TY - JOUR

T1 - Statics and dynamics of quantum XY and Heisenberg systems on graphs

AU - Osborne, Tobias J.

N1 - Copyright: Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006/9/12

Y1 - 2006/9/12

N2 - We consider the statics and dynamics of distinguishable spin- 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 HXY, which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg Hamiltonian HHeis, 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 Gk, k=0,...,N of graphs derived from G (with G1 =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 Gk. 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.

AB - We consider the statics and dynamics of distinguishable spin- 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 HXY, which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg Hamiltonian HHeis, 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 Gk, k=0,...,N of graphs derived from G (with G1 =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 Gk. 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.

UR - http://www.scopus.com/inward/record.url?scp=33748647929&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.74.094411

DO - 10.1103/PhysRevB.74.094411

M3 - Article

AN - SCOPUS:33748647929

VL - 74

JO - Physical Review B - Condensed Matter and Materials Physics

JF - Physical Review B - Condensed Matter and Materials Physics

SN - 1098-0121

IS - 9

M1 - 094411

ER -