimensions in the presence of a quenched disorder. The homogeneous theory flows to an interacting fixed point describing a unitary fermion system. In the presence of the disorder, we find that the system has an interesting phase structure in the space of the coupling constants and exhibits an interacting disorder fixed point in ϵ-expansion. The correlation function obeys Lifshitz scaling behaviour at the disorder fixed point with the anisotropic exponent being z = 2 + γE. We also study the disorder system at finite temperature and compute the leading contribution to the 1PI effective action.

DOI
10.1007/jhep07(2023)003

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Abstract Non-relativistic conformal field theory is significant to understand various aspects of an ultra-cold system. In this paper, we study a non-relativistic system of two-component fermions interacting with a complex boson with Yukawa-like interactions near d = 4-spatial dimensions in the presence of a quenched disorder. The homogeneous theory flows to an interacting fixed point describing a unitary fermion system. In the presence of the disorder, we find that the system has an interesting phase structure in the space of the coupling constants and exhibits an interacting disorder fixed point in ϵ-expansion. The correlation function obeys Lifshitz scaling behaviour at the disorder fixed point with the anisotropic exponent being z = 2 + γE. We also study the disorder system at finite temperature and compute the leading contribution to the 1PI effective action.
Quantum electrodynamics in 2+1 dimensions (QED3{\mathrm{QED}}_{3}QED3​) is a strongly coupled conformal field theory (CFT) of a U(1) gauge field coupled to 2N2N2N two-component massless fermions. The N=2N=2N=2 CFT has been proposed as a ground state of the spin-1/2 kagome Heisenberg antiferromagnet. We study QED3{\mathrm{QED}}_{3}QED3​ in the presence of weak quenched disorder in its two spatial directions. When the disorder explicitly breaks the fermion flavor symmetry from SU(2N)\mathrm{SU}(2N)SU(2N) $\ensuremath{\rightarrow}$ U(1) $\ifmmode\times\else\texttimes\fi{}$ SU(N)\mathrm{SU}(N)SU(N) but preserves time-reversal symmetry, we find that the theory flows to a nontrivial fixed line at nonzero disorder with a continuously varying dynamical critical exponent z>1z>1z>1. We determine the zero-temperature flavor (spin) conductivity along the critical line. Our calculations are performed in the large-NNN limit, and the disorder is handled using the replica method.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
Understanding the influence of quenched random potential is crucial for comprehending the exotic electronic transport of non-Fermi liquid metals near metallic quantum critical points. In this study, we identify a stable fixed point governing the quantum critical behavior of two-dimensional non-Fermi liquid metals in the presence of a random potential disorder. By performing renormalization group analysis on a dimensional-regularized field theory for Ising-nematic quantum critical points, we systematically investigate the interplay between random potential disorder for electrons and Yukawa-type interactions between electrons and bosonic order-parameter fluctuations in a perturbative epsilon expansion. At the one-loop order, the effective field theory lacks stable fixed points, instead exhibiting a runaway flow toward infinite disorder strength. However, at the two-loop order, the effective field theory converges to a stable fixed point characterized by finite disorder strength, termed the “disordered non-Fermi liquid (DNFL) fixed point”. Our investigation reveals that two-loop vertex corrections induced by Yukawa couplings are pivotal in the emergence of the DNFL fixed point, primarily through screening disorder scattering. Additionally, the DNFL fixed point is distinguished by a substantial anomalous scaling dimension of fermion fields, resulting in pseudogap-like behavior in the electron’s density of states. These findings shed light on the quantum critical behavior of disordered non-Fermi liquid metals, emphasizing the indispensable role of higher-order loop corrections in such comprehension.
We study a Dirac fermion model with three kinds of disorder as well as a marginal interaction which forms the critical line of c=1c=1c=1 conformal field theory. Computing scaling equations by the use of a perturbative renormalization group method, we investigate how such an interaction affects the universality classes of disordered systems with non-interacting fermions. We show that some specific fixed points are stable against an interaction, whereas others are unstable and flow to new random critical points with a finite interaction.
Using a functional renormalization group approach we derive the renormalization group (RG) flow of a dissipative variant of the Yukawa-Sachdev-Ye-Kitaev model describing NNN fermions on a quantum dot, which interact via a disorder-induced Yukawa coupling with MMM bosons. The inverse Euclidean propagator of the bosons is assumed to exhibit a nonanalytic term proportional to the modulus of the Matsubara frequency. We show that, to leading order in 1/N1/N1/N and 1/M1/M1/M, the hierarchy of formally exact flow equations for the irreducible vertices of the disorder-averaged model can be closed at the level of the two-point vertices. We find that the RG flow exhibits a non-Fermi liquid fixed point characterized by a finite fermionic anomalous dimension $\ensuremath{\eta}$, which is related to the bosonic anomalous dimension $\ensuremath{\gamma}$ via the scaling law $2=2\ensuremath{\eta}+\ensuremath{\gamma}$ with $0<\ensuremath{\eta}<1/2$. We explicitly calculate $\ensuremath{\eta}$ and the critical exponents characterizing the linearized RG flow in the vicinity of the fixed point as functions of N/MN/MN/M.
A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions d>2d>2d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions d>1d>1d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.