Statics and Dynamics of Quantum XY and Heisenberg Systems on Graphs

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.

5 pages, minor corrections and solution of new model, added figures

References in corpus (5)

• The density-matrix renormalization group
• Perfect state transfer in quantum spin networks
• Lieb-Schultz-Mattis in Higher Dimensions
• Spectral Gap and Exponential Decay of Correlations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem

Cited by in corpus (14)

• The Basics of Perfect Communication through Quantum Networks
• Control by quantum dynamics on graphs
• Interfacing with Hamiltonian Dynamics
• Spin Star as Switch for Quantum Networks
• Quantum Control Theory for State Transformations: Dark States and their Enlightenment
• Structured singular value analysis for spintronics network information transfer control
• The bilinear-biquadratic model on the complete graph
• Perfect state transfer on hypercubes and its implementation using superconducting qubits
• Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
• Quantum transfer through small networks coupled to phonons: effects of topology vs phonons
• Perfect state transfer using Markovian quantum walk
• Perfect State Transfer on Signed Graphs
• A switching approach for perfect state transfer over a scalable and routing enabled network architecture with superconducting qubits
• Which Exterior Powers are Balanced?