Nonperturbative strong-coupling structure of the replica-limit nonlinear sigma model for the two-dimensional symplectic Anderson transition

The two-dimensional symplectic Anderson transition — the metal–insulator transition of disordered 2D electrons with spin-orbit coupling and time-reversal symmetry (class AII) — has a nonlinear sigma-model description in which proliferation of $\mathbb{Z}_{2}$ vortices drives localization into trivial or topological insulating phases. Fu and Kane computed the critical conductivity and correlation-length exponent in an $N=1-\epsilon$ replica expansion, but the physical replica limit $N\to0$ at strong coupling, which fixes the renormalization-group phase diagram and the true universal critical data, remains undetermined: the sign of $\beta(t)$ along the vortex-free line, the existence of nonperturbative fixed points $t_{m}$ and $t_{s}$ (the latter possibly governing a direct topological-to-trivial-insulator transition), the fate of the first-order line found for $N>1$, and the reconciliation of the extrapolated $\sigma^{*}\sim2.5e^{2}/h$ and $\nu\sim1.4$ with the numerical $\sigma^{*}\approx1.4e^{2}/h$ and $\nu\approx2.7$ are all open. Later literature has confirmed the average-symmetry delocalization mechanism and the shared universality class of the two metal–insulator transitions, but no analytic determination of the strong-coupling fixed points or of the critical exponents exists.

Background

In a noninteracting two-dimensional electron system, quantum interference can localize all states for arbitrarily weak disorder (Anderson localization), and the one-parameter scaling theory predicts exactly this for spinless time-reversal-symmetric disorder. When strong spin-orbit coupling coexists with time-reversal symmetry the system belongs instead to the symplectic symmetry class (class AII of the Altland–Zirnbauer classification of disordered systems): weak antilocalization — the spin-orbit-induced interference correction that enhances rather than suppresses conductance — reverses the scaling tendency, a genuinely metallic phase becomes stable, and increasing disorder produces a metal–insulator transition, the two-dimensional symplectic Anderson transition. Its long-distance theory is a nonlinear sigma model (NL$\sigma$M), a field theory whose fields take values in the coset $O(2N)/O(N)\times O(N)$ (a target space that, once $\mathbb{Z}_{2}$ vortices are retained, effectively lifts to its double cover $SO(2N)/SO(N)\times SO(N)$), where $N$ is the number of fermion replicas used to represent disorder averaging and the physical theory is obtained in the replica limit $N\to0$. The renormalization-group (RG) beta function $\beta(t)=dt/d\ln L$ of the dimensionless conductance $t$ — which measures disorder strength through $\sigma=(2\pi t)^{-1}e^{2}/h$ and describes how $t$ flows with system size $L$ — begins $\beta(t)=2(N-1)t^{2}+\ldots$, which is positive for $N>1$ (weak coupling unstable, localized) and negative for $N<1$ (metal stable, weak antilocalization).

A second ingredient is topological. For the integer quantum Hall effect, Pruisken's NL$\sigma$M acquires a topological theta term, which weights field configurations by their winding number, and the proliferation of vortex-like instantons drives the plateau transition. Fu and Kane (2012) showed that in the symplectic class the relevant topological defects are $\mathbb{Z}_{2}$ vortices of the sigma-model field, and that their fugacity $v$ — the statistical weight with which a single vortex enters the disorder-averaged partition sum — must be kept as a second coupling of the theory: the sign of $v$ determines the $\mathbb{Z}_{2}$ topological class of the localized phase (trivial versus topological insulator), so vortex proliferation describes localization into either phase. They analyzed the theory in an $N=1-\epsilon$ expansion: at $N=1$ the vortex-free theory reduces to the XY model of a planar spin, whose vortex-unbinding transition is the Kosterlitz–Thouless (KT) transition (the vortex sector is equivalently described by a dual sine-Gordon model), and the critical points $(t^{*},\pm v^{*})$ governing the metal–insulator transitions are perturbatively connected to the KT critical point for small $\epsilon$. For $N>1$ they found that the $v=0$ line is a first-order transition between the two vortex phases, and conjectured that it does not persist to $N=0$.

The same vortex picture explains delocalized surface states of weak topological insulators (WTIs) and topological crystalline insulators. A symmetry that is broken by every disorder realization but restored by disorder averaging (an average, or statistical, symmetry) forbids isolated $\mathbb{Z}_{2}$ vortices: an odd-vorticity vortex binds zero modes that force the single-vortex fugacity to vanish, while for multiple vortices the zero modes are split by an amount exponentially small in their separation, which confines the vortices by a linear potential. Delocalization protected only on average is now a numerically well-confirmed mechanism.

Extrapolating the $N=1-\epsilon$ results to the physical value $\epsilon=1$, Fu and Kane obtained a critical conductivity $\sigma^{*}=(8/\pi)e^{2}/h\sim2.5e^{2}/h$ and a correlation-length exponent $\nu\sim1.4$ for the symplectic Anderson transition. These differ substantially from numerical transfer-matrix and network-model studies — the latter being simulations on directed lattice models with random scattering at the nodes, the standard discretization of a disordered continuum — which give $\sigma^{*}\approx1.4e^{2}/h$ and $\nu\approx2.7$ (Asada, Slevin and Ohtsuki 2002; Obuse, Furusaki, Ryu and Mudry 2007), while a metal–topological-insulator transition in a different model gave $\nu\approx1.6$ (Onoda, Avishai and Nagaosa 2007). The origin of the discrepancy depends on the behavior of the $N=0$ sigma model at strong coupling, which the $\epsilon$ expansion cannot access.

Problem Statement

Determine, nonperturbatively, the renormalization-group structure of the two-dimensional class-AII (symplectic) nonlinear sigma model in the replica limit $N\to0$, including the $\mathbb{Z}_{2}$-vortex fugacity $v$, and thereby the phase diagram and universal critical data of the two-dimensional symplectic Anderson transition. Specifically: (i) the sign and full functional form of $\beta(t)$ along the $v=0$ line for $N\to0$ — is $\beta(t)<0$ for all $t$, or does it change sign at a finite nonperturbative critical point $t_{m}$? (ii) Does a strong-coupling fixed point $t_{s}$ exist that controls a continuous direct transition between the trivial and the $\mathbb{Z}_{2}$ topological insulating phases, and what is the fate of the first-order transition found at $v=0$ for $N>1$ as $N\to0$ (conjectured by Fu and Kane not to persist to $N=0$, but never demonstrated)? (iii) What are the resulting universal critical conductivity $\sigma^{*}$ and correlation-length exponent $\nu$ of the symplectic Anderson transition — in particular, are the $N=1-\epsilon$ extrapolations $\sigma^{*}\sim2.5e^{2}/h$ and $\nu\sim1.4$ or the numerical values $\sigma^{*}\approx1.4e^{2}/h$ and $\nu\approx2.7$ correct, and does the discrepancy reflect slow convergence of the $\epsilon$ expansion or crossover effects associated with the putative strong-coupling fixed points $t_{m}$ and $t_{s}$? (iv) As a minor quantitative component of the same framework, compute the multi-vortex interaction — the linear confinement of $\mathbb{Z}_{2}$ vortices generated by the exponentially small splitting of bound zero modes — and its dependence on microscopic parameters, together with its role in finite-size crossover phenomena near the $v=0$ line. The four items are independent, so a solution that settles only a stated subset resolves that subset and leaves the remaining items open.

References

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