Quantum-critical transport in marginal Fermi liquids

TL;DR

This work develops an exact low-temperature transport theory for fermions on a lattice coupled to quantum-critical fluctuations of a loop-current (QXY-F) order, within a marginal Fermi-liquid framework. By solving the Kubo equations with a conserving memory-matrix approach, it isolates the impact of vertex corrections and shows that Maki–Thompson diagrams vanish while Aslamazov–Larkin diagrams yield an Umklapp factor that can be computed for a circular Fermi surface; mass renormalization does not enhance electrical or thermal conductivities but induces a $T\ln(\omega_c/T)$ term in the Seebeck coefficient. The theory predicts linear-in-$T$ resistivity with a geometry-dependent, $T$-independent Umklapp factor and a temperature-independent thermal conductivity, plus a logarithmic mass contribution to the Seebeck coefficient, all consistent with Planckian dissipation observed in cuprates, heavy fermions, and related 2D materials. It further demonstrates $\,\omega/T$ scaling of transport in the critical regime and provides numerically exact results for a circular Fermi surface, with qualitative agreement expected for general Fermi surfaces. The results unify transport phenomena across diverse quantum-critical metals and offer a concrete framework for interpreting experimental data on strange metals and related loop-current candidates, including Moiré graphene systems.

Abstract

We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe$_2$. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization $\propto \ln ω_c/T$ appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.

Figures (10)

• Figure 1: Diagrammatic representation of the Kubo equation for the electrical conductivity. The external field coupling to fermion charge times their velocity is shown as a dotted line. The lines are exact single-particle Green's functions $G$. $\Lambda$ is the renormalized current vertex.
• Figure 2: Diagrammatic representation of (a) the irreducible vertex $I$ in terms of the bare-irreducible vertex $I_0$ which is given by the first diagram in which the wiggly line is given by (b), the renormalized collective mode propagator. In our case, the exact collective mode propagator [the left-hand side of Eq. (b)] is already available from quantum Monte Carlo calculation and renormalization group calculations for the quantum XY model coupled to fermions, summarized in Appendix A. $\Pi$ is the exact polarizability of the fermions which occurs in the self-energy of the collective mode propagator. $G$ is the exact fermion Green's function. It is calculable at momenta close to Fermi surface because the triangular vertex in its last part is $1$ to leading order in $\omega/E_F$ and $({\bf p-p}_F)/p_F$, as shown in Appendix B. (d) The total vertex that is used in the transport equations.
• Figure 3: Diagrammatic representation for the integral equation for the external field vertices in calculation of conductivity. This is a sum of three parts shown successively in the three lines. First is the vertex coupling to the renormalized fermion propagators, second and third are the vertex coupling to the collective modes. The second line gives what may be called "Maki-Thompson" diagrams and the third corresponds to the "Aslamazov-Larkin" diagrams. We show that the first two give no correction for the present problem and the third comes from satisfying conservation laws and gives only a numerical correction
• Figure 4: Regions on the complex energy plane for the analytic continuations of (a) $\Lambda({\bf p}, z; i\omega_m)$ with $i \epsilon_n = z$ and (b) $I({\bf p}, i \epsilon_n, {\bf p}', i \epsilon_{n'}; i \omega_m)$ with $i \epsilon_n = z$ and $i \epsilon_{n'} = z'$.
• Figure 5: Reduction factor of electrical resistivity owing to the vertex corrections, i.e. the Umklapp factor, for a circular Fermi surface of radius $k_F$ in a square lattice of unit cell ${a_L}$, calculated as a function of $k_F{a_L}/\pi$ ($1/2 < k_F{a_L}/\pi < 1/\sqrt{2})$ for different number of $L_{max}$. The inset shows the $L_{max}$ dependence of the Umklapp factor for $k_F{a_L}/\pi = 2-\sqrt{2}$.