arXiv subjects

Hrant Topchyan

Publications and source records attributed to Hrant Topchyan.

12 recordsLinked to original sources

arXiv2026

Spin Quantum Hall Effect: the Critical Exponents

The spin quantum Hall effect (SQHE) provides one of the few examples of an Anderson localization transition for which exact critical exponents are known, making it an important testing ground for theories of disordered topological systems and conformal field theory. The corresponding network model, obtained by replacing the random $U(1)$ phases of the Chalker--Coddington model with random $SU(2)$ matrices, belongs to symmetry class C of the Altland--Zirnbauer classification and is believed to describe quasiparticle transport in two-dimensional disordered superconductors with broken time-reversal symmetry. In this work, we perform high-precision numerical calculations of the localization-length exponent ($\nu$) and the boundary critical exponent ($\mu$) for the SQHE network model using the recently developed $S$-matrix approach to random networks.

cond-mat.dis-nn↗

arXiv2026

Statistical properties of quadrangular surfaces

We investigate the statistical properties of random quadrangular surfaces generated by different randomization procedures: the Gruzberg-Kl\"umper-Nuding-Sedrakyan (GKNS) construction, and two newly introduced generalizations of dynamical triangulations (DT), dynamical (DQ) and general quadrangulations (GQ). We formulate these surfaces within a unified graph-theoretic framework and establish the relationships between the elementary operations defining the different ensembles. For GKNS surfaces, we demonstrate that the construction is equivalent to two mutually constrained percolation processes and determine the associated critical point and critical exponents, revealing deviations from ordinary percolation. For DQ and GQ, we analyze the underlying Markov chains and determine the scaling of mixing and relaxation times. We further analyze all three ensembles through their degree distributions, degree correlations, distance statistics, and Hausdorff dimensions. While DQ exhibits exponentially decaying degree distributions and geometric properties similar to DT, GKNS and GQ display broad, scale-free degree distributions. Moreover, GKNS surfaces possess an asymptotic Hausdorff dimension $\Delta_H\approx 2$, whereas DQ and GQ approach $\Delta_H\approx 4$, similarly to DT. This indicates that DQ is compatible with the universal behavior of DT, while GKNS and GQ define distinct classes of random geometry, implying a different underlying measure in the space of random surfaces and a possible geometric framework for a new class of noncritical string theories.

cond-mat.stat-mech↗

arXiv2026

Nucleation of Sachdev-Ye-Kitaev Clusters in One Spatial Dimension

We study how Sachdev-Ye-Kitaev (SYK) interactions can arise from localized single-particle states on a system that is effectively one dimensional. If a local interaction is projected onto coarse localized orbitals, the resulting couplings do not immediately follow the standard SYK distribution. Instead, they have a finite probability of being exactly zero, a broad non-Gaussian distribution for the nonzero values, and strong correlations coming from the geometry of the localized states. We then show that this changes when each localization volume is resolved into $M>1$ smaller microscopic pieces with random phases. As $M$ increases, the distribution of the nonzero couplings moves toward the complex-Gaussian SYK form. At the same time, the large-$M$ limit is a sparse but asymptotically canonical SYK network: the nonzero couplings create SYK clusters, while the pattern of missing or very weak couplings is still determined by the real-space overlap of the localized orbitals. Finally, we map the interaction tensor to a graph in pair space. This makes it possible to follow the formation, merger, and growth of SYK clusters, which we characterize using connected components and clique/simplex counts. The result is a minimal real-space phenomenological theory of SYK-cluster formation, providing clear experimental criteria.

cond-mat.str-el↗

arXiv2026

The $\mathbb{Z}_N^{\times 3}$ symmetry protected boundary modes in two-dimensional Potts paramagnets

We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with $\mathbb{Z}_N^{\times 3}$ symmetry on a triangular lattice. Using a cohomology-based transformation, the lattice models for the edge modes are explicitly obtained, and their structure is shown to be governed by the arithmetic properties of $N$. For prime $N$, the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of $N$, the models exhibit hierarchical or factorized structures. We demonstrate that all phases can be understood in terms of primary models augmented by local defect degrees of freedom that partition the system into independent segments. Finally, the global symmetry is realized on the boundary in a non-on-site and anomalous manner via a projective representation, directly realizing the corresponding 't Hooft anomaly.

cond-mat.str-el↗

arXiv2025

Topological edge states in two-dimensional $\mathbb{Z}_4$ Potts paramagnet protected by the $\mathbb{Z}_4^{\times 3}$ symmetry

We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site $G=\mathbb{Z}_4^{\times 3}$ symmetry, starting from a three-component $\mathbb{Z}_4$ Potts paramagnet on a triangular lattice. Within the group-cohomology framework, $H^{3}(G,U(1))\cong \mathbb{Z}_4^{\times 7}$, we focus on a "colorless" cocycle representative obtained by antisymmetrizing the basic $\mathbb{Z}_4$ three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced nonlocal unitary transformation followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting $\mathbb{Z}_4$ chain with next-to-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as $1/L$ and an entanglement-entropy scaling consistent with a conformal field theory of central charge $c=2.191(4)\simeq 11/5$. The rational value $c=11/5$ matches the coset $SU(3)_3/SU(2)_3$, making it a candidate for the continuum description of the $\mathbb{Z}_4^{\times 3}$ edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.

cond-mat.str-el↗

arXiv2024

Harris-Luck criterion in the plateau transition of the Integer Quantum Hall Effect

The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension $d$ and localization length exponent $\nu$. It states that the new disorder can be relevant only if $d \nu < 2$. We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Kl\"umper, W. Nuding, and A. Sedrakyan, Phys. Rev. B 95, 125414 (2017)] and show that the fluctuations of the geometry are relevant despite $d \nu> 2$, implying that Harris criterion should be modified. We have observed that the fluctuations of the critical point in different quenched configurations of disordered network blocks is of order $L^0$, i.e.~it does not depend on block size $L$ in contrast to the expectation based on the Harris criterion that they should decrease as $L^{-d/2}$ according to the central limit theorem. Since $L^0 > (x-x_c)$ is always satisfied near the critical point, the mentioned network disorder is relevant and the critical indices of the system can be changed. We have also shown that the GKNS disordered network is fundamentally different from Voronoi-Delaunay and dynamically triangulated random lattices: the probability of higher connectivity in the GKNS network decreases in a power law as opposed to an exponential, indicating that we are dealing with a ``scale free" network, such as the Internet, protein-protein interactions, etc.

cond-mat.dis-nn↗

arXiv2024

The integer quantum Hall transition: an $S$-matrix approach to random networks

In this paper we propose a new $S$-matrix approach to numerical simulations of network models and apply it to random networks that we proposed in a previous work 10.1103/PhysRevB.95.125414. Random networks are modifications of the Chalker-Coddington (CC) model for the integer quantum Hall transition that more faithfully capture the physics of electrons moving in a strong magnetic field and a smooth disorder potential. The new method has considerable advantages compared to the transfer matrix approach, and gives the value $\nu \approx 2.4$ for the critical exponent of the localization length in a random network. This finding confirms our previous result and is surprisingly close to the experimental value $\nu_{\text{exp}} \approx 2.38$ observed at the integer quantum Hall transition but substantially different from the CC value $\nu_\text{CC} \approx 2.6$.

cond-mat.dis-nn↗

arXiv2023

Two-dimensional topological paramagnets protected by $\mathbb{Z}_3$ symmetry: Properties of the boundary Hamiltonian

We systematically study gapless edge modes corresponding to $\mathbb{Z}_3$ symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge $c=1$, given by the coset $SU_k(3)/SU_k(2)$ CFT at level k=1.

cond-mat.str-el↗

arXiv2023

SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase

This paper focuses on the creation of a model with explicitly defined symmetry-protected topological (SPT) phases on a triangular lattice as an extension of $Z_2$ Ising model's ferromagnetic phase. Unlike in previously known similar works, this model is based on an initially interacting system which is known to describe experimentally realizable physical systems. The Hamiltonian for these edge states contains four-point spin interactions between next-to-next nearest neighbors. As an initially interacting A generic technique for creating SPT models is developed, allowing for the construction of translation-invariant edge models.

cond-mat.str-el↗

arXiv2022

$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by $(\times Z_3)^3\equiv Z_3\times Z_3\times Z_3$ symmetry and smaller $Z_3$ symmetry. We derive microscopic models for the gapless edge, uncover their symmetries, and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue that the low energy limit of our edge modes is defined by the $SU_k(3)/SU_k(2)$ coset conformal field theory with the level $k=2$. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

cond-mat.str-el↗

arXiv2022

Deep Lake: a Lakehouse for Deep Learning

Traditional data lakes provide critical data infrastructure for analytical workloads by enabling time travel, running SQL queries, ingesting data with ACID transactions, and visualizing petabyte-scale datasets on cloud storage. They allow organizations to break down data silos, unlock data-driven decision-making, improve operational efficiency, and reduce costs. However, as deep learning usage increases, traditional data lakes are not well-designed for applications such as natural language processing (NLP), audio processing, computer vision, and applications involving non-tabular datasets. This paper presents Deep Lake, an open-source lakehouse for deep learning applications developed at Activeloop. Deep Lake maintains the benefits of a vanilla data lake with one key difference: it stores complex data, such as images, videos, annotations, as well as tabular data, in the form of tensors and rapidly streams the data over the network to (a) Tensor Query Language, (b) in-browser visualization engine, or (c) deep learning frameworks without sacrificing GPU utilization. Datasets stored in Deep Lake can be accessed from PyTorch, TensorFlow, JAX, and integrate with numerous MLOps tools.

cs.DC↗

arXiv2020

Geometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions

In the context of the Integer Quantum Hall plateau transitions, we formulate a specific map from random landscape potentials onto 2D discrete random surfaces. Critical points of the potential, namely maxima, minima and saddle points uniquely define a discrete surface $S$ and its dual $S^*$ made of quadrangular and $n-$gonal faces, respectively, thereby linking the geometry of the potential with the geometry of discrete surfaces. The map is parameter-dependent on the Fermi level. Edge states of Fermi lakes moving along equipotential contours between neighbour saddle points form a network of scatterings, which define the geometric basis, in the fermionic model, for the plateau transitions. The replacement probability characterizing the network model with geometric disorder recently proposed by Gruzberg, Kl\"umper, Nuding and Sedrakyan, is physically interpreted within the current framework as a parameter connected with the Fermi level.

cond-mat.dis-nn↗