Statics and dynamics of quantumXYand 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 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 G k , 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 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. The techniques developed to solve interacting manybody quantum systems have led to the discovery of many new intriguing nonclassical phenomena. A canonical example is the discovery of quantum phase transitions 5,6 , phase transitions which occur in the ground state -a pure state -which are driven by quantum rather than thermal fluctuations. However, these techniques can typically only be applied to systems which possess a great deal of symmetry. Hence it is extremely desirable to develop new approaches that can be applied in more general situations.There is a superficial similarity between the mathematics of distinguishable quantum spins and the spectral theory of graphs 7,8,9 , which pertains to the dynamics of a single quantum particle hopping on a discrete graph. In both cases there is a graph structure and a notion of locality. In the case of graphs, locality can be characterised by the support of the particle wavefunction, i.e., the position of the quantum particle. A localised particle remains, for small times, approximately localised. (There is a natural UV cutoff given by the graph structure, hence there is a resulting bound on the propagation speed of the particle.) In the case of spin systems the notion of locality emerges in the Heisenberg picture where the support of operators takes the role of defining local physics. Under dynamics local operators remain approximately local for short times. (This locality result is a consequence of the Lieb-Robinson bound 10,11,12,13 .) In both the case of quantum spin systems and spectral graph theory we are interested in the eigenvalues and the eigenvectors of the generator of time translations: the hamiltonian for the spin system; and the ad...

Which Exterior Powers are Balanced?

“…An equivalent algebraic formulation will follow thereafter. Our treatment follows closely the description given by Osborne [8] (see also [2]). We fix an arbitrary total ordering ≺ on the vertex set V of G. Consider a k-tuple u ∈ V k , say u = (u 1 , .…”

“…They are equivalent to the so-called k-tuple vertex graphs, introduced by Zhu, Liu, Dick and Alavi [10], which generalizes the well-known double vertex graphs (see Alavi, Behzad, Erdös, and Lick [1]). Subsequently, these k-tuple vertex graphs were studied by Osborne [8] in the framework of spin networks in quantum information.…”

Quantum control theory for state transformations: Dark states and their enlightenment

“…Examining the graph produced by the Hamiltonian for the 2-excitation subspace (Fig. 3, formed as in [20]), we can see that there are no eigenstates of the Hamiltonian for the left-hand component (i.e. the connected component to our initial state) that have no overlap with all the states our control connects to.…”

“…From [20], we have Definition. The matrix representing the second excitation subspace of H 0 is given by…”

Speed and entropy of an interacting continuous time quantum walk

“…The idea is to follow the motion of a swarm of several clocking agents (cursor spins in the 'up' state) acting on the register. Stated otherwise, with reference for simplicity to the case N 3 = 2, we allow the clock to perform a quantum walk on the graph having the vertices (x 1 , x 2 ), with 1 x 1 < x 2 s, with edges between nearest neighbours [24].…”