Publications

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Random matrices and the free energy of Ising-like models with disorder

Nils Gluth, Thomas Guhr, Alfred Hucht

SciPost Phys. 17, 122 (2024) · published 30 October 2024 | · pdf

We consider an Ising model with quenched surface disorder, the disorder average of the free energy is the main object of interest. Explicit expressions for the free energy distribution are difficult to obtain if the quenched surface spins take values of $± 1$. Thus, we choose a different approach and model the surface disorder by Gaussian random matrices. The distribution of the free energy is calculated. We chose skew-circulant random matrices and analytically compute the characteristic function of the free energy distribution. From the characteristic function we numerically calculate the distribution and show that it becomes log-normal for sufficiently large dimensions of the disorder matrices, and in the limit of infinitely large matrices tends to a Gaussian. Furthermore, we establish a connection to the central limit theorem.
Bethe vectors and recurrence relations for twisted Yangian based models

Vidas Regelskis

SciPost Phys. 17, 126 (2024) · published 5 November 2024 | · pdf

We study Olshanski twisted Yangian based models, known as one-dimensional "soliton non-preserving" open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is $\mathfrak{gl}_{2n}$ and the boundary symmetry is $\mathfrak{sp}_{2n}$ or $\mathfrak{so}_{2n}$, was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is $\mathfrak{gl}_{2n+1}$ and the boundary symmetry is $\mathfrak{so}_{2n+1}$. We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and $Y(\mathfrak{gl}_n)$-type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.
Irreducible representations of $\mathbb{Z}_2^2$-graded supersymmetry algebra and their applications

Naruhiko Aizawa

SciPost Phys. Proc. 14, 016 (2023) · published 23 November 2023 | · pdf

We give a brief review on recent developments of $\mathbb{Z}_2^n$-graded symmetry in physics in which hidden $\mathbb{Z}_2^n$-graded symmetries and $\mathbb{Z}_2^n$-graded extensions of known systems are discussed. This elucidates physical relevance of the $\mathbb{Z}_2^n$-graded algebras. As an example of physically interesting algebra, we take $\mathbb{Z}_2^2$-graded supersymmetry (SUSY) algebras and consider their irreducible representations (irreps). A list of irreps for ${\cal N} = 1, 2$ algebras is presented and as an application of the irreps, $\mathbb{Z}_2^2$-graded SUSY classical actions are constructed.
Fusion of irreducible modules in the periodic Temperley–Lieb algebra

Yacine Ikhlef, Alexi Morin-Duchesne

SciPost Phys. 17, 132 (2024) · published 12 November 2024 | · pdf

We propose a new family $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ of modules over the enlarged periodic Temperley–Lieb algebra $\mathsf{\mathcal EPTL}_N(\beta)$. These modules are built from link states with two marked points, similarly to the modules $\mathsf{X}_{k,\ell,x,y,z}$ that we constructed in a previous paper. They however differ in the way that defects connect pairwise. We analyse the decomposition of $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ over the irreducible standard modules $\mathsf{W}_{k,x}$ for generic values of the parameters $z$ and $w$, and use it to deduce the fusion rules for the fusion $\mathsf{W} × \mathsf{W}$ of standard modules. These turn out to be more symmetric than those obtained previously using the modules $\mathsf{X}_{k,\ell,x,y,z}$. From the work of Graham and Lehrer, it is known that, for $\beta = -q-q^{-1}$ where $q$ is not a root of unity, there exists a set of non-generic values of the twist $y$ for which the standard module $\mathsf{W}_{\ell,y}$ is indecomposable yet reducible with two composition factors: a radical submodule $\mathsf{R}_{\ell,y}$ and a quotient module $\mathsf{Q}_{\ell,y}$. Here, we construct the fusion products $\mathsf{W}×\mathsf{R}$, $\mathsf{W}×\mathsf{Q}$ and $\mathsf{Q} × \mathsf{Q}$, and analyse their decomposition over indecomposable modules. For the fusions involving the quotient modules $\mathsf{Q}$, we find very simple results reminiscent of $\mathfrak{sl}(2)$ fusion rules. This construction with modules $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ is a good lattice regularization of the operator product expansion in the underlying logarithmic bulk conformal field theory. Indeed, it fits with the correspondence between standard modules and connectivity operators, and is useful for the calculation of their correlation functions. Remarkably, we show that the fusion rules $\mathsf{W}×\mathsf{Q}$ and $\mathsf{Q}×\mathsf{Q}$ are consistent with the known fusion rules of degenerate primary fields.
The impact of the diffusion parameter on the passage time of the folding process

Marcelo Tozo Araujo, Jorge Chahine, Elso Drigo Filho, Regina Maria Ricotta

SciPost Phys. Proc. 14, 015 (2023) · published 23 November 2023 | · pdf

Recently, a mathematical method to solve the Fokker Plank equation (FPE) enabled the analysis of the protein folding kinetics, through the construction of the temporal evolution of the probability density. A symmetric tri-stable potential function was used to describe the unfolded and folded states of the protein as well as an intermediate state of the protein. In this paper, the main points of the methodology are reviewed, based on the algebraic Supersymmetric Quantum Mechanics (SQM) formalism, and new results on the kinetics of the evolution of the system characterized in terms of the diffusion parameter are presented.
Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries

Siyuan Wang, Yanyan Chen, Hongyu Wang, Yuting Hu, Yidun Wan

SciPost Phys. 19, 018 (2025) · published 17 July 2025 | · pdf

In this paper, we apply the method of Fourier transform and basis rewriting developed in [H. Wang et al., J. High Energy Phys. 02, 030 (2020)] for the two-dimensional quantum double model of topological orders to the three-dimensional gauge theory model (with a gauge group $G$) of three-dimensional topological orders. We find that the gapped boundary condition of the gauge theory model is characterized by a Frobenius algebra in the representation category $Rep(G)$ of $G$, which also describes charge splitting and condensation on the boundary. We also show that our Fourier transform maps the three-dimensional gauge theory model with input data $G$ to the Walker-Wang model with input data $Rep(G)$ on a trivalent lattice with dangling edges, after truncating the Hilbert space by projecting all dangling edges to the trivial representation of $G$. This Fourier transform also provides a systematic construction of the gapped boundary theory of the Walker-Wang model. This establishes a correspondence between two types of topological field theories: the extended Dijkgraaf-Witten and extended Crane-Yetter theories.
Boundary chaos: Spectral form factor

Felix Fritzsch, Tomaž Prosen

SciPost Phys. 17, 142 (2024) · published 22 November 2024 | · pdf

Random matrix spectral correlations is a defining feature of quantum chaos. Here, we study such correlations in a minimal model of chaotic many-body quantum dynamics where interactions are confined to the system's boundary, dubbed boundary chaos, in terms of the spectral form factor and its fluctuations. We exactly calculate the latter in the limit of large local Hilbert space dimension $q$ for different classes of random boundary interactions and find it to coincide with random matrix theory, possibly after a non-zero Thouless time. The latter effect is due to a drastic enhancement of the spectral form factor, when integer time and system size fulfill a resonance condition. We compare our semiclassical (large $q$) results with numerics at small local Hilbert space dimension ($q=2,3$) and observe qualitatively similar features as in the semiclassical regime.
Symmetry-enforced minimal entanglement and correlation in quantum spin chains

Kangle Li, Liujun Zou

SciPost Phys. 19, 020 (2025) · published 17 July 2025 | · pdf

The interplay between symmetry, entanglement and correlation is an interesting and important topic in quantum many-body physics. Within the framework of matrix product states, in this paper we study the minimal entanglement and correlation enforced by the $SO(3)$ spin rotation symmetry and lattice translation symmetry in a quantum spin-$J$ chain, with $J$ a positive integer. When neither symmetry is spontaneously broken, for a sufficiently long segment in a sufficiently large closed chain, we find that the minimal Rényi-$\alpha$ entropy compatible with these symmetries is $\min\{ -\frac{2}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), 2\ln(J+1) \}$, for any $\alpha∈\mathbb{R}^+$. In an infinitely long open chain with such symmetries, for any $\alpha∈\mathbb{R}^+$ the minimal Rényi-$\alpha$ entropy of half of the system is $\min\{ -\frac{1}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), \ln(J+1) \}$. When $\alpha→ 1$, these lower bounds give the symmetry-enforced minimal von Neumann entropies in these setups. Moreover, we show that no state in a quantum spin-$J$ chain with these symmetries can have a vanishing correlation length. Interestingly, the states with the minimal entanglement may not be a state with the minimal correlation length.
Two entangled and scientifically impactful lives: Jiří Patera, Pavel Winternitz and the Montréal school of mathematical physics

Luc Vinet

SciPost Phys. Proc. 14, 012 (2023) · published 23 November 2023 | · pdf

This text offers a personal account of the scientific legacy of two giants of mathematical physics at the turn of the Millenium and their heritage in Canada, their land of adoption.
Refined cyclic renormalization group in Russian doll model

Vedant Motamarri, Ivan M. Khaymovich, Alexander Gorsky

SciPost Phys. 17, 157 (2024) · published 6 December 2024 | · pdf

Focusing on Bethe-Ansatz integrable models, robust to both time-reversal symmetry breaking and disorder, we consider the Russian Doll Model (RDM) for finite system sizes and energy levels. Suggested as a time-reversal-symmetry breaking deformation of Richardson's model, the well-known and simplest model of superconductivity, RDM revealed an unusual cyclic renormalization group (RG) over the system size $N$, where the energy levels repeat themselves, shifted by one after a finite period in $\ln N$, supplemented by a hierarchy of superconducting condensates, with the superconducting gaps following the so-called Efimov (exponential) scaling. The equidistant single-particle spectrum of RDM made the above Efimov scaling and cyclic RG to be asymptotically exact in the wideband limit of the diagonal potential. Here, we generalize this observation in various respects. We find that, beyond the wideband limit, when the entire spectrum is considered, the periodicity of the spectrum is not constant, but appears to be energy-dependent. Moreover, we resolve the apparent paradox of shift in the spectrum by a single level after the RG period, despite the disappearance of a finite fraction of energy levels. We also analyze the effects of disorder in the diagonal potential on the above periodicity and show that it survives only for high energies beyond the energy interval of the disorder amplitude. Our analytic analysis is supported with exact diagonalization.

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