Hall conductivity reveals the nature of quantum coherence in strongly correlated metals

Abstract: Linear-in-temperature resistivity is a hallmark for strange metallic transport, and appears universally in many strongly correlated electron systems. However, the focus on the longitudinal channel often overshadows the profound microscopic insights contained within the transverse response. Here, we utilize numerically exact determinantal quantum Monte Carlo simulations of the doped Hubbard model in a magnetic field to calculate longitudinal and transverse transport. We demonstrate that while the resistivity is robustly $T$-linear across parameter sets, the Hall response is highly sensitive to particle-hole asymmetry, Fermi surface topology, and many-body correlation effects. Specifically, the combination of these effects determine a crossover scale in which the system becomes quantum-coherent, and is reflected in the Hall conductivity. Our results demonstrate that while the $T$-linearity in resistivity appears universal, the Hall response reveals a crossover from semi-classical to quantum-coherent transport otherwise masked in the longitudinal channel.

• The paper shows, using numerically exact DQMC simulations of the doped Hubbard model, that Hall conductivity reveals an emergent quantum-coherence scale that longitudinal resistivity obscures.
• Changing next-nearest-neighbor hopping causes the Hall coefficient to change sign and the inverse Hall angle to follow distinct laws, including T², T, and nearly constant behavior, challenging universal strange-metal scaling.
• The study links Hall-response changes to flux-enclosing hopping loops and the minimum in double occupancy, while showing that interaction-driven spectral-weight redistribution can override bare Fermi-surface geometry.

The Hall response of strongly correlated metals has long been treated as a secondary observable, subordinate to the longitudinal resistivity that defines the strange metal phenomenology. In "Hall conductivity reveals the nature of quantum coherence in strongly correlated metals" (2606.12665), Emily Z. Zhang and Thomas P. Devereaux argue that this hierarchy is misleading: the transverse channel carries microscopic information that the longitudinal channel structurally cannot. Using numerically exact determinantal quantum Monte Carlo (DQMC) simulations of the doped Hubbard model in a magnetic field, the authors show that while the resistivity remains robustly linear in temperature across a wide parameter space, the Hall conductivity undergoes qualitative changes that track the onset of many-body quantum coherence. The result reframes the non-universality of the inverse Hall angle in cuprate-like systems not as a puzzle about scattering rates, but as a direct readout of an emergent coherence scale.

Numerical approach

The study simulates the two-dimensional single-band Hubbard model with next-nearest-neighbor hopping $t'$ on $8\times 8$ clusters in the presence of a weak magnetic field ($B = 0.0625\,\Phi_0/a^2$), at $U/t = 6$ and fillings around $\langle n\rangle = 0.7$. Longitudinal transport is obtained from current-current correlators via maximum entropy analytic continuation; the Hall channel is extracted through a subtraction scheme applied to the Hermitian composite correlator $\chi_{xx} - i\chi_{xy}$, which guarantees positive definiteness and permits MaxEnt inversion, followed by a Kramers–Kronig transformation to the DC limit. A Matsubara first-frequency proxy for $R_H$ provides an independent cross-check, and finite-size tests on $6\times 6$ through $10\times 10$ clusters support convergence. The sign problem is mitigated by averaging over hundreds of independently seeded Markov chains. This methodological apparatus matters because the conclusions rest on resolving temperature dependences in $\sigma_{xy}$ that are small relative to $8\times 8$0.

Divergence of longitudinal and transverse channels

The central empirical finding is a clean separation between the two transport channels. The longitudinal resistivity $8\times 8$1 is approximately linear in $8\times 8$2 over all dopings and values of $8\times 8$3 studied, persisting beyond the Mott–Ioffe–Regel limit and showing only weak sensitivity to band structure — consistent with earlier field-free DQMC results on the doped Hubbard model [huang_strange_2019-3]. The Hall coefficient $8\times 8$4, by contrast, changes sign and alters its temperature dependence qualitatively as $8\times 8$5 varies from $8\times 8$6 to $8\times 8$7. Only one parameter set exhibits the $8\times 8$8 scaling of $8\times 8$9 required to produce the canonical $B = 0.0625\,\Phi_0/a^2$0.

This non-universality propagates directly into the inverse Hall angle: log-log analysis shows extended regimes described by simple integer powers — $B = 0.0625\,\Phi_0/a^2$1 near $B = 0.0625\,\Phi_0/a^2$2, $B = 0.0625\,\Phi_0/a^2$3 for $B = 0.0625\,\Phi_0/a^2$4, and approximately constant for $B = 0.0625\,\Phi_0/a^2$5 — rather than the non-integer exponents expected from critical scaling or the universal quadratic law often assumed from cuprate phenomenology [hwang_scaling_1994, ayres_incoherent_2021]. The implication is that the quadratic inverse Hall angle occupies only a narrow region of the Hubbard–Hofstadter parameter space, so its frequent appearance in experiments must reflect material-specific conditions rather than a generic property of strange metals.

Sign of the Hall coefficient beyond Boltzmann geometry

To determine what controls the sign of $B = 0.0625\,\Phi_0/a^2$6, the authors compare it against the momentum distribution contour $B = 0.0625\,\Phi_0/a^2$7 used as a proxy for the Fermi surface in the absence of well-defined quasiparticles. In most parameter sets the sign follows the semi-classical expectation of Ong's geometric picture: electron-like closure about $B = 0.0625\,\Phi_0/a^2$8 gives negative $B = 0.0625\,\Phi_0/a^2$9, hole-like closure about $U/t = 6$0 gives positive $U/t = 6$1, even deep in the correlated regime.

One case violates this rule decisively: both the bare and interacting contours are electron-like, yet $U/t = 6$2 remains positive, accompanied by strong broadening of spectral weight near the antinodes $U/t = 6$3. This demonstrates that interaction-induced redistribution of low-energy spectral weight can override bare Fermi-surface geometry in setting the Hall sign — an effect strongest for $U/t = 6$4 and entirely absent from any Boltzmann-level description. The practical consequence is that inferring carrier density or Fermi surface topology from $U/t = 6$5 in strongly correlated materials is unreliable without accounting for correlation effects.

Coherence scale from flux-enclosing loops

The paper's key mechanistic result concerns the temperature dependence of $U/t = 6$6. At high temperatures ($U/t = 6$7), all curves of $U/t = 6$8 collapse onto a common constant asymptote, while $U/t = 6$9 remains flat down to low temperatures. Upon cooling, each $\langle n\rangle = 0.7$0 departs from the high-temperature behavior at a distinct temperature, ordered by the sign and magnitude of $\langle n\rangle = 0.7$1: deviations occur near the scale $\langle n\rangle = 0.7$2 for $\langle n\rangle = 0.7$3 but are pushed down toward the exchange scale $\langle n\rangle = 0.7$4 for $\langle n\rangle = 0.7$5.

Crucially, these departure temperatures coincide with the minimum of the average double occupancy $\langle n\rangle = 0.7$6. The authors interpret this minimum as an empirical proxy $\langle n\rangle = 0.7$7 for a crossover: above it, double occupancy is thermodynamically suppressed by Coulomb repulsion (a semi-classical regime); below it, doublon fluctuations stabilize singlet formation and coherent many-body motion emerges. Because the Hall conductivity requires virtual hopping processes enclosing magnetic flux — triangular plaquettes with amplitude proportional to $\langle n\rangle = 0.7$8 for finite $\langle n\rangle = 0.7$9, or higher-order processes at $\chi_{xx} - i\chi_{xy}$0 — it is sensitive precisely to this coherent loop motion, whereas the leading longitudinal response involves no such closed trajectories. The Hall channel therefore functions as a detector of the semi-classical-to-quantum-coherent crossover that is invisible in $\chi_{xx} - i\chi_{xy}$1.

A high-temperature expansion in inverse temperature supports this interpretation. Expanding the Lehmann representation of $\chi_{xx} - i\chi_{xy}$2 yields $\chi_{xx} - i\chi_{xy}$3, with coefficients organized by hopping-loop topology and controlled by combinations of $\chi_{xx} - i\chi_{xy}$4, $\chi_{xx} - i\chi_{xy}$5, and $\chi_{xx} - i\chi_{xy}$6. Since $\chi_{xx} - i\chi_{xy}$7 can be parametrically smaller than $\chi_{xx} - i\chi_{xy}$8 and $\chi_{xx} - i\chi_{xy}$9, nominally subleading terms may dominate over extended temperature ranges, and different $R_H$0 select different dominant terms. Combined with $R_H$1, this naturally produces the observed hierarchy of integer power laws in $R_H$2 without invoking quasiparticles or phenomenological two-lifetime models of the Anderson–Ong type.

Limitations and open questions

Several caveats bear directly on the strength of these conclusions. First, the integer-power-law identification relies on analytic continuation of noisy Monte Carlo data; although error bars are reported as smaller than marker sizes and proxy methods agree qualitatively, the precision of exponent extraction over finite temperature windows remains a systematic concern. Second, the high-temperature expansion coefficients $R_H$3 are presented at the level of scaling structure only — the prefactors $R_H$4 are not evaluated, and the authors state that a systematic strong-coupling expansion will appear in future work. Third, the coherence scale $R_H$5 is defined operationally through the double-occupancy minimum; whether this quantity coincides with a more fundamental definition of many-body coherence (e.g., from entanglement or spin correlations) is left open. Fourth, simulations are restricted to $R_H$6, moderate doping, and weak fields on small clusters, so the extent to which the $R_H$7-controlled crossover survives at stronger coupling, lower temperatures, or in the pseudogap regime is untested. Finally, connecting the computed crossovers quantitatively to specific cuprate or moiré materials would require accounting for multi-band physics, phonons, and disorder absent from the model.

Conclusion

This work establishes, within a numerically exact framework, that the Hall conductivity of the doped Hubbard model encodes an emergent coherence scale that the longitudinal resistivity systematically conceals. The robust $R_H$8-linear resistivity coexists with sharply non-universal, integer-power-law Hall angles whose exponents are selected by $R_H$9 through the topology of flux-enclosing hopping loops, and whose onset tracks the double-occupancy minimum marking the semi-classical-to-coherent crossover. The results suggest that classifying strange metals by their transverse response — rather than their resistivity alone — offers a route to a microscopic taxonomy of quantum coherence formation, though confirming this program against experimental data and extending the expansion beyond its current scaling form remain open tasks.