Renormalization-group fate of the localization nonlinear $\sigma$ model for extreme long-range power-law correlated disorder ($m\le\epsilon$ in $d=2+\epsilon$)

For wave or electron propagation in a random potential with power-law spatial correlations $B(x)=\gamma^{2}(x^{2}+a^{2})^{-m}$ , John and Stephen (Phys. Rev. B 28, 6358 (1983)) showed within a replica field theory that, in $d=2+\epsilon$ dimensions, the localization transition is still governed by the standard $K^{2}$ nonlinear $\sigma$ model with the same critical exponents as for uncorrelated disorder — but only provided $m>\epsilon$ . For the excluded regime $m\le\epsilon$ , the higher-order gradient terms generated in the fluctuation expansion about the saddle point may cease to be irrelevant under the renormalization group, and the fate of the $\sigma$ -model description was explicitly left open. This problem asks for the controlled field-theoretic treatment of that extreme long-range regime: the boundary in $(m,d)$ at which higher-order gradient terms become relevant, the resulting effective action and renormalization-group flows, and whether the lower critical dimension and the critical exponents of the localization transition are modified.

Background

Anderson localization is the suppression of wave diffusion by multiple coherent scattering: in a disordered medium the eigenmodes of a wave or Schrödinger equation decay exponentially with a finite localization length $\xi$ , rather than extending across the sample. In the scaling picture the transition between extended and localized states is controlled by the dimensionless conductance $\Sigma$ , and there is a mobility edge — a frequency (or energy) separating localized from extended states — in $d>2$ , while all states are localized for $d\le 2$ (lower critical dimension 2). A standard microscopic route to this criticality is the replica field theory: the disorder average of Green's functions is performed by introducing $n$ copies (replicas) of the system, expanding the fluctuation fields about a replica-diagonal saddle point (which reproduces the coherent-potential approximation (CPA) self-energy, a self-consistent single-site effective-medium estimate of the scattering rate), and integrating out massive fluctuation modes. What survives at long wavelength is a nonlinear $\sigma$ model: a field $Q$ constrained to a saddle-point manifold (pseudo-orthogonal rotations mixing retarded and advanced replica sectors), whose action in momentum $K$ begins as $\Sigma(\omega)\int d^{d}K\,K^{2}Q_{0}(K)Q_{0}(-K)$ , where the coupling $\Sigma(\omega)=D(\omega)\rho(\omega)$ is the conductance (diffusivity times density of states). The zero mode of this action (the $l=0$ mode in the labeling of the fluctuation spectrum) is the diffusive Goldstone mode — the massless mode arising from spontaneously broken retarded-advanced symmetry (energy conservation). The renormalization group (RG) analysis of this $\sigma$ model in $d=2+\epsilon$ dimensions reproduces the localization transition: whether a term in the effective action is relevant (grows under coarse-graining and must be kept) or irrelevant (dies out) is decided by power counting of its momentum dependence.

John and Stephen (1983) applied this machinery to a scalar wave medium whose random index of refraction has correlations of range $a$ decaying either exponentially or as a power law $B(x)=\gamma^{2}(x^{2}+a^{2})^{-m}$ , and showed that, beyond the usual diffusive Goldstone mode, the fluctuation spectrum contains discrete nondiffusive modes associated with approximate wave-vector conservation. When these modes are separated from the Goldstone sector by a finite gap, the low-energy action is the same $K^{2}$ nonlinear $\sigma$ model as for uncorrelated disorder. Consequently, in $d=2+\epsilon$ , the mobility edge and its exponents (localization length $\xi\sim|\omega-\omega_{*}|^{-1/\epsilon}$ , dc conductance vanishing continuously at the mobility edge) coincide with the uncorrelated case, and all states are localized for $d\le 2$ — but, as the published abstract states explicitly, only "provided $m>\epsilon$ ". The reason is that the derivation of the effective action retained only the quadratic-gradient term $(\nabla Q)^{2}$ ; for extremely slowly decaying correlations ( $m\le\epsilon$ ) the higher-order gradient terms generated by expanding about the saddle point may cease to be RG-irrelevant, and nonanalytic low- $K$ dependence or additional relevant operators may enter. That this extreme long-range regime can genuinely change critical behavior is known from the analogous classical problem: Weinrib and Halperin (Phys. Rev. B 27, 413 (1983)) showed that quenched disorder correlated as $x^{-a}$ with $a<d$ alters critical phenomena through a new long-range-disorder fixed point with correlation-length exponent $\nu=2/a$ , replacing the Harris criterion $\nu\ge 2/d$ ; later functional-RG work confirmed this for $O(N)$ models with $1/x^{d-\sigma}$ correlated random fields. Whether an analogous modification occurs for the localization transition of the $\sigma$ model — as opposed to classical spin systems — was left unresolved by the 1983 wave-localization analysis.

Problem Statement

For wave (or electron) propagation in a random potential with stationary power-law spatial correlations $B(x)=\gamma^{2}(x^{2}+a^{2})^{-m}$ (scattering strength $\gamma$ , correlation length $a$ , exponent $m>0$ ), analyzed in $d=2+\epsilon$ dimensions via the replica field theory whose low-energy description is, for $m>\epsilon$ , the $K^{2}$ nonlinear $\sigma$ model with coupling equal to the conductance: determine the theory in the extreme long-range regime $m\le\epsilon$ that John and Stephen (1983) left open. Specifically: (1) establish the precise boundary in $(m,d)$ space at which the higher-order gradient terms generated by the fluctuation expansion about the saddle point cease to be renormalization-group irrelevant; (2) for $m\le\epsilon$ , construct the modified low-energy effective action — determining which (possibly several, possibly none) of the following appear: nonanalytic low- $K$ terms (e.g. momentum dependences weaker than $K^{2}$ ), higher-order gradient terms $K^{2n}$ , and additional relevant operators, and whether the truncation to the $l=0$ (Goldstone) sector remains legitimate given that the spectral gap separating it from the nondiffusive bound-state modes is expected to close in the extreme long-range limit; (3) derive the resulting renormalization-group flows; and (4) determine whether the $K^{2}$ nonlinear $\sigma$ model continues to control criticality or whether the lower critical dimension and the critical exponents (localization-length exponent at the mobility edge, conductance exponent) are modified, characterizing the critical behavior in this extreme long-range regime. Items (1)-(4) are the successive steps of one coherent objective: a controlled field-theoretic characterization of the localization transition in the extreme long-range correlated regime.

References

Sajeev John and Michael J. Stephen, "Wave propagation and localization in a long-range correlated random potential", Phys. Rev. B 28, 6358 (1983), pages 6358-6368. DOI: 10.1103/PhysRevB.28.6358, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.28.6358
Abel Weinrib and B. I. Halperin, "Critical phenomena in systems with long-range-correlated quenched disorder", Phys. Rev. B 27, 413 (1983). DOI: 10.1103/PhysRevB.27.413, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.27.413
Francisco A. B. F. de Moura and Marcelo L. Lyra, "Delocalization in the 1D Anderson Model with Long-Range Correlated Disorder", Phys. Rev. Lett. 81, 3735 (1998). DOI: 10.1103/PhysRevLett.81.3735, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.3735
F. A. B. F. de Moura, M. D. Coutinho-Filho, M. L. Lyra, E. P. Raposo, "Delocalization and ballistic dynamics in the two-dimensional Anderson model with long-range correlated disorder", Europhys. Lett. 66, 585 (2004). DOI: 10.1209/epl/i2003-10238-4, https://doi.org/10.1209/epl/i2003-10238-4
I. F. dos Santos, F. A. B. F. de Moura, M. L. Lyra, M. D. Coutinho-Filho, "Critical behavior of the two-dimensional Anderson model with long-range correlated disorder", J. Phys.: Condens. Matter 19, 476213 (2007). DOI: 10.1088/0953-8984/19/47/476213, https://doi.org/10.1088/0953-8984/19/47/476213
Andrei A. Fedorenko and Florian Kühnel, "Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study", Phys. Rev. B 75, 174206 (2007). DOI: 10.1103/PhysRevB.75.174206, https://doi.org/10.1103/PhysRevB.75.174206, arXiv: cond-mat/0701256, https://arxiv.org/abs/cond-mat/0701256
Simon Thébaud, Lucas Lindsay, and Tom Berlijn, "Breaking Rayleigh's law with spatially correlated disorder to control phonon transport", Phys. Rev. Lett. 131, 026301 (2023). DOI: 10.1103/PhysRevLett.131.026301, https://doi.org/10.1103/PhysRevLett.131.026301, arXiv: 2308.10836, https://arxiv.org/abs/2308.10836

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Renormalization-group fate of the localization nonlinear $\sigma$ model for extreme long-range power-law correlated disorder ( $m\le\epsilon$ in $d=2+\epsilon$ )

Background

Problem Statement

References