Quantum Monte Carlo simulations of dimension-tunability phenomena in the quantum spin system

Abstract

Dimension, referring to the number of spatial dimensions, has always been central to modern physics, particularly condensed matter physics. Capturing and understanding the various phenomena related to dimension have always served as a topic of great interest. With the rapid development of quantum Monte Carlo (QMC) methods, these phenomena in quantum spin systems can now be numerically simulated and investigated. This thesis is a collection of my works aimed at broadening our understanding about the role that dimension plays in quantum spin systems. This thesis collaborates my works from three perspectives: the dimension of the quantum spin system, the dimension of quasiparticle mobility, and the dimension of system symmetry.

In the first part, by focusing on a quasi-one-dimensional geometry, we investigate the amplitude mode in a system of weakly coupled spin chains with the help of QMC simulations, stochastic analytic continuation (SAC), and a chain-mean field approach combined with a mapping to the field-theoretic sine-Gordon model. When the coupled spin chain tending to one-dimensional spin chains, the amplitude mode emerges in the longitudinal spin susceptibility while the goldstone model appears in the transverse channel in the presence of a weak symmetry-breaking staggered field. To capture the spectral information in transverse channel, we develop our measurement method of their corresponding imaginary time correlation by tracing the update process in the QMC simulation. Using this measurement method, we successfully identify these two excitations with the second (first) breather of the sine-Gordon theory, correspondingly.

In the second part, as an unconventional realization of topological orders with an interplay of topology and geometry, fracton (topological) orders feature subextensive topological ground state degeneracy and subdimensional excitations that are movable only within a certain subspace. A well-known example of the type-I fracton orders is the exactly solvable three-dimensional X-cube model, that mobility constraints on subdimensional excitations originate from the absence of spatially deformable string-like operators. To unveil the interplay of topology and geometry, we study both real-space correlation functions and dynamic structure factors of subdimensional excitations in the fracton phase and their evolution into the trivial paramagnetic phase by increasing external fields. We find in the fracton phase, that the correlation functions and the spectral functions show clear anisotropy exactly caused by the underlying mobility constraints.

In the third part, we employ strange correlators to detect 2D subsystem symmetry protected topological (SSPT) phases which are nontrivial topological phases protected by subsystem symmetries. Specifically, we construct efficient strange correlators in the 2D cluster model in the presence of uniform magnetic field, and then perform the projector QMC simulation within the quantum annealing scheme. We find that strange correlators show long range correlation in the SSPT phase, from which we define strange order parameters to characterize the topological phase transition between the SSPT phase at low fields and the trivial paramagnetic phase at high fields. We also find interesting spatial anisotropy of a strange correlator, which can be traced back to the nature of spatial anisotropy of subsystem symmetries that protect SSPT order in the 2D cluster model.