Fate of the non-Fermi-liquid fixed point of a two-dimensional Fermi surface coupled to gauge fluctuations: the endpoints of the Nayak-Wilczek $(1-x)$ expansion and its physical applications

A two-dimensional Fermi surface coupled to a transverse gauge field, with the gauge fluctuations enslaved to fermion density fluctuations by a nonlocal $1/k^{x}$ interaction, admits a controlled non-Fermi-liquid fixed point in the $(1-x)$ expansion of Nayak and Wilczek (1994): at one loop, $\beta(\alpha)=-(1-x)\alpha+8\alpha^{2}+\mathcal{O}(\alpha^{3})$, so $\alpha^{*}=(1-x)/8$, the quasiparticle residue $Z\to0$, and the Green function becomes a branch cut. This expansion controls only $(1-x)\ll1$. The fate of this fixed point at the physical endpoints $x=0$ (Maxwell/Landau-damped gauge field, the universality class of the $\nu=1/2$ composite Fermi liquid and of spinon metals) and $x=1$ (Coulomb, marginal), at large flavor number $n$, for the physical bare coupling $g=2\pi$, and for transport, remains open. Every controlled scheme proposed since (epsilon=3-d, $z_{b}-2$, where $z_{b}$ is the gauge-field dynamical exponent and equals 3 physically, dimensional and codimension regularization, matrix large-$N$) deforms away from the physical two-dimensional problem; the large-$N$ limit itself is strongly coupled; the marginal-Fermi-liquid basin appears to shrink to measure zero under ultraviolet-infrared mixing; the most recent functional-renormalization-group treatment of the model yields exponents differing from all earlier ones; and microscopic calculations question the gauge-fluctuation signature $q^{3}\ln q$ at $\nu=1/2$. The $(1-x)\ll1$ limitation of Nayak and Wilczek (1994) is unresolved; the infrared fate of the two-dimensional gauge-coupled Fermi surface remains one of the central open problems of compressible quantum matter.

Background

In the Landau theory of Fermi liquids, the low-energy excitations of a generic interacting metal are weakly interacting quasiparticles: the single-particle Green function has a sharp pole at the Fermi surface (the closed curve in momentum space separating occupied from empty single-particle states at zero temperature), with finite residue $Z$ (the weight of that pole) and a decay rate much smaller than the excitation energy. In renormalization-group (RG) language, the free Fermi gas is an infrared-stable fixed point with essentially no relevant local perturbations. The known exceptions are special: the one-dimensional Luttinger liquid, where kinematic constraints make interactions marginal and produce spin-charge separation (independent propagation of spin and charge degrees of freedom) and branch-cut Green functions, and the overscreened multichannel Kondo impurity (a magnetic impurity coupled to several conduction-electron channels, so that screening overcompensates and no quasiparticle ground state exists). Yet several two-dimensional systems of intense interest are 'almost' but not quite Fermi liquids: the compressible quantum Hall state at filling $\nu=1/2$ and the anomalous normal ('strange metal') state of the copper-oxide superconductors.

Nayak and Wilczek (1994) addressed whether a genuinely two-dimensional non-Fermi-liquid metal can be described by a controlled fixed point. The construction builds on flux attachment: a Chern-Simons gauge field, a statistical gauge field in $2+1$ dimensions whose action is first order in derivatives and implements the binding of fictitious magnetic flux to fermions, is coupled to fermionic excitations about a Fermi surface, together with a nonlocal density interaction with Fourier-space kernel $1/k^{x}$. Because the temporal component of the Chern-Simons field obeys a constraint that enslaves transverse gauge fluctuations to density fluctuations, increasing $x$ suppresses both; $x=0$ gives a Maxwell-like ($k^{2}$) gauge kinetic term while $x=1$ corresponds to Coulomb interactions. Near a point on the Fermi surface, with the momentum measured from that point and decomposed into a component perpendicular to the surface ($k_{y}$) and a component tangential to it ($k_{x}$), the dispersion scales as $\epsilon(\mathbf{k})=v_{F}(k_{y}+ak_{x}^{2})$, and the RG rescaling $k_{x}\to s^{1/2}k_{x}$, $k_{y}\to sk_{y}$, $\omega\to s\omega$ gives the gauge coupling tree-level dimension $[g]=-(1-x)/2$: the Fermi-liquid fixed point is unstable for $x<1$ and marginal at $x=1$. Extracting pole parts in $(1-x)$, in close analogy with the epsilon-expansion of critical phenomena with $(1-x)$ playing the role of $\epsilon$, gives the one-loop beta function $\beta(\alpha)=-(1-x)\alpha+8\alpha^{2}+\mathcal{O}(\alpha^{3})$ for the dimensionless coupling $\alpha=g^{2}v_{F}/(2\pi)$, with an infrared-stable fixed point at $\alpha^{*}=(1-x)/8$. There the anomalous dimension of the Fermi velocity is $\eta_{v_{F}}=(1-x)/2$, so the Green function acquires a branch cut instead of a pole, $Z\to0$; observable properties acquire corresponding non-Fermi-liquid corrections, e.g. the tunneling conductance takes the form $g(V)\sim g_{0}+|V|^{1-\eta_{v_{F}}}$. At $x=1$ the coupling is marginally irrelevant, $\alpha(\mu)\sim1/(8\ln\mu)$, giving logarithmically corrected Fermi-liquid behavior close to the marginal-Fermi-liquid phenomenology of Varma et al. For $n$ fermion species the fixed point moves to $\alpha^{*}=n(1-x)/8$.

The physical motivation is twofold. Halperin, Lee and Read argued that at $\nu=1/2$ the electron fractionalizes into a 'composite fermion' (electron plus two flux quanta) forming a Fermi surface coupled to a fluctuating Chern-Simons gauge field; their one-loop self-energy is non-Fermi-liquid-like but uncontrolled at higher orders. Polchinski studied a Fermi surface of electrically neutral spinons (the spin-carrying fragments of the electron in a spin-charge-separated description, whose charge is carried by bosonic holons) coupled to a Maxwell gauge field, as in spin-charge-separated descriptions of the copper oxides, invoking a large-$n$ limit whose validity Nayak and Wilczek (1994) questioned. In both physical problems the gauge field is Landau-damped: particle-hole excitations of the Fermi surface add a nonanalytic $|\omega|/|q|$ term to the gauge inverse propagator, giving dynamical exponent $z=3$ (the scaling relation $\omega\sim q^{z}$ between frequency and momentum) and the familiar self-energy $\Sigma\sim\omega^{2/3}$. Whether the weak-coupling fixed point of the $(1-x)$ expansion controls these physical cases at $x=0$ and $x=1$ is precisely what the construction leaves open, and is the subject of this problem.

Problem Statement

Determine the infrared fate and physical content of the two-dimensional Fermi surface coupled to gauge fluctuations at and beyond the endpoints of the Nayak-Wilczek $(1-x)$ expansion. Specifically: (a) does a stable non-Fermi-liquid fixed point exist for the two-dimensional Fermi surface coupled to a transverse (Landau-damped or Maxwell) gauge field, the universality class relevant to the $\nu=1/2$ composite Fermi liquid and to spinon gauge theories of the copper oxides, and if so is it the strong-coupling continuation of the weak-coupling fixed point, and what are its exponents and low-energy properties? (b) At the Coulomb endpoint $x=1$, is the logarithmically slow marginally irrelevant flow the correct infrared behavior, or do non-perturbative effects qualitatively change it? (c) What is the fate of the fixed point at large flavor number $n$, where $\alpha^{*}=n(1-x)/8$ is not small and all orders contribute to $\beta$? (d) Do systems with physical bare couplings, such as the $\nu=1/2$ state with $g=2\pi$, or spin-charge-separated models with condensed holons, lie in the basin of attraction of this fixed point? (e) What is the controlled electrical resistivity of such a state, including the role of umklapp processes (scattering in which crystal momentum is transferred to the lattice, required for current relaxation in an otherwise translationally invariant metal) and the relation between transport and single-particle lifetimes? (f) Does the universality class survive confining gauge dynamics and holon condensation? The problem is to establish, by a demonstrable control parameter or non-perturbative argument, whether a stable non-Fermi-liquid fixed point exists in the physical regimes and, if so, its exponents, basin of attraction, and transport laws, or else to refute its existence in a controlled way; results restricted to deformed models (higher dimensions, non-physical gauge dynamics, or restricted diagram classes) address a different problem from the physical two-dimensional one. The lettered parts depend on (a): if the existence of the fixed point is refuted, (b)-(f) become moot, whereas if it is affirmed they are addressed together with (a).

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