The Directed-Loop Algorithm

Abstract

The directed-loop scheme is a framework for generalized loop-type updates in quantum Monte Carlo, applicable both to world-line and stochastic series expansion methods. Here, the directed-loop equations, the solution of which gives the probabilities of the various loop-building steps, are discussed in the context of the anisotropic S = 1/2 Heisenberg model in a uniform magnetic field. This example shows how the directed-loop concept emerges as a natural generalization of the conventional loop algorithm, where the loops are selfavoiding, to cases where selfintersection must be allowed in order to satisfy detailed balance.The directed-loop algorithm

Method to Characterize Spinons as Emergent Elementary Particles

“…The ground state of this system is in the class of the standard critical Heisenberg chain for g < g c and is a doubly-degenerate VBS for g > g c . Using Lanczos diagonalization to extract the lowest singlet and triplet excitations and studying their crossings in the standard way for this kind of transition (see, e.g., [17]), we obtain g c ≈ 0.1645 (in agreement with a recent QMC study of the critical properties of the same model [18]). We have also studied the J-Q 2 model, i.e., using two singlet projectors in the Q-term in (1), for which g c ≈ 0.84831.…”

Machine learning phases of matter

“…Among the most powerful are Monte Carlo simulations, which consist of two steps: a stochastic importance sampling over state space, and the evaluation of estimators for physical quantities calculated from these samples [3]. These estimators are constructed based on a variety of physical impetuses; e.g.…”

“…We thus anticipate adoption to the field of quantum technology [23], such as quantum error correction protocols and quantum state tomography [24]. The ability of machine learning algorithms to generalize to situations beyond their original design anticipates future applications such as the detection of phases and phase transitions in models vexed with the Monte Carlo sign problem [3], as well as in experiments with single-site resolution capabilities such as the modern quantum gas microscopes [25,26]. As in all other areas of "big data", we expect the rapid adoption of machine learning techniques as a basic research tool in condensed matter and statistical physics in the near future.…”

“…Despite this curse, the machine learning community has developed a number of techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. In light of this success, it is natural to ask whether such techniques could be applied to the arena of condensed-matter physics, particularly in cases where the microscopic Hamiltonian contains strong interactions, where numerical simulations are typically employed in the study of phases and phase transitions [2,3]. We demonstrate that modern machine learning architectures, such as fully-connected and convolutional neural networks [4], can provide a complementary approach to identifying phases and phase transitions in a variety of systems in condensed matter physics.…”

Thermodynamic properties of the 2D frustrated Heisenberg model for the entire J1–J2 circle

“…Note, that different alternative states are considered to be competitive in the locality of LRO disappearance at T = 0. These are in particular columnar and box phases which preserve the SU(2) symmetry, but brake the translational one (see [61] for recent review). The disordered state in SSSA is always spin-liquid-like in the above noted sense and it can not be distinguished from the mentioned alternatives with nearby energies.…”