The study of spins and particles on graphs has broad applications, from the dynamics of interacting systems on networks to combinatorial problems. Here, we study the large-$n$ limit of the $O(n)$ model on graphs, which is considerably more challenging than on regular lattices, as the loss of translational invariance gives rise to an infinite set of saddle point constraints in the thermodynamic limit. We show that the free energy at low and high temperature $T$ is determined by the spectrum of two fundamental graph-theoretic objects: the Laplacian matrix at low $T$ and the Adjacency matrix at high $T$. Their interplay is studied across several classes of graphs. For regular lattices the two coincide. We obtain an exact solution on trees, where the Lagrange multipliers interestingly depend solely on the number of nearest neighbors. We further contrast these classical results with those for a quantum spin model on an exemplary tree. For decorated lattices, the singular part of the free energy is governed by the Laplacian spectrum, whereas this is true for the full free energy only in the zero temperature limit. Finally, we discuss a bipartite fully connected graph to highlight the importance of a finite coordination number in these results.
The study of spins and particles on graphs has broad applications, from the dynamics of interacting systems on networks to combinatorial problems. Here, we study the large-n limit of the O(n) model on graphs, which is considerably more challenging than on regular lattices, as the loss of translational invariance gives rise to an infinite set of saddle point constraints in the thermodynamic limit. We show that the free energy at low and high temperature T is determined by the spectrum of two fundamental graph-theoretic objects: the Laplacian matrix at low T and the Adjacency matrix at high T. Their interplay is studied across several classes of graphs. For regular lattices the two coincide. We obtain an exact solution on trees, where the Lagrange multipliers interestingly depend solely on the number of nearest neighbors. We further contrast these classical results with those for a quantum spin model on an exemplary tree. For decorated lattices, the singular part of the free energy is governed by the Laplacian spectrum, whereas this is true for the full free energy only in the zero temperature limit. Finally, we discuss a bipartite fully connected graph to highlight the importance of a finite coordination number in these results.
We consider a system of qubits coupled via nearest-neighbor interaction governed by the Heisenberg Hamiltonian. We further suppose that all coupling constants are equal to 1. We are interested in determining which graphs allow for a transfer of quantum state with fidelity equal to 1. To answer this question, it is enough to consider the action of the Laplacian matrix of the graph in a vector space of suitable dimension. Our main result is that if the underlying graph is a tree with more than two vertices, then perfect state transfer does not happen. We also explore related questions, such as what happens in bipartite graphs and graphs with an odd number of spanning trees. Finally, we consider the model based on the XY-Hamiltonian, whose action is equivalent to the action of the adjacency matrix of the graph. In this case, we conjecture that perfect state transfer does not happen in trees with more than three vertices.
We consider a system of qubits coupled via nearest-neighbour in-teraction governed by the Heisenberg Hamiltonian. We further sup-pose that all coupling constants are equal to 1. We are interested in determining which graphs allow for a transfer of quantum state with fidelity equal to 1. To answer this question, it is enough to consider the action of the Laplacian matrix of the graph in a vector space of suitable dimension. Our main result is that if the underlying graph is a tree with more than two vertices, then perfect state transfer does not happen. We also explore related questions, such as what happens in bipartite graphs and graphs with an odd number of spanning trees. Finally, we consider the model based on the XY-Hamiltonian, whose action is equivalent to the action of the adjacency matrix of the graph. In this case, we conjecture that perfect state transfer does not happen in trees with more than three vertices. 1
A fundamental principle in condensed matter physics is the effective description in terms of quasiparticles. The high-energy part of the system is accounted for by renormalized properties of the quasiparticles, providing both, an accurate description and interpretation of the relevant low-energy physics. We consider two classes of methods suited to provide effective quasiparticle descriptions in the thermodynamic limit: (non-perturbative) linked-cluster expansions ((N)LCEs) and continuous unitary transformations (CUTs). We focus specifically on the combination of both methods, providing a different perspective and solutions to existing challenges. LCEs provide high-order series expansions in a perturbation parameter, by combining effective Hamiltonians determined on finite clusters. We introduce a white-graph expansion for the method of perturbative continuous unitary transformations when implemented as an LCE. The essential idea behind an expansion in white graphs is to perform an optimized bookkeeping during the calculation by exploiting the model-independent effective Hamiltonian in second quantization and the associated inherent cluster additivity. This approach is shown to be especially well suited for microscopic models with many coupling constants, since the total number of relevant graphs is drastically reduced. The white-graph expansion is exemplified for a two-dimensional quantum spin model, illustrating its efficiency. In NLCEs, the perturbative treatment of finite clusters is replaced by a numerical exact (block) diagonalization. While LCEs are restricted due to their perturbative nature, the non-perturbative variant is not. In graph-based continuous unitary transformations (gCUTs), the block diagonalization is achieved by a non-perturbative CUT performed on finite clusters. The central objective of this thesis is a modification of the gCUT scheme, allowing to treat two kinds of major issues, denoted by pseudo and genuine decay, occuring for NLCEs due to the non-perturbative treatment of clusters. Indeed, one finds surprising effects caused by the non-perturbative renormalization. In particular, we identify a fundamental challenge for any non-perturbative approach based on finite clusters resulting from the reduced symmetry on graphs, most importantly the breaking of translational symmetry when targeting the properties of excited states. This can be traced back to the appearance of intruder states in the low-energy spectrum, which represent a major obstacle in quasi-degenerate perturbation theory. Here, a generalized notion of cluster additivity is introduced, which is used to formulate an optimized scheme of gCUTs, allowing to solve and to physically understand this major issue. Most remarkably, our improved scheme demands to go beyond the paradigm of using the exact eigenvectors on graphs. We demonstrate that the modified scheme is correct in the non-perturbative regime. Even at quantum criticality, the scheme gives valid results. To determine the critical behavior, one must rely on extrapolation schemes. We introduce a generic approach to extract critical properties from sequences of numerical data which is directly relevant for NLCEs. The scheme is applied to the quantum phase transition between the dimerized and the isotropic spin 1/2 Heisenberg chain. Finally, we investigate a scenario where the gapped quasiparticle excitation is not stable for all momenta, i.e., the one-particle mode merges with the continuum for certain momenta and one observes quasiparticle decay. In this case, intruder states merging with the low-energy spectrum on finite clusters represent genuine physics of the system. Again, a proper renormalization on finite clusters satisfies the generalized cluster additivity. Our adjusted renormalization approach performed on finite clusters does not necessarily lead to a full decoupling of the quasiparticle subspaces and the remaining interactions are part of the effective description. The resulting effective Hamiltonian can be analyzed to describe quasiparticle decay in the thermodynamic limit. The modified scheme is applied to four-leg spin 1/2 Heisenberg ladders, providing insights into the spectral properties. Overall, this thesis presents several important developments for the derivation of effective quasiparticle pictures in quantum spin models via perturbative and non-perturbative LCEs, opening various opportunities for future investigations.
We consider the statics and dynamics of distinguishable spin-21 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.
The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems, however, the Hamiltonian is proportional to the adjacency matrix instead. It is well known that these quantum walks are equivalent when the graph is regular, i.e., when each vertex has the same number of neighbors. If the graph is irregular, however, the quantum walks evolve differently. In this paper, we show that for some irregular graphs, if the particle is initially localized at a certain vertex, the probability distributions of the two quantum walks are identical, even though the amplitudes differ. We analytically prove this for a graph with five vertices and a graph with six vertices. By simulating the walks on all 1 018 689 568 simple, connected, irregular graphs with 11 vertices or less, we found 64 graphs with this notion of equivalence. We also give eight infinite families of graphs supporting these equivalent walks.
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.