We introduce a white-graph expansion for the method of perturbative continuous unitary transformations when implemented as a linked-cluster expansion. 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 of coupled two-leg XXZ ladders.
We introduce a white-graph expansion for the method of perturbative continuous unitary transformations when implemented as a linked-cluster expansion. 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 of coupled two-leg XXZ ladders.
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 propose a hypergraph expansion which facilitates the direct treatment of quantum spin models with many-site interactions via perturbative linked cluster expansions. The main idea is to generate all relevant subclusters and sort them into equivalence classes essentially governed by hypergraph isomorphism. Concretely, a reduced König representation of the hypergraphs is used to make the equivalence relation accessible by graph isomorphism. During this procedure we determine the embedding factor for each equivalence class, which is used in the final resummation in order to obtain the final result. As an instructive example we calculate the ground-state energy and a particular excitation gap of the plaquette Ising model in a transverse field on the three-dimensional cubic lattice.
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 graph-based continuous unitary transformations 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 construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians \mathcal{H}=\mathcal{H}_0 +V ℋ = ℋ 0 + V . Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of \mathcal{H}_0 ℋ 0 . Our transformation generalizes the minimal transformation known amongst others under the names Takahashi’s transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of \mathcal{H} ℋ , that is adiabatically connected to the eigenspace of \mathcal{H}_0 ℋ 0 with eigenvalue e_0^n e 0 n , solely depends on the eigenspaces of \mathcal{H} ℋ connected to e_0^m e 0 m with e_0^m\leq e_0^n e 0 m ≤ e 0 n . In contrast, other cluster-additive transformations like the multi-block orthogonalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.
The established approach of perturbative continuous unitary transformations (pCUTs) constructs effective quantum many-body Hamiltonians as perturbative series that conserve the number of one quasiparticle type. We extend the pCUT method to similarity transformations---dubbed pcst++---allowing for multiple quasiparticle types with complex-valued energies. This enlarges the field of application to closed and open quantum many-body systems with unperturbed operators corresponding to arbitrary superimposed ladder spectra. To this end, a generalized counting operator is combined with the quasiparticle generator for open quantum systems recently introduced by Schmiedinghoff and Uhrig [G. Schmiedinghoff and G. S. Uhrig, SciPost Phys. 13, 122 (2022)]. The pcst++ then yields model-independent block-diagonal effective Hamiltonians and Lindbladians, allowing a linked-cluster expansion in the thermodynamic limit similar to the conventional pCUT method. We illustrate the application of the pcst++ method by discussing representative closed, open, and non-Hermitian quantum systems.
We introduce a generic scheme to perform nonperturbative linked cluster expansions in long-range ordered quantum phases. Clusters are considered to be surrounded by an ordered reference state leading to effective edge fields in the exact diagonalization on clusters, which break the associated symmetry of the ordered phase. Two approaches, based either on a self-consistent solution of the order parameter or on minimal sensitivity with respect to the ground-state energy per site, are formulated to find the optimal edge field in each NLCE order. Furthermore, we investigate the scaling behavior of the NLCE data sequences towards the infinite-order limit. We apply our scheme to gapped and gapless ordered phases of XXZ Heisenberg models on various lattices and for spins 1/2 and 1 using several types of cluster expansions ranging from a full-graph decomposition, rectangular clusters, up to more symmetric square clusters. It is found that the inclusion of edge fields allows to regularize nonperturbative linked-cluster expansions in ordered phases yielding convergent data sequences. This includes the long-range spin-ordered ground state of the spin-1/2 and spin-1 Heisenberg model on the square and triangular lattice as well as the trimerized valence bond crystal of the spin-1 Heisenberg model on the kagome lattice.