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arXiv:quant-ph/0312126 · Physical Review B

Statics and dynamics of quantum XY and Heisenberg systems on graphs

Tobias J. Osborne

Published Sep 12, 2006• 24 citations
arXivHTMLPDF

Abstract

We consider the statics and dynamics of distinguishable spin-$\frac{1}{2}$ systems on an arbitrary graph $G$ with $N$ vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the $XY$ Hamiltonian ${H}_{XY}$, which contains an $XY$ interaction for every pair of spins connected by an edge in $G$; and (ii) the Heisenberg Hamiltonian ${H}_{\mathrm{Heis}}$, which contains a Heisenberg interaction term for every pair of spins connected by an edge in $G$. We find that the action of the $XY$ (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence ${G}_{k}$, $k=0,\dots{},N$ of graphs derived from $G$ (with ${G}_{1}=G$). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs ${G}_{k}$. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the $XY$ model in a magnetic field on the path consisting of $N$ vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the $XY$ model and Heisenberg model in a magnetic field on the complete graph.

DOI
10.1103/physrevb.74.094411

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Exactly solvable models are essential in physics. For many-body spin- 1 / 2 systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions.
This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.
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