Details
| Original language | English |
|---|---|
| Article number | 094411 |
| Journal | Physical Review B - Condensed Matter and Materials Physics |
| Volume | 74 |
| Issue number | 9 |
| Publication status | Published - 12 Sept 2006 |
| Externally published | Yes |
Abstract
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.
ASJC Scopus subject areas
- Materials Science(all)
- Electronic, Optical and Magnetic Materials
- Physics and Astronomy(all)
- Condensed Matter Physics
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In: Physical Review B - Condensed Matter and Materials Physics, Vol. 74, No. 9, 094411, 12.09.2006.
Research output: Contribution to journal › Article › Research › peer review
}
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 -