In this thesis I will outline various quantum phase transitions in 2D models of magnets that are amenable to simulation with quantum Monte Carlo techniques. The key player in this work is the theory of deconfined criticality, which generically allows for zero temperature quantum phase transitions between phases that break distinct global symmetries. I will describe models with different symmetries including SU(N), SO(N), and "easy-plane" SU(N) and I will demonstrate how the presence or absence of continuous transitions in these models fits together with the theory of deconfined criticality.
In this thesis I will outline various quantum phase transitions in 2D models of magnets that are amenable to simulation with quantum Monte Carlo techniques. The key player in this work is the theory of deconfined criticality, which generically allows for zero temperature quantum phase transitions between phases that break distinct global symmetries. I will describe models with different symmetries including SU(N), SO(N), and "easy-plane" SU(N) and I will demonstrate how the presence or absence of continuous transitions in these models fits together with the theory of deconfined criticality.
Quantum Criticality
Phase transitions can be caused by temperature fluctuations or, more exotically, by quantum fluctuations at zero temperature. To describe some of these quantum phase transitions, researchers came up with a complex theory called deconfined quantum criticality. However, subsequent numerical simulations were inconsistent with some of the predictions of the theory, leading to a debate on its validity. By using quantum Monte Carlo simulations, Shao et al. show that it is possible to reconcile numerics with the theory for a specific model of 2D quantum magnetism.
Science , this issue p. [213][1]
[1]: /lookup/doi/10.1126/science.aad5007
The antiferromagnetic to valence-bond-solid phase transition in the two-dimensional J-Q model (an S=1/2 Heisenberg model with four-spin interactions) is studied using large-scale quantum Monte Carlo simulations. The results support a continuous transition of the ground state, in agreement with the theory of "deconfined" quantum criticality. There are, however, large corrections to scaling, of logarithmic or very slowly decaying power-law form, which had not been anticipated. This suggests that either the SU(N) symmetric noncompact CP;{N-1} field theory for deconfined quantum criticality has to be revised or that the theory for N=2 (as in the system studied here) differs significantly from N-->infinity (where the field theory is analytically tractable).
I will give an overview of unbiased numerical work on the Neelvalence bond solid (VBS) phase transition in d = 2 antiferromagnets. This progress has been possible due to the discovery of a new class of Hamiltonians of SU(N) spins that are free of the sign problem of quantum Monte Carlo. I will show through extensive numerical studies of the quantum phase transition on a variety of bipartite systems: square, rectangular, honeycomb and square bilayer, for a number of values of N (2 ≤ N ≤ 10), that a close to complete picture of an unusual “deconfined critical point” has emerged. Significantly, no direct evidence for first order behavior has been found on the largest simulations with 256 × 256 spins, the crucial role of Berry phases at the critical point has been verified, strong evidence for non-compact CPN−1 universality is evident for a range of N values, the inferred “dangerous” (ir)relevance of lattice anisotropy at the critical point is consistent with various limiting analytic calculations on the CPN−1 field theory and close to the critical point dramatic signatures of the emergent photon excitation have been detected in VBS correlation functions. I will conclude with some open theoretical issues that remain to be resolved and possible experimental realizations.
To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU(N) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU(N) N\'eel and valence-bond solid phases is characterized for N=2, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size (L=256) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y\ensuremath{\equiv}1/\ensuremath{\nu}$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field ${\ensuremath{\Psi}}^{2}$ for the SU(3) model.
Unraveling Deconfined Quantum Criticality in Non-Hermitian Easy-Plane J-Q Model
Deconfined quantum critical point (DQCP) characterizes the continuous transition beyond Landau-Ginzburg-Wilson paradigm, occurring between two phases that exhibit distinct symmetry breaking. The debate over whether genuine DQCP exists in physical SU(2) spin systems or the transition is weakly first-order has persisted for many years. In this letter, we construct a non-Hermitian easy-plane J-Q model and perform sign-problem-free quantum Monte Carlo (QMC) simulation to explore the impact of non-Hermitian microscopic interactions on the transition that potentially features a DQCP. Our results demonstrate that the intensity of the first-order transitions significantly diminishes with the amplification of non-Hermitian interactions, serving as numerical evidence to support the notion that the transition in J-Q model is quasi-critical, possibly in the vicinity of the fixed point governing DQCP in the complex plane, described by a non-unitary conformal field theory (CFT). The non-Hermitian interaction facilitates the approach towards such a complex fixed point in the parameter regime. Furthermore, our QMC study on the non-Hermitian J-Q model opens a new route to numerically investigating the nature of complex CFT in the microscopic model.
Strongly correlated low-dimensional quantum spin models provide a well-established frame- \nwork to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. \n \nFirst, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. \n \nSecond, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. \n \nThird, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.