Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension

Universal scaling in real dimension

The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its scaling theory is controlled by a single parameter, the spectral dimension ds, which plays the role of the relevant parameter on complex geometries. The graph under consideration allows us to tune the value of the spectral dimension continuously also to noninteger values and to find the universal exponents as continuous functions of the dimension. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of O ( N ) symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in real dimensions.

Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions

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.

Finite-size scaling of the random-field Ising model above the upper critical dimension

Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.

Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.

Logarithmic finite-size scaling of the self-avoiding walk at four dimensions

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z_{c}=0.147622380(2), improving over the existing result z_{c}=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z_{c}, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L^{2}(lnL)^{2y[over ̂]_{h}} and C∼(lnL)^{2y[over ̂]_{t}}, and the logarithmic exponents are determined as y[over ̂]_{h}=0.251(2) and y[over ̂]_{t}=0.25(3), in excellent agreement with the field theoretical prediction y[over ̂]_{h}=y[over ̂]_{t}=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.

Complete graph and Gaussian fixed-point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries

We present an extensive Markov chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that both scaling predictions, i.e., the complete graph and the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.

Finite-size scaling of O(n) systems at the upper critical dimensionality

Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (d _c = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows [Formula: see text] governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as [Formula: see text] with [Formula: see text] a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.

Random-Length Random Walks and Finite-Size Scaling in High Dimensions

We address a long-standing debate regarding the finite-size scaling (FSS) of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behavior previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behavior of the corresponding Green's function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.

Size effects in finite systems with long-range interactions

Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)FRACEG0218-348X10.1142/S0218348X95000473], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as 1/r^{1+σ} in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.

Lifted worm algorithm for the Ising model

We design an irreversible worm algorithm for the zero-field ferromagnetic Ising model by using the lifting technique. We study the dynamic critical behavior of an energylike observable on both the complete graph and toroidal grids, and compare our findings with reversible algorithms such as the Prokof'ev-Svistunov worm algorithm. Our results show that the lifted worm algorithm improves the dynamic exponent of the energylike observable on the complete graph and leads to a significant constant improvement on toroidal grids.

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behavior for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.