Publications

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Introduction to dynamical mean-field theory of randomly connected neural networks with bidirectionally correlated couplings

Wenxuan Zou, Haiping Huang

SciPost Phys. Lect. Notes 79 (2024) · published 20 February 2024 | · pdf

Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.
Integrability and complexity in quantum spin chains

Ben Craps, Marine De Clerck, Oleg Evnin, Philip Hacker

SciPost Phys. 16, 041 (2024) · published 7 February 2024 | · pdf

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.

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Probabilistic deconstruction of a theory of gravity, part II: Curved space

Sunok Josephine Suh

SciPost Phys. 16, 012 (2024) · published 17 January 2024 | · pdf

We propose that the underlying context of holographic duality and the Ryu-Takayanagi formula is that the volume measure of spacetime is a probability measure constrained by quantum dynamics. We define quantum stochastic processes using joint quantum distributions which are realized in a quantum system as expectation values of products of projectors. In anti-de Sitter JT gravity, we show that Einstein's equations arise from the evolution of probability under the quantum stochastic process induced by the boundary, with the area of compactified space in the gravitational theory identified as a probability density evolving under the quantum process. Extrapolating these and related results in flat JT gravity found in [SciPost Phys. 15, 174 (2023)], we conjecture that general relativity arises in the semi-classical limit of the evolution of probability with respect to quantum stochastic processes.
Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states

Márton Mestyán, Balázs Pozsgay, Ian M. Wanless

SciPost Phys. 16, 010 (2024) · published 9 January 2024 | · pdf

We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are arranged in a spatially symmetric pattern and the states have maximal entanglement for all bipartitions that follow from the reflection symmetries of the given geometry. We consider those cases where the state itself is invariant with respect to the geometrical symmetry group. The simplest examples are those dual unitary operators which are also self dual and reflection invariant, but we also consider the generalizations in the hexagonal, cubic, and octahedral geometries. We provide a number of constructions and concrete examples for these objects for various local dimensions. All of our examples can be used to build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple equivalent choices for the "direction of time".

1 citation
Modified rational six vertex model on the rectangular lattice

Samuel Belliard, Rodrigo Alves Pimenta, Nikita A. Slavnov

SciPost Phys. 16, 009 (2024) · published 9 January 2024 | · pdf

We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified Izergin determinant. The proofs are based on the quantum inverse scattering method and its representation theory together with elementary linear algebra.
Invariant approach to the driven Jaynes-Cummings model

Ivan Alejandro Bocanegra-Garay, L. Hernández-Sánchez, Irán Ramos-Prieto, Francisco Soto-Eguibar, Héctor Manuel Moya-Cessa

SciPost Phys. 16, 007 (2024) · published 8 January 2024 | · pdf

We investigate the dynamics of the driven Jaynes-Cummings model, where a two-level atom interacts with a quantized field and both, atom and field, are driven by an external classical field. Via an invariant approach, we are able to transform the corresponding Hamiltonian into the one of the standard Jaynes-Cummings model. Subsequently, the exact analytical solution of the Schrödinger equation for the driven system is obtained and employed to analyze some of its dynamical variables.
On correlation functions in models related to the Temperley-Lieb algebra

Kohei Fukai, Raphael Kleinemühl, Balázs Pozsgay, Eric Vernier

SciPost Phys. 16, 003 (2024) · published 5 January 2024 | · pdf

We deal with quantum spin chains whose Hamiltonian arises from a representation of the Temperley-Lieb algebra, and we consider the mean values of those local operators which are generated by the Temperley-Lieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of Temperley-Lieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are non-degenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the so-called golden chain, the $Q$-state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.
Lifting integrable models and long-range interactions

Marius de Leeuw, Ana Lucia Retore

SciPost Phys. 15, 241 (2023) · published 15 December 2023 | · pdf

In this paper we discuss a constructive approach to check whether a constant Hamiltonian is Yang-Baxter integrable. We then apply our method to long-range interactions and find the Lax operator and R-matrix of the three-loop SU(2) sector in N=4 SYM. We show that all known integrable long-range deformations of the 6-vertex models of this type can be obtained from a Lax operator and an R-matrix. Finally we discuss what happens at higher loops and highlight some general structures that these models exhibit.
A new integrable structure associated to the Camassa-Holm peakons

Jean Avan, Luc Frappat, Eric Ragoucy

SciPost Phys. 15, 228 (2023) · published 6 December 2023 | · pdf

We provide a closed Poisson algebra involving the Ragnisco-Bruschi generalization of peakon dynamics in the Camassa-Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a one-parameter integrable extension of their structure. It leads to a new $N$-body peakon solution to the Camassa-Holm shallow-water equation depending on two parameters. We present two explicit constructions of a (non-dynamical) $r$-matrix formulation for this new Poisson algebra. The first one relies on a tensorization of the $N$-dimensional auxiliary space by a 4-dimensional space. We identify a family of Poisson commuting quantities in this framework, including the original ones. This leads us to constructing a second formulation identified as a spectral parameter representation.
Covariant canonical formulations of classical field theories

Francois Gieres

SciPost Phys. Lect. Notes 77 (2023) · published 5 December 2023 | · pdf

We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories and general relativity) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws (notably related to geometric symmetries) within the different approaches. Moreover, for each of these approaches, the impact of space-time boundaries is also addressed. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.

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