Magnetic properties of interacting, disordered electron systems in d=2 dimensions

Abstract: We compute the magnetic susceptibilities of interacting electrons in the presence of disorder on a two-dimensional square lattice by means of quantum Monte Carlo simulations. Clear evidence is found that at sufficiently low temperatures disorder can lead to an enhancement of the ferromagnetic susceptibility. We show that it is not related to the transition from a metal to an Anderson insulator in two dimensions, but is a rather general low temperature property of interacting, disordered electronic systems.

• The paper demonstrates that disorder can enhance ferromagnetic susceptibility at low temperatures in interacting 2D electron systems, diverging from non-interacting behavior.
• The paper employs determinantal quantum Monte Carlo simulations with extensive disorder averaging to analyze spin correlations and finite-size effects in the Anderson-Hubbard model.
• The paper reveals that the ferromagnetic enhancement is not driven by local moment formation or increased density of states, challenging conventional Stoner and single-particle theories.

Magnetic Susceptibility in the 2D Anderson-Hubbard Model with Disorder and Interaction

Introduction

The study investigates the magnetic properties of strongly correlated electrons in the presence of quenched disorder on a two-dimensional lattice, using determinantal quantum Monte Carlo (DQMC) simulations of the Anderson-Hubbard model. The principal focus is the interplay between electronic correlations (parametrized by on-site interaction $U$) and disorder strength ($\Delta$) and their impact on both ferromagnetic (FM) and antiferromagnetic (AF) susceptibilities. The analysis is motivated by unresolved questions surrounding the stability and enhancement of magnetic order in systems where electron interactions coexist with random potential fluctuations, which are of central interest in condensed matter and cold atom physics, as well as for understanding quantum phase transitions in correlated electron systems.

Model and Methodology

The Hamiltonian considered consists of the standard Anderson-Hubbard model on a square lattice: $$H = -t \sum_{\langle ij \rangle, \sigma} c_{i\sigma}^\dagger c_{j\sigma} + \sum_{i\sigma} (\epsilon_i - \mu) n_{i\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow}$$ where $t$ is the hopping, $U$ the on-site repulsion, and $\epsilon_i$ are random on-site energies uniformly distributed within $[-\Delta/2, \Delta/2]$. The parameters $(U, \Delta, T)$ are varied, and simulations are performed at fixed fillings away from half-filling.

Key observables include the (equal-time) real-space spin-spin correlation function $C(r)$, its Fourier-transformed structure factor $S(q)$, and corresponding uniform ($q=0$) and AF ($q=(\pi,\pi)$) susceptibilities. Extensive disorder averaging is performed to ensure statistical accuracy, and lattice-size dependence is checked for robustness against finite-size effects.

Results

Non-Interacting Regime

For $U=0$, both FM and AF susceptibilities decrease monotonically with increasing disorder at all temperatures. The origin is the suppression of local moments and spin correlations due to electrons localizing at random potential minima.

Interacting Regime

A key finding is that with $U>0$, at sufficiently low temperatures, an increase in disorder leads to a robust enhancement of the uniform (FM) susceptibility $\chi_F$. This contrasts with the strictly monotonic suppression observed in the non-interacting limit and is observed over a broad range of fillings and interaction strengths. The AF susceptibility $\chi_{AF}$, in contrast, is always suppressed by disorder, consistent with the established view that AFLRO is destabilized by random potential fluctuations away from half-filling.

The enhancement of $\chi_F$ with increasing $\Delta$ emerges only at low $T$, and at higher temperatures, the behavior reverts to disorder suppression. Notably, this FM susceptibility enhancement is not accompanied by an enhancement of the local moment (which monotonically decreases with $\Delta$), nor is it linked to an increase of the density of states at the Fermi level—which is demonstrated to decrease with disorder as shown by the charge susceptibility.

Connection to the Metal-Insulator Transition

A direct analysis of $\chi_F$ near the disorder-driven metal-to-Anderson insulator transition reveals no singular behavior or enhancement in proximity to the quantum critical point. The enhancement is not an indicator of criticality at the MIT, but a generic low-$T$ feature of the interplay between interactions and disorder.

Stoner Criterion and Localization

The observed FM enhancement cannot be explained by the Stoner criterion, as increasing $\Delta$ suppresses $N(E_F)$. Nor can it be attributed to a proliferation of local moments, which decrease with $\Delta$. Furthermore, the effect is found in both metallic and Anderson-insulating phases and has been observed in DMFT calculations in infinite dimensions, indicating that Anderson localization is not its origin.

The charge and spin sectors are demonstrated to decouple in their disorder response. While the charge compressibility and conductivity are monotonically suppressed by disorder, spin susceptibility shows nontrivial, disorder-induced enhancement, highlighting the nuanced many-body effects in the spin channel that are absent in single-particle formulations.

Implications

These results provide numerical evidence for a disorder-driven enhancement of FM susceptibility at low $T$ in correlated electron systems. This effect is not tied to traditional mechanisms such as local moment formation, Stoner instability, or transitions at the metal-insulator boundary. Instead, it implies a distinct low-temperature many-body mechanism where interactions and disorder conspire to increase FM correlations despite suppressed local moments and charge response.

From a theoretical standpoint, this challenges the adequacy of mean-field and single-particle approaches, necessitating explicit non-perturbative consideration of both disorder and interaction. It also calls for a revision of "standard wisdom" that disorder generically suppresses magnetism in correlated electron systems—at least for the FM channel under certain thermodynamic conditions.

On the experimental side, the results suggest that enhanced FM fluctuations should be a generic feature of disordered, strongly correlated two-dimensional electron systems at low temperatures, with possible relevance to quantum critical phenomena in semiconductor heterostructures and ultracold atomic systems with engineered disorder and interactions.

Future Directions

Further investigation is warranted to quantitatively characterize the mechanism responsible for the FM enhancement, including detailed analysis of real- and momentum-space spin correlations, entanglement structure, and relevant collective excitations. Theoretical extensions could consider longer-range interactions, dimensional crossovers, or inclusion of spin-orbit coupling. Experimentally, targeted studies in tunable cold atom or solid-state platforms could aim to observe this anomalous enhancement of FM susceptibility.

The results underscore the importance of advanced many-body numerical methods for exploring subtle quantum effects at the intersection of disorder and strongly correlated physics, and they highlight open questions regarding the low-energy collective phenomena and possible quantum ordered states in disordered, interacting electron systems.

Conclusion

This study provides strong numerical evidence that, in two-dimensional correlated electron systems, disorder at low temperatures can enhance the ferromagnetic susceptibility in a regime inaccessible to single-particle or mean-field physics. This enhancement is distinct from both local moment and Stoner mechanisms and is not directly linked to the metal-insulator transition. The phenomenon broadens the conceptual landscape for quantum magnetism in disordered materials, emphasizing the need for further theoretical and experimental efforts to elucidate the underlying many-body mechanism and its consequences for quantum criticality and electronic phase behavior (1108.1078).