dorsal/arxiv
View SchemaStatics and Dynamics of Quantum XY and Heisenberg Systems on Graphs
| Authors | Tobias J. Osborne |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0312126 |
| URL | https://arxiv.org/abs/quant-ph/0312126 |
| DOI | 10.1103/PhysRevB.74.094411 |
| Journal | Phys. Rev. B 74, 094411 (2006) |
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 H_XY, which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg hamiltonian H_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,..., 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 utilise results from spectral graph theory to solve new spin models: the XY model and heisenberg model in a magnetic field on the complete graph.
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"abstract": "We consider the statics and dynamics of distinguishable spin-1/2 systems on\nan arbitrary graph G with N vertices. In particular, we consider systems of\nquantum spins evolving according to one of two hamiltonians: (i) the XY\nhamiltonian H_XY, which contains an XY interaction for every pair of spins\nconnected by an edge in G; and (ii) the Heisenberg hamiltonian H_Heis, which\ncontains a Heisenberg interaction term for every pair of spins connected by an\nedge in G. We find that the action of the XY (respectively, Heisenberg)\nhamiltonian on state space is equivalent to the action of the adjacency matrix\n(respectively, combinatorial laplacian) of a sequence G_k, k=0,..., N of graphs\nderived from G (with G_1=G). This equivalence of actions demonstrates that the\ndynamics of these two models is the same as the evolution of a free particle\nhopping on the graphs G_k. Thus we show how to replace the complicated dynamics\nof the original spin model with simpler dynamics on a more complicated\ngeometry. A simple corollary of our approach allows us to write an explicit\nspectral decomposition of the XY model in a magnetic field on the path\nconsisting of N vertices. We also use our approach to utilise results from\nspectral graph theory to solve new spin models: the XY model and heisenberg\nmodel in a magnetic field on the complete graph.",
"arxiv_id": "quant-ph/0312126",
"authors": [
"Tobias J. Osborne"
],
"categories": [
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"cond-mat"
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"doi": "10.1103/PhysRevB.74.094411",
"journal_ref": "Phys. Rev. B 74, 094411 (2006)",
"title": "Statics and Dynamics of Quantum XY and Heisenberg Systems on Graphs",
"url": "https://arxiv.org/abs/quant-ph/0312126"
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