arXiv subjects

Alexander Altland

Publications and source records attributed to Alexander Altland.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Chaos with a Macroscopic Zero-Mode Sector

Chaotic many-body spectra are expected to densely fill their energy window. We show that constrained spin chains with chiral symmetry evade this expectation by hosting an exponentially large manifold of symmetry-protected exact zero modes separated from the surrounding spectrum by a sharp gap at zero energy. The gap is generated by chaotic level repulsion, with width set by the number of zero modes times the mean level spacing. We verify this mechanism in an East-West kinetically constrained chain, develop a minimal random-matrix description, and show how the gap can be detected through linear-response spectroscopy.

cond-mat.str-el↗

arXiv2026

Theory of Anderson localization on the hyperbolic plane

The two-dimensional hyperbolic plane, $\mathbb{H}^2$, is an unusual system in that dimensionality changes with scale: locally two-dimensional and planar at short distances, but effectively infinite-dimensional at large scales, it provides an interesting paradigm for the study of (quantum) phase transitions, notably the disorder-driven Anderson transition. Generalizing previous work, which treated short and large distance scales separately, we develop a unified framework interpolating between the principles of low- and high-dimensional Anderson localization. As a main result, we derive a two-parameter flow in a plane spanned by scale-dependent curvature (setting the system's effective dimensionality) and conductivity, with an extended critical line separating metallic and insulating phases.

cond-mat.dis-nn↗

arXiv2026

Quantum chaos and the holographic principle

Recent years have witnessed tremendous progress in developing a fine-grained low-dimensional holographic correspondence, specifically the construction of quantum mechanical boundary theories as holographic duals of two-dimensional gravity. In these developments, quantum chaos played a crucial role, both as source of universality and as a guiding principle for the matching of bulk and boundary signatures of gravity. In this article we review the construction of the chaos-assisted low-dimensional holographic correspondence for non-experts. We open with an introductory discussion of the two main protagonists of the theory, the SYK model and two-dimensional Jackiw-Teitelboim gravity. Within this framework we will discuss two independent 'bridges' between bulk and boundary physics, one pertaining to early time chaotic instabilities, the other to late time quantum chaos up to and including time scales of the order of the gravitational quantum level spacing. We will demonstrate that the resolution of these fine-grained quantum scales requires the extension of semiclassical gravity by elements of string theory. We conclude with an outlook towards higher dimensional generalizations of the chaotic holographic correspondence.

quant-ph↗

arXiv2026

Continuum field theory of matchgate tensor network ensembles

Tensor networks provide discrete representations of quantum many-body systems, yet their precise connection to continuum field theories remains relatively poorly understood. Invoking a notion of typicality, we develop a continuum description for random ensembles of two-dimensional fermionic matchgate tensor networks with spatially fluctuating parameters. As a diagnostic of the resulting universal physics, we analyze disorder-averaged moments of fermionic two-point functions, both in flat geometry and on a hyperbolic disk, where curvature reshapes their long-distance structure. We show that disorder drives universal long-distance behavior governed by a nonlinear sigma-model of symmetry class D with a topological term, placing random matchgate networks in direct correspondence with the thermal quantum Hall problem. The resulting phase structure includes localized phases, quantum Hall criticality, and a robust thermal metal with diffusive correlations and spontaneous replica-symmetry breaking. Weak non-Gaussian deformations reduce the symmetry to discrete permutations, generate a mass for the Goldstone modes, and suppress long-range correlations. In this way, typicality offers a route from ensembles of discrete tensor networks to continuum quantum field theories.

cond-mat.dis-nn↗

arXiv2025

Freeness Reined in by a Single Qubit

Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension $D0 \gg 1$. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order $O(1)$. These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to non-uniformly distributed stationary quantum states, which we characterize analytically and confirm numerically.

quant-ph↗

arXiv2025

Path integral approach to quantum thermalization

We introduce a quasiclassical Green function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear $\sigma$-model of disordered systems, the $G \Sigma$-formalism for strong correlations, and real time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective `Thouless time', up to the many-body Heisenberg time. We illustrate the approach with two case studies, (i) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries, and (ii) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.

cond-mat.mes-hall↗

arXiv2025

The universality class of the first levels in low-dimensional gravity

We investigate the physics of a small group of quantum states defined above the sharply defined ground state of a chaotic ensemble. This `universality class of the first levels' (UFL) is realized in the majority of `synthetic' random matrix models but, for all we know, in only one microscopically defined system: low-dimensional gravity. We discuss the physical properties of these states, notably their exceptional rigidity against external perturbations, as quantified by the so-called quantum state fidelity. Examining these structures through the lenses of random matrix and string theory, we highlight their relevance to the physics of low-dimensional holographic principles.

hep-th↗

arXiv2025

The chiral SYK model in three-dimensional holography

A celebrated realization of the holographic principle posits an approximate duality between the $(0+1)$-dimensional quantum mechanical SYK model and two-dimensional Jackiw-Teitelboim gravity, mediated by the Schwarzian action as an effective low energy theory common to both systems. We here propose a generalization of this correspondence to one dimension higher. Starting from different microscopic realizations of effectively chiral $(1+1)$-dimensional generalizations of the SYK model, we derive a reduction to the Alekseev-Shatashvilli (AS)-action, a minimal extension of the Schwarzian action which has been proposed as the effective boundary action of three-dimensional gravity. In the bulk, we show how the same action describes fluctuations around the Euclidean BTZ black hole configuration, the dominant stationary solution of three-dimensional gravity. These two constructions allow us to match bulk and boundary coupling constants, and to compute observables. Specifically, we apply semiclassical techniques inspired by condensed matter physics to the computation of out-of-time-order correlation functions (OTOCs), demonstrating maximal chaos in the chiral SYK chain and its gravity dual.

hep-th↗

arXiv2024

A minimal tensor network beyond free fermions

This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a tricritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.

cond-mat.str-el↗

arXiv2024

Disordered topological crystalline phases

The imposition of crystalline symmetries is known to lead to a rich variety of insulating and superconducting topological phases. These include higher-order topological phases and obstructed atomic limits with and without filling anomalies. We here comprehensively classify such topological crystalline phases (TCPs) with mirror, twofold rotation, and inversion symmetries in the presence of disorder that preserves the crystalline symmetry on average. We find that the inclusion of disorder leads to a simplification of the classification in comparison to the clean case. We also find that, while clean TCPs evade a general bulk-boundary principle, disordered TCPs admit a complete bulk-boundary correspondence, according to which (bulk) topological phases are topologically equivalent if and only if they have the same anomalous boundary states and filling anomaly. We corroborate the stability of disordered TCPs by way of field-theoretic, numerical and symmetry-based analyses in various case studies. While the boundary signatures of most disordered TCPs are similar to their clean counterparts, the addition of disorder to certain mirror-symmetric TCPs results in novel higher-order statistical topological phases, in which zero-energy hinge states have critical wavefunction statistics, while remaining protected from Anderson localization.

cond-mat.mes-hall↗

arXiv2024

Topology in the Random Scattering of Light

Light scattering in random media is usually considered within the framework of the three-dimensional Anderson universality class, with modifications for the vector nature of electromagnetic waves. We propose that the linear dispersiveness of light introduces topological aspects into the picture. The dynamics of electromagnetic waves follow the same differential equations as those of a spin-$1$ Weyl semimetal. In the presence of disorder, this equivalence leads to a range of phenomena explored in this paper. These include topological protection against localization when helicity hybridization is weak, the emergence of exotic phases in weakly scattering media, and anomalies in optical transparency in the presence of synthetic `magnetic fields'. We argue that some of these effects should be visible and investigated already in weakly disordered optical materials.

cond-mat.dis-nn↗

arXiv2024

Tensor product random matrix theory

The evolution of complex correlated quantum systems such as random circuit networks is governed by the dynamical buildup of both entanglement and entropy. We here introduce a real-time field theory approach -- essentially a fusion of the $G \Sigma$-functional of the SYK-model and the field theory of disordered systems -- enigneered to microscopically describe the full range of such crossover dynamics: from initial product states to a maximum entropy ergodic state. To showcase this approach in the simplest nontrivial setting, we consider a tensor product of coupled random matrices, and compare to exact diagonalization.

cond-mat.mes-hall↗

arXiv2024

Quantum Chaos on Edge

In recent years, the physics of many-body quantum chaotic systems close to their ground states has come under intensified scrutiny. Such studies are motivated by the emergence of model systems exhibiting chaotic fluctuations throughout the entire spectrum (the Sachdev-Ye-Kitaev (SYK) model being a renowned representative) as well as by the physics of holographic principles, which likewise unfold close to ground states. Interpreting the edge of the spectrum as a quantum critical point, here we combine a wide range of analytical and numerical methods to the identification and comprehensive description of two different universality classes: the near edge physics of ``sparse'' and the near edge of ``dense'' chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension, which is exponentially small or algebraically small in the sparse and dense case, respectively. Notable representatives of the two classes are generic chaotic many-body models (sparse) and single particle systems, invariant random matrix ensembles, or chaotic gravitational systems (dense). While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different. Considering the SYK model as a representative of the sparse class, we apply a combination of field theory and exact diagonalization to a detailed discussion of its edge spectrum. Conversely, Jackiw-Teitelboim gravity is our reference model for the dense class, where an analysis of the gravitational path integral and random matrix theory reveal universal differences to the sparse class, whose implications for the construction of holographic principles we discuss.

hep-th↗

arXiv2023

Chaotic fluctuations in a universal set of transmon qubit gates

Transmon qubits arise from the quantization of nonlinear resonators, systems that are prone to the buildup of strong, possibly chaotic, fluctuations. Such instabilities will likely affect fast gate operations which involve the transient population of higher excited states outside the computational subspace. Here we show that a statistical analysis of the instantaneous eigenphases of the time evolution operator, in particular of their curvatures, allows for identifying the subspace affected by chaotic fluctuations. Our analysis shows that fast entangling gates, operating at speeds close to the so-called quantum speed limit, contain transient regimes where the dynamics indeed becomes partially chaotic for just two transmons.

quant-ph↗

arXiv2023

Fragility of surface states in non-Wigner Dyson topological insulators

Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a ``bridge'' between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, $E=0$. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number $\nu=1$ as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrum-wide quantum critical delocalization of all states vs. full localization except at the $E=0$ critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.

cond-mat.mes-hall↗

arXiv2023

Classical Chaos in Quantum Computers

The development of quantum computing hardware is facing the challenge that current-day quantum processors, comprising 50-100 qubits, already operate outside the range of quantum simulation on classical computers. In this paper we demonstrate that the simulation of classical limits can be a potent diagnostic tool potentially mitigating this problem. As a testbed for our approach we consider the transmon qubit processor, a computing platform in which the coupling of large numbers of nonlinear quantum oscillators may trigger destabilizing chaotic resonances. We find that classical and quantum simulations lead to similar stability metrics (classical Lyapunov exponents vs. quantum wave function participation ratios) in systems with $\mathcal{O}(10)$ transmons. However, the big advantage of classical simulation is that it can be pushed to large systems comprising up to thousands of qubits. We exhibit the utility of this classical toolbox by simulating all current IBM transmon chips, including the recently announced 433-qubit processor of the Osprey generation, as well as future devices with 1,121 qubits (Condor generation). For realistic system parameters, we find a systematic increase of Lyapunov exponents with system size, suggesting that larger layouts require added efforts in information protection.

quant-ph↗

arXiv2023

Experimental Observation of Topological Quantum Criticality

We report on the observation of quantum criticality forming at the transition point between topological Anderson insulator phases in a one-dimensional photonic quantum walk with spin. The walker's probability distribution reveals a time-staggered profile of the dynamical spin-susceptibility, recently suggested as a smoking gun signature for topological Anderson criticality in the chiral symmetry class AIII. Controlled breaking of phase coherence removes the signal, revealing its origin in quantum coherence.

quant-ph↗

arXiv2022

Topological quantum criticality of the disordered Chern insulator

We consider the two-dimensional topological Chern insulator in the presence of static disorder. Generic quantum states in this system are Anderson localized. However, topology requires the presence of a subset of critical states, with diverging localization length (the Chern insulator analog of the `center of the Landau band states' of the quantum Hall insulator.) We discuss geometric criteria for the identification of these states at weak disorder, and their extension into the regime of strong disorder by analytical methods. In this way, we chart a critical surface embedded in a phase space spanned by energy, topological control parameter, and disorder strength. Our analytical predictions are supplemented by a numerical analysis of the position of the critical states, and their multifractal properties.

cond-mat.mes-hall↗