Symmetric squares of graphs

Audenaert, Katrien; Godsil, Chris; Royle, Gordon F.; Rudolph, Terry

doi:10.1016/j.jctb.2006.04.002

Cited by 54 publications

(83 citation statements)

References 2 publications

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“…The first crucial feature to be exploited is that XXZ models are particle number preserving, meaning that the subspaces generated by states with an arbitrary but fixed number N of (say) down-spins are invariant under the XXZ Hamiltonian. The restrictions of the Hamiltonian to the N-particle subspaces turn out to be discrete Schrödinger-type operators on the N-th symmetric product graph G N of G. This concept and some of its graph theoretic implications have been used in the literature on spin systems before, at least for the isotropic XXX (or Heisenberg) model [4,29,30] (and [9,3] provide counterexamples to an iso-spectrality problem posed in [4]).…”

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confidence: 99%

Droplet states in quantum XXZ spin systems on general graphs

Fischbacher

Stolz

2018

Journal of Mathematical Physics

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We study XXZ spin systems on general graphs. In particular, we describe the formation of droplet states near the bottom of the spectrum in the Ising phase of the model, where the Z-term dominates the XX-term. As key tools we use particle number conservation of XXZ systems and symmetric products of graphs with their associated adjacency matrices and Laplacians. Of particular interest to us are strips and multi-dimensional Euclidean lattices, for which we discuss the existence of spectral gaps above the droplet regime. We also prove a Combes-Thomas bound which shows that the eigenstates in the droplet regime are exponentially small perturbations of strict (classical) droplets.

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confidence: 99%

Droplet states in quantum XXZ spin systems on general graphs

Fischbacher

Stolz

2018

Journal of Mathematical Physics

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“…The quotient disconnected graph components for identical hard-core bosons are shown in (c). [38]. Because D and A G k do not commute, the eigenvalues of symmetric powers are not trivially related to those of the Cartesian powers from which they are derived.…”

Section: Hard-core Bosons On Path Graphsmentioning

confidence: 99%

Perfect quantum state transfer of hard-core bosons on weighted path graphs

2015

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The ability to accurately transfer quantum information through networks is an important primitive in distributed quantum systems. While perfect quantum state transfer (PST) can be effected by a single particle undergoing continuous-time quantum walks on a variety of graphs, it is not known if PST persists for many particles in the presence of interactions. We show that if single-particle PST occurs on one-dimensional weighted path graphs, then systems of hard-core bosons undergoing quantum walks on these paths also undergo PST. The analysis extends the Tonks-Girardeau ansatz to weighted graphs using techniques in algebraic graph theory. The results suggest that hard-core bosons do not generically undergo PST, even on graphs which exhibit single-particle PST.

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“…See [1] for further properties. The trace of the walk generating function is a graph invariant, and we denote it by…”

Section: Powers Of Graphsmentioning

confidence: 99%

“…Assume I G,2k (t) = I H,2k (t). By Proposition 4, there is a bijection σ from the set of 2k-tuples of G to the set of 2k-tuples of H , such that for every 2k-tuple i1 . .…”

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confidence: 99%

Spectra of symmetric powers of graphs and the Weisfeiler–Lehman refinements

Alzaga

Iglesias

Pignol

2010

Journal of Combinatorial Theory, Series B

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The k-th power of an n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power -or the spectrum of the k-th symmetric power -is a complete graph invariant for small values of k, for example, for k = O (1) or k = O (log n).In this paper, we answer this question in the negative: we prove that if the well-known 2k-dimensional Weisfeiler-Lehman method fails to distinguish two given graphs, then their k-th powers -and their k-th symmetric powers -are cospectral. As it is well known, there are pairs of non-isomorphic n-vertex graphs which are not distinguished by the k-dim WL method, even for k = Ω(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.

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Symmetric squares of graphs

Cited by 54 publications

References 2 publications

Droplet states in quantum XXZ spin systems on general graphs

Droplet states in quantum XXZ spin systems on general graphs

Perfect quantum state transfer of hard-core bosons on weighted path graphs

Spectra of symmetric powers of graphs and the Weisfeiler–Lehman refinements

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