We present an overview of the quantum criticality and finite-size scaling properties (FSS) of the Ising model in a transverse field with algebraically decaying interactions. We summarize our numerical studies based on stochastic series expansion quantum Monte Carlo simulations and linked-cluster expansions using white graphs in combination with Monte Carlo integration for graph embedding. We focus on ferromagnetic interactions in a linear chain and present a complete set of critical exponents as functions of the decay exponents of the long-range interactions. We identify the three regimes predicted by field theory, ranging from the nearest-neighbour Ising to the long-range Gaussian universality classes, with an intermediate regime showing a range of critical exponents. In the long-range Gaussian regime, the system's dimension is above the upper critical dimension of the transition, causing the hyperscaling relation and standard FSS to break down due to irrelevant variables. We present 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. The presented formalism recovers a generalized hyperscaling relation and FSS forms for observables. In an outlook, we present our recent developments for antiferromagnetic interactions and frustrated lattice geometries.
In this talk, we present a reduced basis method for quantum spin eigenvalue problems. Within the reduced basis methods approach, an effective low-dimensional subspace of a quantum many-body Hilbert space is constructed in order to investigate, e.g., the ground-state phase diagram. The basis of this subspace is built from solutions of snapshots, i.e., ground states corresponding to particular and well-chosen parameter values. Here, we show how a greedy strategy to assemble the reduced basis and thus to select the parameter points can be implemented based on matrix-product-states (MPS) calculations using DMRG-optimization. Once the reduced basis has been obtained, observables required for the computation of phase diagrams can be computed with a computational complexity independent of the underlying Hilbert space for any parameter value. We illustrate the efficiency and accuracy of this approach for different quantum spin-chains.
In this talk I will discuss quantum analogues of classical reaction-diffusion models. Here, fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes, such as pair annihilation, A + A -> 0, or coagulation, A + A -> A. It is known, that in classical systems of that kind, the interplay between dissipative events and diffusive particle hopping leads to critical collective dynamics, e.g. manifesting in a power-law decay of the particle density. In quantum systems, where diffusion is replaced by coherent hopping, the competition with dissipation also leads to such collective effects. However, the particulars may be different. In this talk I will focus on the so-called reaction-limited regime, where spatial density fluctuations are quickly smoothed out due to fast hopping. By exploiting the time-dependent generalized Gibbs ensemble method, I will discuss how coherent hopping and quantum statistics alter reaction-diffusion behaviour.
I will discuss the influence of boundaries on the critical temperature of superconductors in Bardeen-Cooper-Schrieffer (BCS) theory. I will present recent results showing that the critical temperature on half-spaces in dimensions \(d=1,2,3\) is strictly higher than on \(\mathbb R^d\), at least at weak coupling. These results are based on a well-known criterion providing a lower bound on the critical temperature. Furthermore, I will present a new linear criterion providing an upper bound on the critical temperature and show that the relative difference of the critical temperatures on half-space and \(\mathbb R^d\) vanishes in the weak-coupling limit.
This work analyses the decay behavior of the Bardeen-Cooper-Schrieffer (BCS) superconducting gap function in the presence of power-law long-range electron-electron interactions on two-dimensional (2D) and three-dimensional (3D) lattices. For s-wave and other non-nodal cases, the gap function exhibits the typical exponential decay in real space. In contrast, we numerically demonstrate and analytically prove that nodal d-wave states in \(d\) dimensions with interactions following a \(1/\vert x\vert^\nu\) dependence exhibit a gap function that decays algebraically at zero temperature, with an exponent \(d+\nu\). Therefore, the decay rate of the gap function directly reflects the exponent of the long-range interactions and is critically dependent on the system's dimensionality, thus allowing the characterization of interaction range from the superconducting gap behavior.
Magnetic skyrmions and antiskyrmions are characterised by an integer topological charge, describing the winding of the magnetic orientation. Half-integer winding numbers, can be obtained for magnetic vortices (merons). Here, we discuss the physics of magnetic textures with fractional topological charge which is neither integer nor half-integer [1,2].
For example, long-ranged Coulomb interactions in quantum anomalous Hall systems as realized in twisted bilayer graphene favour the formation of textures best described as a lattice of fractionally charged objects [2]. Furthermore,
fractional textures naturally arise when three or more magnetic domains meet in magnets with, e.g., cubic anisotropy [1]. We also show that a single magnetic skyrmion can explode into four fractional defects [1], each carrying charge -1/4.
Only defects with fractional charge lead to an Aharonov-Bohm effect for magnons characterized by highly singular scattering.
Power-law interactions with long-range behavior are ubiquitous in nature. However, the exact representation of these long-range interactions on the macroscopic scale is often challenging, as standard approaches like the continuum approximation do not yield the correct quantitative behavior. In this talk, we present the Singular Euler-Maclaurin Expansion as a powerful tool for the analytical and numerical treatment of long-range interacting lattice systems for any dimension, any lattice, and any power-law exponent. Using methods from analytic number theory and applied functional analysis, we rigorously derive lattice corrections to the continuum representation. We apply this approach to highly relevant systems, such as spin crystals with a multi-atomic basis in 3D with dipolar interactions. We conclude the presentation with an outlook on unconventional superconductivity in multiband systems in the presence of long-range interactions.
At the beginning we will discuss new experimental and theoretical results on the magnetic hopfions [1]. These three-dimensional topological states were found to form combinations with the skyrmion strings, whereas as a subject of homotopy theory such combinations are shown to be classified by the free abelian group \(\mathbb Z\times \mathbb Z\). We will briefly consider this group of quadratic growth and explain why the systems classified by homotopy groups of exponential growth may be of further interest [2]. Finally, we will discuss our efficient methods in numerical micromagnetics based on a novel rigorously justified representation [3] of magnetostatic energy.
[1] F. Zheng, et al., Nature 623, 718 (2023)
[2] F.N. Rybakov & O. Eriksson, arXiv:2205.15264 (2022)
[3] G. Di Fratta, et al., SIAM J. Math. Anal. 52, 3580 (2020)
We introduce a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with \(3\times 10^{23}\) particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
In my talk I will present some examples of superconducting quantum networks embedded in a dissipationless transmission line. I will concentrate on two examples: vertex-sharing frustrated Kagome lattice and frustrated saw-tooth chains composed of Josephson junctions. The frustration is provided by periodically arranged 0- and π- Josephson junctions. We will derive effective Ising-type spin Hamiltonian with strongly anisotropic long-range interaction and discuss quantum and classical phases of these models.
In this talk, I will present a new method for the numerical simulation of the Gross–Pitaevskii (GP) equation, for which a well-known feature is the appearanc e of quantized vortices with core size of the order of a small parameter ε. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit ε → 0, through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite dimensional GP equation for small, but finite, ε. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of ε typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.
Heavy Fermion materials have captivated researchers for decades due to their rich phase diagrams and the appearance of non-Fermi-liquid critical points, exhibiting intriguing phenomena such as linear-in-temperature resistivity, Fermi-surface reconstructions, and unconventional energy-over-temperature scaling. Amazingly, these strange metal phenomena manifest across a diverse set of material platforms, including flat band systems and optimally doped transition metal compounds, indicating the existence of a new type of robust quantum phase of matter [1].
In this talk, I will present results obtained from an optimized DMFT(NRG) algorithm for a multi-orbital extension of the periodic Anderson model (Kondo superlattices). Two critical aspects will be addressed:
(I) Demonstrating the existence of a novel critical point associated with Kondo breakdown, characterized by purely local fluctuations arising from destructive hybridization interference [2].
(I) Uncovering a novel "below-Kondo" energy scale of the PAM, emerging from recently discovered reentrant Kondo physics in the single impurity context [3].
We compare our numerical findings with predictions derived from the Lieb-Mattis theorem (LMT) and show the necessity of the new energy scale to consistently reconcile the predictions of LMT with the conventional single-impurity limit for well-separated local moments.
In this talk, we discuss numerical methods for computing the probability distributions of random matrix theory. We begin by describing some applications, and give a brief overview of previously proposed numerical methods. We then describe a new class of numerical methods, which takes advantage of the fact that the compact integral operators appearing in determinantal formulas for these distributions commute with certain differential operators (i.e., these compact operators are bispectral). These methods exploit these differential operators to facilitate extremely rapid and accurate calculations, even deep in the distributions' tails. We also discuss open problems and current research.
The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. Beyond its numerous applications in pure mathematics, it has recently been identified as a key component in simulating exotic quantum materials. In this work, we derive a compact and efficiently computable representation of the Epstein zeta function and examine its analytical properties across all arguments. We introduce a superexponentially convergent algorithm, including error bounds, for computing the Epstein zeta function in arbitrary dimensions. To facilitate the computation of integrals involving the Epstein zeta function, we decompose it into a power-law singularity and a regularized Epstein zeta function, which is analytic in the first Brillouin zone. We present the first implementation of the Epstein zeta function and its regularization for arbitrary real arguments in EpsteinLib, a high-performance C library with Python bindings, and rigorously benchmark its precision and performance against known formulas, achieving full precision across the entire parameter range. Finally, we apply our library to the computation of Casimir energies in multidimensional geometries.
Topological phases in superconductors are of great interest due to their potential use for topological quantum computing. They are usually characterized by quasiparticles located at the edges of the superconductor. In this talk we show results for the 1d long-range superconductor with spin solved self-consistently with the mean-field approximation. They exhibit massive edge states that exponentially decay to the short-range massless edge states with increasing long-range power law. This mass is not removed by lifting the spin degeneracy by a magnetic field.
Discrete Time Crystals (DTC) featuring long-range spatio-temporal order have been observed recently on noisy intermediate-scale (NISQ) quantum hardware. However, the intrinsic decoherence times of these devices restrict the observation of DTC oscillations to a low number cycles in the application of the Floquet unitary in a quantum circuit computation. Focusing on a paradigmatic DTC model, in this talk we will explore a scheme based on in-circuit measurements and feedback-assisted evolution of these quantum states, reporting how DTC correlations can be enhanced beyond intrinsic decoherence time scales of a few simple noise models, opening the path to explore methods leading to a longer observation of this genuine non-equilibrium phase of matter on realistic quantum hardware.