Do true metals exist in two dimensions? Thermodynamic fate of the interaction-delocalized disordered two-dimensional electron gas

Whether a disordered two-dimensional electron gas (2DEG) with Coulomb interactions can form a genuine thermodynamic metallic phase — one with a localization length that diverges in the infinite-size, zero-temperature limit — is the central unresolved question of the two-dimensional metal-insulator transition (2D MIT) field. The non-interacting scaling theory forbids such a metal, yet experiments since 1994 show robust metallic behavior, and the finite-size quantum Monte Carlo study of Fleury and Waintal (Phys. Rev. Lett. 101, 226803 (2008)) found that interactions delocalize multi-component (spin- and valley-degenerate, M=4) systems so strongly that the extracted inverse localization length vanishes within statistical resolution — while showing no deviation from one-parameter scaling, leaving the existence of a true 2D metal undecided. The task asks for the thermodynamic-limit resolution of this question, together with the linked issues that define what experiments actually probe: possible deviations from one-parameter scaling outside the numerically explored window, the modification of sample trajectories $\eta=r_s\sqrt{k_F l}$ by correlated disorder and screening, and the finite-temperature competition between interaction delocalization and hopping through localized excited states.

Background

In the scaling theory of localization (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979), the dimensionless conductance $g$ of a disordered electronic system changes with system size $L$ according to a scaling function $\beta(g)=d\log g/d\log L$ . In two dimensions (2D), and in the absence of spin-orbit coupling (the 'orthogonal' symmetry class), $\beta(g)$ is negative for arbitrarily weak disorder, so a non-interacting 2D electron gas is always Anderson-localized in the thermodynamic limit: its conductance decays as $g\propto e^{-L/\xi}$ with a finite localization length $\xi$ . This 'no metals in 2D' result was directly challenged in 1994 by transport experiments in silicon metal-oxide-semiconductor field-effect transistors (Si MOSFETs), in which the resistivity stopped decreasing and started increasing sharply as temperature was lowered below a density-dependent crossover (Kravchenko et al., 1994); similar apparent metallic behavior was subsequently found in many high-mobility 2D systems and is reviewed in Abrahams, Kravchenko and Sarachik (2001). The problem is theoretically hard because both disorder and electron-electron interactions must be treated non-perturbatively at low density. Two dimensionless parameters control the physics: $r_s$ , the ratio of Coulomb to Fermi kinetic energy (growing as density falls), and $1/k_F l$ , the disorder strength, where $k_F$ is the Fermi momentum and $l$ the mean free path; as the density $n_s$ decreases both grow in a fixed ratio $\eta=r_s\sqrt{k_F l}$ that characterizes a given sample. The number of internal components $M$ (spin, and valley degeneracy in Si-like materials) also matters: $M=2$ for GaAs heterostructures, $M=4$ for Si MOSFETs (spin $\times$ valley). Theoretical proposals for the apparent metal range from interaction-corrected scaling theories, which describe data through two-parameter scaling but contain no accepted stable 2D metallic fixed point (Punnoose and Finkel'stein, 2005; Anissimova et al., 2007), to phase-separation scenarios in which transport is carried by an intermediate state between a Fermi liquid and a Wigner crystal — the crystalline state of electrons stabilized by Coulomb repulsion — such as the electronic microemulsion of Spivak (2003). In this context, Fleury and Waintal (2008) performed the first quantitative calculation of the localization length of a disordered 2DEG including many-body correlations, using ground-state quantum Monte Carlo on a lattice model with Ewald-summed Coulomb interactions (a standard periodic-image lattice-sum treatment of the long-ranged interaction) combined with finite-size scaling (method in Fleury and Waintal, Phys. Rev. Lett. 100, 076602 (2008)). They found that correlations delocalize the system: for $M=2$ the effect is a finite increase of $\xi$ (still an insulator), while for $M=4$ the data flow to $\beta=0$ and the extracted $a/\xi$ vanishes within statistical resolution ( $a$ is the mean interparticle spacing), i.e. the system is 'practically' metallic at the sizes studied. Crucially, they observed no deviation from the non-interacting one-parameter scaling curve — meaning scaling is described by the single parameter $\xi(r_s,k_F l)$ — within the explored window $0.5<k_F l<4$ , $0\le r_s\le 10$ , so they could not conclude that the system becomes a 'true' metal in the thermodynamic sense (a divergence of $\xi$ ), and stated that $M=4$ systems 'are not "true" thermodynamic metals and would become insulating at ultrasmall temperature (2-C systems are unambiguously insulators)'. They further proposed that the observed metallic behavior of Si MOSFETs is a finite-temperature effect of interaction delocalization: the ground state is delocalized while the excited states (populated by temperature or spin polarization) are localized, which accounts semi-quantitatively for the resistivity increase with $T$ , its destruction by an in-plane magnetic field, and the special role of valley degeneracy; they noted that mapping experiments onto the $(r_s,k_F l)$ plane requires understanding correlated disorder (which raises $\eta$ at low density) and screening (which lowers the effective disorder). Note that the source title's phrase 'many-body localization' (2008) predates the modern many-body-localization literature and refers to localization properties of the interacting many-body ground state, not to the eigenstate-phase problem. More recent experiments have confirmed the persistence of metallic behavior in ultra-clean two-valley 2D systems even under full spin polarization (Melnikov et al., 2020), and the phenomenon, its Wigner-Mott interpretation, and the low-density phases are reviewed by Shashkin and Kravchenko (2021) — but no experimental or theoretical work has settled the zero-temperature, infinite-size question.

Problem Statement

Determine the thermodynamic-limit fate of the disordered, interacting, low-density two-dimensional electron gas in the orthogonal symmetry class: does the interaction-induced delocalization found at finite system sizes — strongest for multi-component systems with spin and valley degeneracy ( $M=4$ , as in Si MOSFETs), where the extracted inverse localization length vanishes within statistical resolution — extrapolate to a genuine metallic phase, i.e. a localization length $\xi(r_s,k_F l)$ that diverges as system size grows (in practice signaled by a metallic fixed point with $\beta\ge 0$ or by a demonstrated breakdown of the non-interacting one-parameter scaling curve), or does one-parameter scaling ultimately prevail so that every such system is an insulator in the zero-temperature thermodynamic limit (at ultrasmall temperature for $M=4$ ; unambiguously for $M=2$ )? This single overarching objective comprises the linked sub-questions that jointly characterize it — settling the thermodynamic-limit fate stated above is what completes the task, and (a)–(c) set out the linked issues that a resolution must address: (a) whether small but physically important deviations from one-parameter scaling exist outside the numerically explored window $0.5<k_F l<4$ , $0\le r_s\le 10$ ; (b) how correlated disorder and screening by the electrons modify the sample-specific trajectory $\eta=r_s\sqrt{k_F l}$ at low density, so that experimental systems can be mapped quantitatively onto the $(r_s,k_F l)$ phase diagram; and (c) the quantitative finite-temperature interplay between interaction delocalization of the ground state and the population of localized excited states, including variable-range hopping (thermally activated tunneling between distant localized states), which fixes what temperature window the observed 'metallic' behavior reflects. An answer may take the form of a controlled analytic theory establishing or excluding a stable 2D metallic fixed point, a numerical scaling study whose thermodynamic extrapolation decides the sign of $1/\xi$ at asymptotically large sizes, or an experimental determination reaching the relevant ultralow-temperature regime.

References

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