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arXiv:2501.07792 · Physical review. B./Physical review. B

Hall angle of a spatially random vector model

Yili Wang, Young-Kwon Han, Xian-Hui Ge, Sang-jin Sin A

Published Jun 5, 2025• 4 citations
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Abstract

Strange metals exhibit linear resistivity and anomalous Hall transport, yet a comprehensive theory that accounts for both phenomena is still lacking. Recent studies have shown SYK-like spatially random couplings between a Fermi surface and a bosonic field, either scalar or vector type, can yield linear-$T$ resistivity. In this paper, we continue the investigation on a vector coupling in the presence of a magnetic field. We compute the fermion and boson propagators, along with the self-energy and polarization functions, and determine their dependence on the magnetic field. Although the Hall angle does not exhibit the signature of strange-metal, the linear-in-temperature resistivity remains at low temperatures. Results indicate that random interactions can robustly support linear transport, though additional ingredients may be required to capture the full phenomenology of strange metals.

DOI
10.1103/rfz7-y7f9

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Strange metals exhibit linear resistivity and anomalous Hall transport, yet a comprehensive theory that accounts for both phenomena is still lacking. Recent studies have shown SYK-like spatially random couplings between a Fermi surface and a bosonic field, either scalar or vector type, can yield linear-TTT resistivity. In this paper, we continue the investigation on a vector coupling in the presence of a magnetic field. We compute the fermion and boson propagators, along with the self-energy and polarization functions, and determine their dependence on the magnetic field. Although the Hall angle does not exhibit the signature of strange-metal, the linear-in-temperature resistivity remains at low temperatures. Results indicate that random interactions can robustly support linear transport, though additional ingredients may be required to capture the full phenomenology of strange metals.
Recent studies have shown that a spatially random Yukawa-type interaction between a Fermi surface and critical bosons can produce linear-in-temperature resistivity, the defining signature of strange metals. In this article, we systematically classify all scalar couplings of the form (ψ†ψ)nϕm(ψ^{\dagger}ψ)^nϕ^m(ψ†ψ)nϕm in arbitrary dimensions to identify possible candidates for strange-metal behaviour within this disordered framework. We find that only spatially random Yukawa-type interaction in (2+1)(2+1)(2+1) dimensions can yield linear resistivity. This indicates that linear resistivity is not a universal property of all spatially random scalar coupling, and strange-metal property replies on both dimensions and interaction type.
Recently, Patel et al. Science 381, 790 (2023) introduced a higher-dimensional version of the SYK model with random coupling in a Yukawa interaction to explain the linear-TTT resistivity. We test the universality of this mechanism by replacing the scalar field with a vector field in various dimensions. We find that it applies to vector and scalar interactions, although the details differ significantly. However, this mechanism for linear-TTT resistivity works only in (2+1)(2+1)(2+1) dimensions and not in higher dimensions, regardless of the interaction type. Based on these results, we explore the role of spatial random disorder and provide a simple explanation of how such random scattering transforms a Fermi liquid into a strange metal by modifying the self-energies of the involved bosons and fermions.
The strange metal phase of correlated electrons materials was described in a recent theory by a model of a Fermi surface coupled a two-dimensional quantum critical bosonic field with a spatially random Yukawa coupling. With the assumption of self-averaging randomness, similar to that in the Sachdev-Ye-Kitaev model, numerous observed properties of a strange metal were obtained for a wide range of intermediate temperatures, including the linear in temperature resistivity. The Harris criterion implies that spatial fluctuations in the local position of the critical point must dominate at lower temperatures. For an [Formula: see text]-component boson with [Formula: see text], we use multiple graphics processing units (GPUs) to compute the real frequency spectrum of the boson propagator in a self-consistent mean-field treatment of the boson self-interactions, but an exact treatment of multiple realizations of the spatial randomness from the random boson mass. We find that Landau damping from the fermions leads to the emergence of the physics of the random transverse-field Ising model at low temperatures, as has been proposed by Hoyos, Kotabage, and Vojta. This regime is controlled by localized overdamped eigenmodes of the bosonic scalar field, also has a resistivity which is nearly linear-in-temperature, and extends into a "quantum critical phase" away from the quantum critical point, as observed in several cuprates. For the [Formula: see text] Ising scalar, the mean-field treatment is not applicable, and so we use Hybrid Monte Carlo simulations running on multiple GPUs; we find a rounded transition and localization physics, with strange metal behavior in an extended region around the transition.
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 ground state. Our model and results provide a controlled theoretical framework to interpret magnetotransport experiments, at linear order in applied magnetic field, in strange-metal phases found in strongly correlated solid-state electron systems.
“Strange” metals that do not follow the predictions of Fermi liquid theory are prevalent in materials that feature superconductivity arising from electron interactions. In recent years, it has been hypothesized that spatial randomness in electron interactions must play a crucial role in strange metals for their hallmark linear-in-temperature ( T ) resistivity to survive down to low temperatures where phonon and umklapp processes are ineffective, as is observed in experiments. However, a clear picture of how this happens has not yet been provided in a realistic model free from artificial constructions, such as large- N limits and replica tricks. We study a realistic model of two-dimensional metals with spatially random antiferromagnetic interactions in a nonperturbative regime, using numerically exact high-performance, large-scale hybrid Monte Carlo simulation and exact averages over the quenched spatial randomness. Our simulations reproduce strange metals’ key experimental signature of linear-in- T resistivity with a universal “Planckian” transport scattering rate Γ tr ∼ k B T / ℏ that is independent of coupling constants. We further find that strange metallicity in these systems is not associated with a quantum critical point and, instead, arises from a phase of matter with gapless antiferromagnetic fluctuations that lacks long-range correlations and spans an extended region of parameter space: A feature that is also observed in several experiments. These gapless antiferromagnetic fluctuations take the form of spatially localized overdamped modes, whose presence could possibly be detected using recently developed nanoscale magnetometry methods. Our work paves the way for an eventual microscopic understanding of the role of spatial disorder in determining important properties of correlated-electron materials.
Strange metals—ubiquitous in correlated quantum materials—transport electrical charge at low temperatures but not by the individual electronic quasiparticle excitations, which carry charge in ordinary metals. In this work, we consider two-dimensional metals of fermions coupled to quantum critical scalars, the latter representing order parameters or fractionalized particles. We show that at low temperatures ( T ), such metals generically exhibit strange metal behavior with a T -linear resistivity arising from spatially random fluctuations in the fermion-scalar Yukawa couplings about a nonzero spatial average. We also find a T ln(1/ T ) specific heat and a rationale for the Planckian bound on the transport scattering time. These results are in agreement with observations and are obtained in the large N expansion of an ensemble of critical metals with N fermion flavors.