Superlinear Hall angle and carrier mobility from non-Boltzmann magnetotransport in the spatially disordered Yukawa-SYK model on a square lattice

TL;DR

The paper addresses magnetotransport in strange metals by embedding a spatially disordered Yukawa-SYK sector on a square lattice and solving the saddle-point equations to linear order in magnetic field. It combines exact imaginary-axis solutions, analytic continuation to real frequencies, and a Kubo-based calculation of $\sigma_{xx}^{(0)}$ and $\sigma_{xy}^{(1)}$ at fixed carrier density, revealing a superlinear $|\cot[\Theta_H(T)]| \propto T^{\alpha}$ with $\alpha$ up to about 1.45 in the crossover regime alongside $T$-linear resistivity. The lattice embedding uniquely drives a decrease in $R_H(T)$ and decouples longitudinal and Hall dynamics, yielding a robust mechanism for enhanced Hall-angle nonlinearity near quantum criticality. This controlled framework connects non-Fermi-liquid behavior to lattice effects and offers qualitative alignment with experimental trends in strange metals, while highlighting avenues for quantitative refinement and extensions to other lattice settings.

Abstract

Exact numerical results for the DC magnetoconductivity tensor of the two-dimensional spatially disordered Yukawa-Sachdev-Ye-Kitaev (2D-YSYK) model on a square lattice, at first order in applied perpendicular magnetic field, are obtained from the self-consistent disorder-averaged solution of the 2D-YSYK saddle-point equations. This system describes fermions endowed with a Fermi surface and coupled to a bosonic scalar field through spatially random Yukawa interactions. The resulting local and energy-dependent fermionic self-energies are employed in the Kubo formalism to calculate the longitudinal and Hall conductivities, the Hall coefficient, the carrier mobility, and the cotangent of the Hall angle, at fixed fermion density. From the interplay between YSYK interactions and square-lattice embedding, and the non-Boltzmann frequency-dependent self energies, we find nontrivial evolution of the magnetotransport coefficients as a function of temperature and YSYK interaction strength, notably a superlinear evolution of the Hall-angle cotangent and the inverse carrier mobility with temperature, concomitant with linear-in-temperature resistivity, in an extended crossover regime above the low-temperature Marginal Fermi Liquid (MFL) ground state. Our model and results provide a controlled theoretical framework to interpret linear magnetotransport experiments in strange-metal phases found in strongly correlated solid-state electron systems.

Figures (14)

• Figure 1: (a) Schematics of our system's geometry and configuration: a 2D-YSYK square lattice with nearest neighbour hopping amplitude $t$ is subjected to an external electric field along $x$ and out-of-plane magnetic field $\vec{B}$ along $z$, thus inducing a longitudinal current $\vec{I}$ as well as a Hall electric field $\vec{E}$ along $y$; at each lattice site, a local self-energy $\Sigma(\epsilon)$ stems from the disorder-averaged YSYK interactions $g_{ij,k}$ between fermion flavours $i=\left\{i,j,l,n\cdots\right\}$ and a scalar bosonic mode $\Phi_k$. (b) Schematic phase diagram of transport regimes as a function of temperature $k_B T/t$ and interaction strength $g'/t^{\frac{3}{2}}$, for fermion density $n=0.45/a^2$ and boson stiffness $J=t$, normalized to the fermion hopping $t$; coloured circles are numerical estimations of the boundaries between different regimes from the saddle-point solutions, while dashed curves are continuous interpolations; dashed blue and green arrows show the temperature paths scanned in panels (c-e) for $g'/t^{\frac{3}{2}}=\left\{2,5\right\}$. (c-e) Inverse longitudinal conductivity (c), Hall coefficient (d) , and cotangent of the Hall angle (e), as a function of $k_B T/t$, for $g'/t^{\frac{3}{2}}=2$ (blue solid curves) and $g'/t^{\frac{3}{2}}=5$ (green solid curves); colour shadings qualitatively identify the regimes in panel (b).
• Figure 2: Visual summary in logarithmic scale of the dependence of the Hall-angle cotangent $\cot[\Theta_H(T)]$ on dimensionless temperature $k_B T/t$, for boson stiffness $J=t$. (a) Results at fixed interaction $g'=2t^{3/2}$ and fermion density $n=\left\{0.3,0.35,0.4,0.45,0.475\right\}/a^2$. (b) Results at fixed fermion density $n=0.45/a^2$$g'=2t^{3/2}$ and interaction $g'=\left\{1,2,3,5\right\}t^{3/2}$. The exponent $\alpha$ of $\left|\cot[\Theta_H(T)]\right|\propto T^\alpha$ is maximized for weak interaction and small doping $\Delta n=0.5/a^2-n$.
• Figure 3: Dependence of the renormalized boson mass $m_b(T)$ and of the chemical potential $\mu(T)$ on dimensionless temperature $k_B T/t$ for boson stiffness $J=t$. (a) $m_b(T)$ and (b) $\mu(T)$ at fixed density $n=0.45/a^2$ and different interactions $g'$. (c) $m_b(T)$ and (d) $\mu(T)$ at fixed interaction $g'=2 t^{3/2}$ and different densities $n$. The insets in panels (a), (b), and (c) zoom on the low-$T$ regime.
• Figure 4: Linear DC magnetotransport coefficients as a function of temperature $k_B T/t$ normalized by the fermion hopping $t$, for fermion density $n=0.45/a^2$ and boson stiffness $J=t$, for different interactions $g'$. The system is tuned to the QCP at $T\rightarrow 0^+$ for each $g'$ at fixed density.
• Figure 5: Linear DC magnetotransport coefficients as a function of temperature $k_B T/t$ normalized by the fermion hopping $t$, for spatially disordered interaction $g'=2 t^{3/2}$ and boson stiffness $J=t$, for different fermion density $n$. The system is tuned to the QCP at $T\rightarrow 0^+$ for each density.