Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions - Adelhardt, Patrick; Koziol, Jan; Langheld, Anja; Schmidt, Kai Phillip (Entropy, 2024)
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.<br
Keywords: quantum spin systems; long-range interactions; Ising interactions; XY interactions; Heisenberg interactions; Monte Carlo; series expansion; perturbative continuous unitary transformation; stochastic series expansion; quantum phase transitions; critical exponents; quantum simulation

Order-by-disorder and long-range interactions in the antiferromagnetic transverse-field Ising model on the triangular lattice—A perturbative point of view - Koziol, Jan; Mühlhauser, Matthias; Schmidt, Kai Phillip (Results in Physics, 2024)
We study the low-field ground-state (GS) properties of the antiferromagnetic transverse-field Ising model with long-range interactions (afLRTFIM) on the triangular lattice. We use the method of perturbative continuous unitary transformations (pCUT) to derive an effective model for the degenerate GS space of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system, by treating the transverse-field (TF) and the long-range interactions (LRI) as a perturbation. We determine a level-crossing between the gapped plain stripe phase at small TF and the gapped clock-ordered phase at intermediate TF at hc≅0.129 for α=6, N=36 spins in order three perturbation theory on the finite system. We discuss the qualitative layout of the quantum phase diagram of the afLRTFIM on the triangular lattice.
CRIS-Team Scopus Importer:2024-06-07
Keywords: Clock order; Long-range interactions; Order-by-disorder; Quantum Ising model; Transverse field; Triangular lattice

Order-by-disorder in the antiferromagnetic J1-J2-J3 transverse-field Ising model on the ruby lattice - Duft, Antonia; Koziol, Jan; Adelhardt, Patrick; Mühlhauser, Matthias; Schmidt, Kai Phillip (Physical Review Research, 2024)
We investigate the quantum phase diagram of the J1-J2-J3 antiferromagnetic transverse-field Ising model on the ruby lattice. In the low-field limit we derive an effective quantum dimer model, analyzing how the extensive ground-state degeneracy at zero field is lifted by an order-by-disorder scenario. We support our analysis by studying the gap closing of the high-field phase using series expansions. For J2>J3, we find a columnar phase at low fields, followed by a clock-ordered phase stabilized by resonating plaquettes at intermediate field values, and an emergent three-dimensional (3D)-XY quantum phase transition to the polarized high-field phase. For J3>J2, an order-by-disorder mechanism stabilizes a distinct k=(0,0) order and a quantum phase transition in the 3D-Ising universality class is observed. Further, we discuss the possible implementation of the columnar- and clock-ordered phase in existing Rydberg atom quantum simulators. When taking into account the full algebraically decaying long-range interactions on the ruby lattice, we find that long-range interactions favor the same ground state as the quantum fluctuations induced by a transverse field, which could make the ruby lattice a promising candidate for the realization of a clock-ordered phase.
CRIS-Team Scopus Importer:2024-10-25

Quantum criticality and excitations of a long-range anisotropic XY chain in a transverse field - Adelhardt, Patrick; Koziol, Jan; Schellenberger, Andreas; Schmidt, Kai Phillip (Physical Review B, 2020)
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 v analytically by a bosonic quantum-field theory.
CRIS-Team WoS Importer:2020-11-27

Quantum criticality of the transverse-field Ising model with long-range interactions on triangular-lattice cylinders - Koziol, Jan; Fey, Sebastian; Kapfer, Sebastian; Schmidt, Kai Phillip (Physical Review B, 2019)
To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically decaying long-range Ising interactions on quasi-one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out of the infinitely degenerate nearest-neighbor Ising ground-state space on the two-dimensional triangular lattice. Further, while cylinders with circumferences 6, 10, 14, etc., are always in the same stripe phase for any decay exponent of the long-range Ising interaction, the family of cylinders with circumferences 4, 8, 12, etc., displays a phase transition between two different types of stripe structures. For the full long-range transverse-field Ising model, we concentrate on cylinders with circumference four and six. The ground-state phase diagram consists of several quantum phases in both cases including an x-polarized phase, stripe-ordered phases, and clock-ordered phases which emerge from an order-by-disorder scenario already present in the nearest-neighbor model. In addition, the generic presence of a potential intermediate gapless phase with algebraic correlations and associated Kosterlitz-Thouless transitions is discussed for both cylinders.
CRIS-Team WoS Importer:2019-10-22

Quantum-critical properties of the long-range transverse-field Ising model from quantum Monte Carlo simulations - Koziol, Jan; Langheld, Anja; Kapfer, Sebastian; Schmidt, Kai Phillip (Physical Review B, 2021)
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 nu and 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.
CRIS-Team WoS Importer:2021-07-09

Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations - Krämer, Calvin; Koziol, Jan; Langheld, Anja; Hörmann, Max; Schmidt, Kai Phillip (SciPost Physics, 2024)
We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents νs and νw as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent β and the critical exponent νav of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.
CRIS-Team Scopus Importer:2024-09-06

Quantum phases of hardcore bosons with repulsive dipolar density-density interactions on two-dimensional lattices - Koziol, Jan; Morigi, G.; Schmidt, Kai Phillip (SciPost Physics, 2024)
We analyse the ground-state quantum phase diagram of hardcore Bosons interacting with repulsive dipolar potentials. The bosons dynamics is described by the extended-Bose-Hubbard Hamiltonian on a two-dimensional lattice. The ground state results from the interplay between the lattice geometry and the long-range interactions, which we account for by means of a classical spin mean-field approach limited by the size of the considered unit cells. This extended classical spin mean-field theory accounts for the long-range density-density interaction without truncation. We consider three different lattice geometries: square, honeycomb, and triangular. In the limit of zero hopping the ground state is always a devil’s staircase of solid (gapped) phases. Such crystalline phases with broken translational symmetry are robust with respect to finite hopping amplitudes. At intermediate hopping amplitudes, these gapped phases melt, giving rise to various lattice supersolid phases, which can have exotic features with multiple sublattice densities. At sufficiently large hoppings the ground state is a superfluid. The stability of phases predicted by our approach is gauged by comparison to the known quantum phase diagrams of the Bose-Hubbard model with nearest-neighbour interactions as well as quantum Monte Carlo simulations for the dipolar case on the square and triangular lattice. Our results are of immediate relevance for experimental realisations of self-organised crystalline ordering patterns in analogue quantum simulators, e.g., with ultracold dipolar atoms in an optical lattice.
CRIS-Team Scopus Importer:2024-10-25

Quantum robustness of the toric code in a parallel field on the honeycomb and triangular lattice - Kott, Viktor; Mühlhauser, Matthias; Koziol, Jan; Schmidt, Kai Phillip (SciPost Physics, 2024)
We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in z-direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising★ universality class for both signs of the field. The same is true for a positive field in x-direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising★. In contrast, for negative x-field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY★ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative x-fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of x- and z-fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising★ and 3D XY★ type with interesting potential multi-critical crossing points. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.
CRIS-Team Scopus Importer:2024-09-13

Scaling at quantum phase transitions above the upper critical dimension - Langheld, Anja; Koziol, Jan; Adelhardt, Patrick; Kapfer, Sebastian; Schmidt, Kai Phillip (SciPost Physics, 2022)
The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above the upper critical dimension following the recently introduced Q-FSS formalism for thermal phase transitions. Contrary to long-standing belief, the correlation sector is affected by dangerous irrelevant variables. The presented formalism recovers a generalized hyperscaling relation and FSS form. Using this new FSS formalism, we determine the full set of critical exponents for the long-range transverse-field Ising chain in all criticality regimes ranging from the nearest-neighbor to the long-range mean-field regime. For the same model, we also explicitly confirm the effect of dangerous irrelevant variables on the characteristic length scale.
CRIS-Team Scopus Importer:2022-12-02

Systematic analysis of crystalline phases in bosonic lattice models with algebraically decaying density-density interactions - Koziol, Jan; Duft, Antonia; Morigi, Giovanna; Schmidt, Kai Phillip (SciPost Physics, 2023)
We propose a general approach to analyse diagonal ordering patterns in bosonic lattice models with algebraically decaying density-density interactions on arbitrary lattices. The key idea is a systematic search for the energetically best order on all unit cells of the lattice up to a given extent. Using resummed couplings we evaluate the energy of the ordering patterns in the thermodynamic limit using finite unit cells. We apply the proposed approach to the atomic limit of the extended Bose-Hubbard model on the triangular lattice at fillings f = 1/2 and f = 1. We investigate the ground-state properties of the antiferromagnetic long-range Ising model on the triangular lattice and determine a six-fold degenerate plain-stripe phase to be the ground state for finite decay exponents. We also probe the classical limit of the Fendley-Sengupta-Sachdev model describing Rydberg atom arrays. We focus on arrangements where the atoms are placed on the sites or links of the Kagome lattice. Our method provides a general framework to treat cristalline structures resulting from long-range interactions.
CRIS-Team Scopus Importer:2023-06-30