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 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.
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 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 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.
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.
We present a numerical, non-perturbative framework for constructing effective Hamiltonians that describe the dynamics of low-energy degrees of freedom within a restricted Hilbert space in quantum many-body systems. The approach is based on block diagonalization guided by a minimal-deformation principle imposed within a selected target sector. The formulation is designed to remain compatible with the numerical linked-cluster expansion. For gapped systems, the relation between minimal deformation and cluster additivity requires careful treatment when excited eigenstates contain finite admixtures of the ground state. After establishing a cluster-additive basis that reproduces the Hörmann-Schmidt construction, the minimal-deformation criterion uniquely determines the effective Hamiltonian within each excitation sector. The same criterion also provides a practical numerical procedure for selecting relevant low-energy eigenstates, including regimes characterized by strong level mixing and avoided crossings. The framework is illustrated using two spin models: the one-dimensional transverse-field Ising model as a benchmark and the two-dimensional Shastry-Sutherland model with Dzyaloshinskii-Moriya interactions, relevant to SrCu _2 2 (BO _3 3 ) _2 2 . In both cases, the resulting effective Hamiltonians accurately capture the excitation dynamics and the associated band structures.
In this work ground and excited electronic states of Heisenberg cluster models, in the form of con- figuration interaction many-body wave functions, are characterized within the spin-adapted Graph- ical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin Slocal ≥ 21 are presented, including J1–J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and bio-mimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reorder- ing, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block- diagonal structure of the Hamiltonian matrix it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route towards wave functions of reduced multi-reference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest.