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.
213In 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.
Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N ¼ 1, 2, and 3.The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm.In particular, one of the prominent phases for N > 1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z 2 gauge field in its deconfined phase.However, unlike a conventional Z 2 gauge theory, we do not impose "Gauss's Law" by hand; instead, it emerges because of spontaneous symmetry breaking.Correspondingly, unlike a conventional Z 2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature.By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N ¼ 2 and a valence-bond solid for N ¼ 3. Furthermore, for N ¼ 3, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.
Using Quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with
N flavors of fermions per site for
N=1,2 and
3. The models have an extensive number of conserved quantities but are not integrable, and have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond Landau-Ginzburg paradigm. In particular, one of the prominent phase for
N>1 corresponds to
2N gapless Dirac fermions coupled to an emergent
Z2 gauge field in its deconfined phase. However, unlike a conventional
Z2 gauge theory, we do not impose the `Gauss's Law' by hand and instead, it emerges due to spontaneous symmetry breaking. Correspondingly, unlike a conventional
Z2 gauge theory in two spatial dimensions, our models have a finite temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short range entangled phase, which corresponds to N\'eel/Superconductor for
N=2 and a Valence Bond Solid for
N=3. Furthermore, for
N=3, the Valence Bond Solid further undergoes a transition to a N\'eel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.
The theory of deconfined quantum critical (DQC) points describes phase transitions at absolute temperature T = 0 outside the standard paradigm, predicting continuous transformations between certain ordered states where conventional theory would require discontinuities. Numerous computer simulations have offered no proof of such transitions, instead finding deviations from expected scaling relations that neither were predicted by the DQC theory nor conform to standard scenarios. Here we show that this enigma can be resolved by introducing a critical scaling form with two divergent length scales. Simulations of a quantum magnet with antiferromagnetic and dimerized ground states confirm the form, proving a continuous transition with deconfined excitations and also explaining anomalous scaling at T > 0. Our findings revise prevailing paradigms for quantum criticality, with potential implications for many strongly correlated materials.
The transition between the N\'eel antiferromagnet and the valence-bond solid state in two dimensions has become a paradigmatic example of deconfined quantum criticality, a non-Landau transition characterized by fractionalized excitations (spinons). We consider an extension of this scenario whereby the deconfined spinons are subject to a magnetic field. The primary purpose is to identify the exotic scenario of a Bose-Einstein condensate of spinons. We employ quantum Monte Carlo simulations of the $J\ensuremath{-}Q$ model with a magnetic field, and we perform a quantum field theoretic analysis of the magnetic field and temperature dependence of thermodynamic quantities. The combined analysis provides evidence for Bose-Einstein condensation of spinons and also demonstrates an extended temperature regime in which the system is best described as a gas of spinons interacting with an emergent gauge field.
Phase transitions have been an active area of research in statistical mechanics for almost a century and have recently been integrated into quantum mechanics. Many phenomena such as superconductivity and unconventional magnetism are understood to arise from exotic quantum phases and at points describing quantum phase transitions. A detailed understanding of these phase transitions requires numerical simulations of models which benchmark realistic models against theoretical frameworks. The topic of this thesis is the implementation of Quantum Monte Carlo simulation, which is a powerful technique to understand quantum condensed matter, in interesting models to illustrate novel phenomena in magnetic systems. The novel features of condensed matter systems described in this thesis consist of emergent symmetries at critical points, interesting dynamical features of such systems and the drastic effects of defects in spin systems used in the field of adiabatic quantum computing. Emergent symmetries are shown by condensed matter systems especially at critical points and are features which cannot be shown by individual or a small number of spins. Examples of this in one and two dimensions are presented in an early chapter of this thesis. In addition to this, spin systems can show excitations which have an interesting spatial structure as a consequence of restricted dynamics which only allow the excitations to spread in a particular region. This is presented in the context of a simple model in the following chapter along with numerical support. The following chapter contains a description of adiabatic quantum computing along with a particular model which we study. The phase transition and the effects on the performance of adiabatic quantum computing are studied in this context.