Quasiparticle properties of quantum magnets with long-range interactions are investigated by high-order linked-cluster expansions in the thermodynamic limit. It is established that perturbative continuous unitary transformations on white graphs are a promising and flexible approach to treat long-range interactions in quantum many-body systems. We exemplify this scheme for the one-dimensional transverse-field Ising chain with long-range interactions. For this model, the elementary quasiparticle gap is determined allowing to access the quantum-critical regime including critical exponents and multiplicative logarithmic corrections for the ferro- and antiferromagnetic case.
Quasiparticle properties of quantum magnets with long-range interactions are investigated by high-order linked-cluster expansions in the thermodynamic limit. It is established that perturbative continuous unitary transformations on white graphs are a promising and flexible approach to treat long-range interactions in quantum many-body systems. We exemplify this scheme for the one-dimensional transverse-field Ising chain with long-range interactions. For this model, the elementary quasiparticle gap is determined allowing to access the quantum-critical regime including critical exponents and multiplicative logarithmic corrections for the ferro- and antiferromagnetic case.
Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.
The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents z and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
Open quantum many-body systems are commonly described by Lindblad master equations, yet the treatment of Lindbladian operators in the thermodynamic limit remains a major challenge. We develop a framework for constructing effective quasiparticle-conserving Lindbladian operators directly in the thermodynamic limit. Our approach extends continuous similarity transformations to non-Hermitian open quantum systems and enables the systematic block diagonalization of Lindbladians with respect to the quasiparticle (qp) number. We formulate two complementary methods. The first, projective continuous similarity transformations (pcst++), generalizes perturbative continuous unitary transformations to Lindblad operators so that a linked-cluster expansion allows us to obtain high-order series expansions of the infinite system. The second, deepCST, extends directly evaluated enhanced perturbative continuous unitary transformations by combining the same qp-conserving generator with a perturbative truncation scheme that yields non-perturbative effective Lindbladian operators directly in thermodynamic limit. We apply pcst++ and deepCST to the dissipative transverse-field Ising chain with local dissipation. We focus on the low-Ising regime. We purify the Lindbladian by splitting each spin into two sites. A spin flip then corresponds to two qps. We derive and analyze the effective, qp-conserving Lindbladian in the sectors with zero to two qps. We show how the Ising interaction renormalizes the decay rates of elementary spin-flip excitations and provides the microscopic mechanism for a competition between coherent interactions and dissipation. Our work establishes perturbative and non-perturbative continuous similarity transformations as a versatile tool for deriving effective qp pictures of open quantum many-body systems in the thermodynamic limit.
We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.
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.