arXiv subjects

Ilya A. Gruzberg

Publications and source records attributed to Ilya A. Gruzberg.

At least 19 recordsLinked to original sources

arXiv2026

Boundary critical behavior of the Gross-Neveu-Yukawa model

We study the critical behavior of the semi-infinite Gross-Neveu-Yukawa model, a quantum field theory describing Dirac fermions interacting with bosonic fields via a Yukawa coupling. We consider Neumann and Dirichlet boundary conditions for the bosonic fields, and the most general boundary conditions for the fermions that preserve unitarity, conformal invariance, and charge conjugation symmetry. We analyze the phase diagram and identify distinct fixed points corresponding to different universality classes of boundary critical behavior. The associated boundary critical exponents, which govern the scaling behavior and crossover phenomena, are computed to one-loop order. We also discuss the relevance of our results to the semi-infinite pseudoscalar Yukawa model.

hep-th↗

arXiv2025

Localization-landscape generalized Mott-Berezinskiĭ formula

We introduce a conceptual reformulation of the Mott-Berezinskiĭ (MB) theory of low-frequency AC conductivity in disordered systems based on localization landscape theory. Instead of assuming uniform localization and fixed hopping distances, transport is described through an effective potential whose geometry encodes the spatial organization and energy-dependent localization of quantum states. Using the associated Agmon metric, we define a generalized Mott scale that replaces the classical hopping length with a geometric criterion set by the disorder landscape. This framework naturally incorporates strong spatial inhomogeneity and yields the AC conductivity directly from the effective potential. The standard MB result is recovered as a limiting case. Our approach extends the conceptual foundation of MB theory to arbitrary disordered media and energies approaching the mobility edge, providing a unified description of AC transport in complex quantum materials.

cond-mat.dis-nn↗

arXiv2024

Surface quantum critical phenomena in disordered Dirac semimetals

We study a non-Anderson disorder driven quantum phase transition in a semi-infinite Dirac semimetal with a flat boundary. The conformally invariant boundary conditions, which include those that are time-reversal invariant, lead to nodal-like surface states on the boundary. In this case the boundary becomes metallic at a critical disorder that is weaker than that for the semimetal-diffusive metal transition in the bulk. The latter transition takes place in the presence of a metallic surface; in the language of surface critical phenomena this corresponds to the so-called extraordinary transition. The lines of the surface and the extraordinary transitions meet at the special transition point. To elucidate universal properties at different transitions on the phase diagram, we employ renormalization group methods and compute the corresponding surface critical exponents using $\varepsilon$-expansion.

cond-mat.dis-nn↗

arXiv2024

Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions

Multifractals arise in various systems across nature whose scaling behavior is characterized by a continuous spectrum of multifractal exponents $Δ_q$. In the context of Anderson transitions, the multifractality of critical wave functions is described by operators $O_q$ with scaling dimensions $Δ_q$ in a field-theory description of the transitions. The operators $O_q$ satisfy the so-called Abelian fusion expressed as a simple operator product expansion. Assuming conformal invariance and Abelian fusion, we use the conformal bootstrap framework to derive a constraint that implies that the multifractal spectrum $Δ_q$ (and its generalized form) must be quadratic in its arguments in any dimension $d \geq 2$.

cond-mat.dis-nn↗

arXiv2023

Generalized multifractality in 2D disordered systems of chiral symmetry classes

We study generalized multifractality that characterizes eigenstate fluctuations and correlations in disordered systems of chiral symmetry classes AIII, BDI, and CII. By using the non-linear sigmamodel field theory, we construct pure-scaling composite operators and eigenfunction observables that satisfy Abelian fusion rules. The observables are labelled by two multi-indices $λ$, $λ'$ referring to two sublattices, at variance with other symmetry classes, where a single multi-index $λ$ (that can be viewed as a generalized version of a Young diagram) is needed. Further, we analyze Weyl symmetries of multifractal exponents, which are also peculiar in chiral classes, in view of a distinct root system associated with the sigma-model symmetric space. The analytical results are supported and complemented by numerical simulations that are performed for a 2D lattice Hamiltonian of class AIII, both in the metallic phase and at the Anderson transition. Both in the metallic phase and at the transition, the numerically obtained exponents satisfy Weyl symmetry relations, confirming that the sigma-model is the right theory of the problem. Furthermore, in the metallic phase, we observe the generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator), as expected in the one-loop approximation. On the other hand, the generalized parabolicity is strongly violated at the metal-insulator transition, implying violation of local conformal invariance.

cond-mat.dis-nn↗

arXiv2022

Metal-insulator transition in a 2D system of chiral unitary class

We perform a numerical investigation of Anderson metal-insulator transition (MIT) in a twodimensional system of chiral symmetry class AIII by combining finite-size scaling, transport, density of states, and multifractality studies. The results are in agreement with the sigma-model renormalization-group theory, where MIT is driven by proliferation of vortices. We determine the phase diagram and find an apparent non-universality of several parameters on the critical line of MIT, which is consistent with the analytically predicted slow renormalization towards the ultimate fixed point of the MIT. The localization-length exponent $ν$ is estimated as $ν= 1.55 \pm 0.10$.

cond-mat.dis-nn↗

arXiv2022

Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Andersonlocalization transitions are considered. By using the non-linear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signalling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related with a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.

cond-mat.dis-nn↗

arXiv2022

Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables

This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al, Annals of Physics, 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling exponents. In particular, all exponents corresponding to polynomial pure-scaling observables up to the order $q=5$ are found numerically. For the observables for which the percolation mapping is derived, analytical and numerical results are in perfect agreement with each other. The analytical and numerical results unambiguously demonstrate that the generalized parabolicity (i.e., proportionality to eigenvalues of the quadratic Casimir operator) does not hold for the spectrum of generalized-multifractality exponents. This excludes Wess-Zumino-Novikov-Witten models, and, more generally, any theories with local conformal invariance, as candidates for the fixed-point theory of the spin quantum Hall transition. The observable construction developed in this work paves a way to investigation of generalized multifractality at Anderson-localization critical points of various symmetry classes.

cond-mat.dis-nn↗

arXiv2022

The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap

This paper studies the critical behavior of the 3d classical $\mathrm{O}(N)$ model with a boundary. Recently, one of us established that upon treating $N$ as a continuous variable, there exists a critical value $N_c > 2$ such that for $2 \leq N < N_c$ the model exhibits a new extraordinary-log boundary universality class, if the symmetry preserving interactions on the boundary are enhanced. $N_c$ is determined by a ratio of universal amplitudes in the normal universality class, where instead a symmetry breaking field is applied on the boundary. We study the normal universality class using the numerical conformal bootstrap. We find truncated solutions to the crossing equation that indicate $N_c \approx 5$. Additionally, we use semi-definite programming to place rigorous bounds on the boundary CFT data of interest to conclude that $N_c > 3$, under a certain positivity assumption which we check in various perturbative limits.

cond-mat.stat-mech↗

arXiv2021

Generalized multifractality at spin quantum Hall transition

Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C. Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then "translate" them to the language of eigenstate observables. Performing numerical simulations on network models for SQH and IQH transitions, we confirm the analytical predictions for scaling observables and determine the corresponding exponents. Remarkably, the generalized-multifractality exponents at the SQH critical point strongly violate the generalized parabolicity of the spectrum, which implies violation of the local conformal invariance at this critical point.

cond-mat.dis-nn↗

arXiv2021

Multicriticality of Two-dimensional Class D Disordered Topological Superconductors

A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We characterize the critical behaviors at various critical points and the tricritical point using numerical evaluation of the localization length, the conductance (or conductivity), and the density of states. We conclude that the two metal-insulator transitions (DTM-TQH and DTM-AI) belong to the same universality class, whereas the tricritical point (TCP) represents a distinct universality class.

cond-mat.dis-nn↗

arXiv2021

Criticality of two-dimensional disordered Dirac fermions in the unitary class and universality of the integer quantum Hall transition

Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E=0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m=0. This observation lead Ludwig et al [Phys. Rev. B 50, 7526 (1994)] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal critical properties. Given the far-reaching implications for the emerging field of the quantum anomalous Hall effect, modern condensed matter physics and our general understanding of disordered critical points, it is surprising that this conjecture has never been tested numerically. Here, we report the results of extensive numerics on the phase diagram and criticality of 2D-DDF in the unitary class. We find a critical line at m=0, with energy-dependent localization length exponent. At large energies, our results for the DDF are consistent with state-of-the-art numerical results ν_IQH = 2.56 - 2.62 from models of the IQHT. At E=0 however, we obtain ν_0 =2.30-2.36 incompatible with ν_IQH. This result challenges conjectured relations between different models of the IQHT, and several interpretations are discussed.

cond-mat.mes-hall↗

arXiv2018

Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law

In one-dimensional disordered wires electronic states are localized at any energy. Correlations of the states at close positive energies and the AC conductivity $σ(ω)$ in the limit of small frequency are described by the Mott-Berezinskii theory. We revisit the instanton approach to the statistics of wave functions and AC transport valid in the tails of the spectrum (large negative energies). Applying our recent results on functional determinants, we calculate exactly the integral over gaussian fluctuations around the exact two-instanton saddle point. We derive correlators of wave functions at different energies beyond the leading order in the energy difference. This allows us to calculate corrections to the Mott-Berezinskii law (the leading small frequency asymptotic behavior of $σ(ω)$) which approximate the exact result in a broad range of $ω$. We compare our results with the ones obtained for positive energies.

cond-mat.dis-nn↗

arXiv2017

On functional determinants of matrix differential operators with multiple zero modes

We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of $r\times r$ matrix second order differential operators $O$ with $0 < n \leqslant 2r$ linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed $r$, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by $2r$. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the $n$ zero modes and other $r-n$ or $2r-n$ (depending on the boundary conditions) solutions of the homogeneous equation $O h=0$, in the spirit of Gel'fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.

cond-mat.dis-nn↗

arXiv2013

Statistics of Conductances and Subleading Corrections to Scaling near the Integer Quantum Hall Plateau Transition

We study the critical behavior near the integer quantum Hall plateau transition by focusing on the multifractal (MF) exponents $X_q$ describing the scaling of the disorder-average moments of the point contact conductance $T$ between two points of the sample, within the Chalker-Coddington network model. Past analytical work has related the exponents $X_q$ to the MF exponents $Δ_q$ of the local density of states (LDOS). To verify this relation, we numerically determine the exponents $X_q$ with high accuracy. We thereby provide, at the same time, independent numerical results for the MF exponents $Δ_q$ for the LDOS. The presence of subleading corrections to scaling makes such determination directly from scaling of the moments of $T$ virtually impossible. We overcome this difficulty by using two recent advances. First, we construct pure scaling operators for the moments of $T$ which have precisely the same leading scaling behavior, but no subleading contributions. Secondly, we take into account corrections to scaling from irrelevant (in the renormalization group sense) scaling fields by employing a numerical technique ("stability map") recently developed by us. We thereby numerically confirm the relation between the two sets of exponents, $X_q$ (point contact conductances) and $Δ_q$ (LDOS), and also determine the leading irrelevant (corrections to scaling) exponent $y$ as well as other subleading exponents. Our results suggest a way to access multifractality in an experimental setting.

cond-mat.mes-hall↗

arXiv2012

Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition

We present a numerical finite size scaling study of the localization length in long cylinders near the integer quantum Hall transition (IQHT) employing the Chalker-Coddington network model. Corrections to scaling that decay slowly with increasing system size make this analysis a very challenging numerical problem. In this work we develop a novel method of stability analysis that allows for a better estimate of error bars. Applying the new method we find consistent results when keeping second (or higher) order terms of the leading irrelevant scaling field. The knowledge of the associated (negative) irrelevant exponent $y$ is crucial for a precise determination of other critical exponents, including multifractal spectra of wave functions. We estimate $|y| > 0.4$, which is considerably larger than most recently reported values. Within this approach we obtain the localization length exponent $2.62 \pm 0.06$ confirming recent results. Our stability analysis has broad applicability to other observables at IQHT, as well as other critical points where corrections to scaling are present.

cond-mat.dis-nn↗

arXiv2012

Exact exponents for the spin quantum Hall transition in the presence of multiple edge channels

Critical properties of quantum Hall systems are affected by the presence of extra edge channels - present, in particular, at higher plateau transitions. We study this phenomenon for the case of the spin quantum Hall transition. Using supersymmetry we map the corresponding network model to a classical loop model, whose boundary critical behavior was recently determined exactly. We verify predictions of the exact solution by extensive numerical simulations.

cond-mat.mes-hall↗

arXiv2010

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

We generalize universal relations between the multifractal exponent α_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude Λ_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude Λ_c^o is universally related to the boundary multifractal exponent α_0^s for the typical wave function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z_2 topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum Hall plateau transition, and (d) the spin quantum Hall plateau transition.

cond-mat.dis-nn↗