arXiv subjects

Martin R. Zirnbauer

Publications and source records attributed to Martin R. Zirnbauer.

At least 19 recordsLinked to original sources

arXiv2026

Multifractality at the Integer Quantum Hall Transition: Acquittal of the Parabolic Law

Stationary wave functions at the integer quantum Hall transition are known to be multifractal, but the exact form of the multifractality spectrum has remained a subject of debate. While conformal field theory arguments predict a parabolic law, numerical simulations show deviations from parabolicity. We resolve this discrepancy by pointing out that powers of the local wave intensity fail to obey the Gaussian Free Field and Abelian Fusion Hypothesis assumed in earlier analysis. Rather, due to the non-Abelian nature of the underlying effective field theory, wave intensity correlators are dressed by insertions of background charge distributed uniformly in space. An exact expression for the $q$-moments of point-contact generated eigenstates is presented. Numerical tests are performed for the critical Chalker-Coddington network model on a rectangular torus. Our results are in precise agreement with the predictions of conformal symmetry realized as a level-4 current algebra.

cond-mat.dis-nn↗

arXiv2024

On the infrared limit of the O(3) nonlinear $σ$-model at $θ= π$

2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.

hep-th↗

arXiv2023

Wegner model in high dimension: U(1) symmetry breaking and a non-standard phase of disordered electronic matter, I. One-replica theory

The Anderson transition between localized and metallic states is traditionally analyzed by assuming a one-parameter scaling hypothesis. Although that hypothesis has been confirmed near two dimensions by epsilon = d-2 expansion of the Wegner-Efetov nonlinear sigma model, there exists mounting evidence that the transition in d=3 or higher may have a second branch and that two relevant parameters are needed in order to describe the universal behavior at criticality. Doubt of the standard hypothesis also comes from field theory. Indeed, increasing the space dimension pushes the Anderson transition towards strong disorder, where a strong-coupling approach very different from the usual weak-coupling analysis of the sigma model is called for. In the present work, we develop a novel field theory of Anderson transitions at strong coupling based on the key observation that the U(1) symmetry which distinguishes retarded from advanced fields may undergo spontaneous symmetry breaking. That symmetry breakdown splits the sigma model coupling into two, thus leading to a natural scenario of two-parameter scaling. While we develop the field theory from the concrete starting point of the Wegner N-orbital model, we believe our results to be of much wider applicability. The first of a series, the present paper offers a pedagogical introduction to the main ideas in the setting of the one-replica theory. Subsequent papers will employ the self-consistent approximation of Abou-Chacra et al. and develop the full supersymmetric theory. The latter establishes the existence of a new renormalization-group fixed point, whose basin of attraction constitutes a third phase of disordered electronic matter.

cond-mat.dis-nn↗

arXiv2021

Color-Flavor Transformation Revisited

The "color-flavor transformation", conceived as a kind of generalized Hubbard-Stratonovich transformation, is a variant of the Wegner-Efetov supermatrix method for disordered electron systems. Tailored to quantum systems with disorder distributed according to the Haar measure of a compact Lie group of any classical type (A, B, C, or D), it has been applied to Dyson's Circular Ensembles, random network models, disordered Floquet dynamical systems, quantum chaotic graphs, and more. We review the method and, in particular, explore its limits of validity. An application to O(N)-Haar expectations of ratios of random characteristic polynomials is given. We also sketch a novel method to treat models where the color-flavor transformation fails.

math-ph↗

arXiv2021

Marginal CFT perturbations at the integer quantum Hall transition

According to recent arguments by the author, the conformal field theory (CFT) describing the scaling limit of the integer quantum Hall plateau transition is a deformed level-4 Wess-Zumino-Novikov-Witten model with Riemannian target space inside a complex Lie supergroup GL. After a summary of that proposal and some of its predictions, the leading irrelevant and relevant perturbations of the proposed CFT are discussed. Argued to be marginal, these result in a non-standard renormalization group (RG) flow near criticality, which calls for modified finite-size scaling analysis and may explain the long-standing inability of numerical work to reach agreement on the values of critical exponents. The technique of operator product expansion is used to compute the RG-beta functions up to cubic order in the couplings. The mean value of the dissipative conductance at the RG-fixed point is calculated for a cylinder geometry with any aspect ratio.

math-ph↗

arXiv2020

Hilbert-space geometry of random-matrix eigenstates

The geometry of multi-parameter families of quantum states is important in numerous contexts, including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characterization of quantum critical points. Here, we discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles, deriving the full probability distribution of the quantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relations to Levy stable distributions and compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.

cond-mat.dis-nn↗

arXiv2020

Particle-Hole Symmetries in Condensed Matter

The term "particle-hole symmetry" is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of "particle-hole conjugation" as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle-hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle-hole conjugation with one or another involution that reverses the sign of the first-quantized Hamiltonian. We illustrate that construction principle with a series of examples including the Su-Schrieffer-Heeger model and the Kitaev-Majorana chain. For an enhanced perspective, we contrast particle-hole symmetries with the charge-conjugation symmetry of relativistic Dirac fermions. We go on to present two major applications in the realm of interacting electrons. For one, we argue that the celebrated Haldane phase of antiferromagnetic quantum spin chains is adiabatically connected to a free-fermion topological phase protected by a particle-hole symmetry. For another, we review the recent proposal by Son for a particle-hole conjugation symmetric effective field theory of the half-filled lowest Landau level, and we comment on the emerging microscopic picture of the composite fermion.

math-ph↗

arXiv2019

Bulk-boundary correspondence for disordered free-fermion topological phases

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture, along with a boundary class, using Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $KR$-theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the "strong" and the "weak" invariants.

math-ph↗

arXiv2019

The integer quantum Hall plateau transition is a current algebra after all

The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model. Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4. Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be q(1-q)/4. The Lagrangian of the RG fixed-point theory for r retarded and r advanced replicas is proposed to be the GL(r|r)_4 Wess-Zumino-Witten model deformed by a truly marginal perturbation. The latter emerges from the non-linear sigma model by a natural scenario of spontaneous symmetry breaking.

math-ph↗

arXiv2015

Of symmetries, symmetry classes, and symmetric spaces: from disorder and quantum chaos to topological insulators

Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.

cond-mat.dis-nn↗

arXiv2015

Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems

Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a "Bott clock" topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K-theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d-dimensional system in symmetry class s to the ground state of a (d+1)-dimensional system in symmetry class s+1. This relation gives a new vantage point on topological insulators and superconductors.

math-ph↗

arXiv2012

Supersymmetry approach to Wishart correlation matrices: Exact results

We calculate the `one-point function', meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No explicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation matrices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation matrices.

math.ST↗

arXiv2010

Symmetry Classes

Physical systems exhibiting stochastic or chaotic behavior are often amenable to treatment by random matrix models. In deciding on a good choice of model, random matrix physics is constrained and guided by symmetry considerations. The notion of 'symmetry class' (not to be confused with 'universality class') expresses the relevance of symmetries as an organizational principle. Dyson, in his 1962 paper referred to as the Threefold Way, gave the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. In this article we review Dyson's Threefold Way from a modern perspective. We then describe a minimal extension of Dyson's setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, where Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.

math-ph↗

arXiv2009

Chevalley's restriction theorem for reductive symmetric superpairs

Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k+k,k) with the flip involution where k is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley's restriction theorem due to Sergeev and Kac, Gorelik. For general symmetric superpairs, the invariants exhibit a new and surprising behaviour. We illustrate this phenomenon by a detailed discussion in the example g=C(q+1)=osp(2|2q,C), endowed with a special involution. In this case, the invariant algebra defines a singular algebraic curve.

math.RT↗

arXiv2009

Quasi-diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a `diffusive' phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green's functions and a family of Ward identities coming from internal supersymmetry.

math-ph↗

arXiv2007

Autocorrelation of ratios of L-functions

We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of L-functions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.

math.NT↗

arXiv2004

The supersymmetry method of random matrix theory

The supersymmetry method pioneered by Wegner and Efetov (1979-1983) is reviewed for the case of Hermitian band random matrices, with special attention given to the original method of Schaefer and Wegner and a recent variant due to Fyodorov. Particular emphasis is placed on correctness of the mathematical reasoning.

math-ph↗

arXiv2004

Symmetry classes in random matrix theory

Dyson's (1962) classification of matrix ensembles is reviewed from a modern perspective, and its recent extension to disordered fermion problems is motivated and described. It is explained in particular why symmetry classes are associated with large families of symmetric spaces.

math-ph↗