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arXiv:1905.08256 · Physical Review Letters

Two-Dimensional Non-Fermi-Liquid Metals: A Solvable Large- N Limit

Jeremías Aguilera Damia, Shamit Kachru A, S. Raghu, Gonzalo Torroba

Published Aug 29, 2019• 56 citations
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Abstract

Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-N limit (N corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU(N) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-N" theory), yields a tractable large N limit. At low energies, the system consists of an overdamped boson with dynamical exponent z=3 coupled to a non-Fermi-liquid with self-energy Σ(ω)∼ω^{2/3}, consistent with previous studies.

DOI
10.1103/physrevlett.123.096402

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The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number, NB{N}_{B}NB​, of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global SU(NB)SU({N}_{B})SU(NB​) flavor symmetry group. The leading large NB{N}_{B}NB​ solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading 1/NB1/{N}_{B}1/NB​ corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large NB{N}_{B}NB​, our results are valid down to d=2d=2d=2. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large NB{N}_{B}NB​ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].
We study the low-energy behavior of metals coupled to gapless bosons. This problem arises in several contexts in modern condensed matter physics; we focus on the theory of metals near continuous quantum phase transitions (where the boson is the order parameter). In the vicinity of d=3 spatial dimensions, the upper critical dimension of the theory, the ratio of fermion and boson speeds, v/c, acts as an additional control parameter, enabling us to access IR fixed points where this ratio vanishes. This limit corresponds to a non-Fermi liquid coupled to bosons with critical exponents governed by the Wilson-Fisher fixed point.
The destruction of Fermi-liquid behavior when a gapless Fermi surface is coupled to a fluctuating gapless boson field is studied theoretically. This problem arises in a number of different contexts in quantum many-body physics. Examples include fermions coupled to a fluctuating transverse gauge field pertinent to quantum spin-liquid Mott insulators, and quantum critical metals near a Pomeranchuk transition. We develop a controlled theoretical approach to determine the low-energy physics. Our approach relies on combining an expansion in the inverse number (N)(N)(N) of fermion species with a further expansion in the parameter $ϵ={z}_{b}\ensuremath{-}2$, where zb{z}_{b}zb​ is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small NNN extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.
We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a z=3z=3z=3 scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors Nc{N}_{c}Nc​. When Nc{N}_{c}Nc​ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large Nc{N}_{c}Nc​, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-Tc{T}_{c}Tc​ materials in a systematic $\ensuremath{\epsilon}$ expansion.
We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory (d=3d=3d=3 spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-NNN limit. In this limit, we find that the theory has no superconducting instability.
Non-Fermi liquids (NFLs) are a class of strongly interacting gapless fermionic systems without long-lived quasiparticle excitations.An important group of NFL models feature itinerant fermions coupled to soft bosonic fluctuations near a quantum-critical point and are widely believed to capture the essential physics of many unconventional superconductors.However, numerically, the direct observation of a canonical NFL behavior in such systems, characterized by a power-law form in the Green's function, has been elusive.Here, we consider a Sachdev-Ye-Kitaev (SYK)-like model with random Yukawa interaction between critical bosons and fermions (dubbed the Yukawa-SYK model).We show that it is immune from the minus-sign problem and hence can be solved exactly via large-scale quantum Monte Carlo simulation beyond the large-N limit accessible to analytical approaches.Our simulation demonstrates that the Yukawa-SYK model features "self-tuned quantum criticality"; namely, the system is critical independent of the bosonic bare mass.We put these results to the test at finite N, and our unbiased numerics reveal clear evidence of these exotic quantum-critical NFL properties-the power-law behavior in the Green's function of fermions and bosons-which propels the theoretical understanding of critical Planckian metals and unconventional superconductors.