Group word dynamics from local random matrix Hamiltonians and beyond

TL;DR

This work develops a novel mapping between local Haar-Ising random-matrix spin chains and a single-particle hopping problem on the Cayley graph of a right-angled Coxeter group, enabled by free probability in the large-q limit. The approach yields a clear, computable link between many-body energy-density dynamics and group-word dynamics on hyperbolic lattices, producing a Gaussian density of states and diffusive energy transport consistent with locality. By introducing braid relations, the authors show that the group structure can generate additional conserved quantities and potential integrability, with exact revivals in some finite cases and rich fluctuation patterns in others. The study thereby connects free probability, hyperbolic lattice quantum mechanics, and the phenomenology of both chaotic and integrable Hamiltonian dynamics, and suggests further avenues for matrix-model realizations on other hyperbolic tilings and for rigorous hydrodynamic analyses.

Abstract

We study one dimensional quantum spin chains whose nearest neighbor interactions are random matrices that square to one. By employing free probability theory, we establish a mapping from the many-body quantum dynamics of energy density in the original chain to a single-particle hopping dynamics when the local Hilbert space dimension is large. The hopping occurs on the Cayley graph of an infinite Coxeter reflection group. Adjacency matrices on large finite clusters of this Cayley graph can be constructed numerically by leveraging the automatic structure of the group. The density of states and two-point functions of the local energy density are approximately computed and consistent with the physics of a generic local Hamiltonian: Gaussian density of states and thermalization of energy density. We then ask what happens to the physics if we modify the group on which the hopping dynamics occurs, and conjecture that adding braid relations into the group leads to integrability. Our results put into contact ideas in free probability theory, quantum mechanics of hyperbolic lattices, and the physics of both generic and integrable Hamiltonian dynamics.

Figures (17)

• Figure 1: Geometry of the local Haar-Ising random matrix spin chain.
• Figure 2: Various random matrix models involving $5$ HI bonds. Examples leading to standard freeness are illustrated in (a) and (b). In (a), the Hamiltonian is more "local" than (b) in the sense that it is vanishingly sparse as $q\rightarrow\infty$ but neither are geometrically local as far as energy density operators are concerned. Panel (c), however, is an $L=5$ geometrically local chain corresponding to $\varepsilon$-freeness for $\varepsilon$ defined as in Eq. \ref{['eq:varepsilon']}; this type of geometry is the focus of the paper.
• Figure 3: Finite portions of the Cayley graphs of $G_L$, with respect to the generators $g_i$ for $L=3,4,5$ in panels $(a),(b),(c)$, respectively. The graphs are embedded in the hyperbolic disk to emphasize their hyperbolic nature. The graphs can thus be viewed as tilings of the hyperbolic disk by polygons. In (a), $3$apeirogons meet at each vertex, in (b) two squares and two apeirogons meet, and in (c) five squares meet.
• Figure 4: The finite state automaton $W$ which recursively enumerates all words in the language $\mathcal{L}$ of short-lex geodesic words for the group $G_4$ corresponding to an $L=4$ local HI RMT chain in PBC. The nodes of the automaton are states and transitions correspond to concatenation of the letter $g_i$ to a previously allowed word $g\in \mathcal{L}$. The recursive step consists of generating all words of radius $d+1$ by concatenating all words of radius $d$ with a single generator $g_i$. Given the state of a radius $d$ word, $W$ dictates which letters $g_i$ can be concatenated to form radius $d+1$ words and the corresponding states. See the text for further details.
• Figure 5: The recursive algorithm producing short-lex geodesic group words and adjacency matrices $V_i$ for the group $G_5$ corresponding to an $L=5$ HI RMT chain. Here we are performing the recursive step on the purple node, the group word $g_5\equiv 5$, and all black nodes and connections have been created. We need to put unit entries into $V_1,V_2,V_3,V_4$ in the correct locations. The green nodes and solid green connections are easily made, as they correspond to new group words generated by the word acceptor $W$. The dashed green connections are more subtle; we need to connect node $n$ to $m$, for example. Here, $n,m,k,l$ are indices for the location of the corresponding group word in the ordered set $\mathcal{L}$, not shorthand for generators. Starting on node $g_5$, $W$ will not allow a transition to $g_5g_2$ since $g_2g_5$ is already an element of $\mathcal{L}$ which represents that group element. Since the transition is rejected by $W$, we know it must be part of a cycle of length $4$ and we can use the previously constructed adjacency matrices to identify the index $m$ that needs to be connected to $n$.

Theorems & Definitions (6)

• Claim 1
• proof
• Claim 2
• proof
• Claim 3
• proof