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Renormalization Group Analysis of 2+1D Quantum XY Model With Dissipation

Changtao Hou

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

This thesis present the recently theoretical and numerical results on 2D dissipative quantum XY model. The two-dimensional quantum XY model is applicable to the quantum critical properties of several experimental systems, such as superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, and loop current order transition in the cuprates. Renormalization group methods are applied to solve the reformulated ac- tion of the original model in terms of two type topological excitations: vortices and warps. The calculations explain the extraordinary properties of the model studied through quan- tum Monte Carlo simulations: the separability of the correlation function in space and time, the correlation length in space proportional to logarithm of the correlation length in time near the transition from disordered phase to ordered phase. The running dynamical critical exponent is introduced to address the anisotropy between time and space. The effects of anisotropy fields have been examined through renormalization group method. The transi- tion from disordered phase to ordered phase of this model has been studied by quantum Monte Carlo. The divergence of temporal correlation length in function of (Kc − K)/Kc is examined by numerical simulation. The logarithmic relation between temporal correlation length and spacial correlation length is further confirmed. Also, the same logarithmic rela- tion for different correlation function with different space separation is found and implicitly confirmed the separability of correlation function in space and time.

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This thesis present the recently theoretical and numerical results on 2D dissipative quantum XY model. The two-dimensional quantum XY model is applicable to the quantum critical properties of several experimental systems, such as superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, and loop current order transition in the cuprates. Renormalization group methods are applied to solve the reformulated ac- tion of the original model in terms of two type topological excitations: vortices and warps. The calculations explain the extraordinary properties of the model studied through quan- tum Monte Carlo simulations: the separability of the correlation function in space and time, the correlation length in space proportional to logarithm of the correlation length in time near the transition from disordered phase to ordered phase. The running dynamical critical exponent is introduced to address the anisotropy between time and space. The effects of anisotropy fields have been examined through renormalization group method. The transi- tion from disordered phase to ordered phase of this model has been studied by quantum Monte Carlo. The divergence of temporal correlation length in function of (Kc − K)/Kc is examined by numerical simulation. The logarithmic relation between temporal correlation length and spacial correlation length is further confirmed. Also, the same logarithmic rela- tion for different correlation function with different space separation is found and implicitly confirmed the separability of correlation function in space and time.
The two-dimensional quantum XY model, with a Caldeira-Leggett form of dissipation, is applicable to the quantum-critical properties of diverse experimental systems, ranging from superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, to the loop-current order transition in cuprates. We solve the reexpression of this model in terms of orthogonal topological excitations, vortices, and a variety of instantons, by renormalization group methods. The calculations explain the extraordinary properties of the model discovered in Monte Carlo calculations: the product form of the quantum-critical fluctuations in space and time, a spatial correlation length proportional to the logarithm of the temporal correlation length near the transition from a disordered to a fully ordered state, and the occurrence of a phase with spatial order without temporal order. They are intimately related to the flow of the metric of time in relation to the metric of space, i.e., of the dynamical critical exponent zzz. These properties appear to be essential in understanding the strange metallic phase found in a variety of quantum-critical transitions as well as the accompanying high-temperature superconductivity.
We use quantum Monte Carlo simulations to calculate the phase diagram and the correlation functions for the quantum phase transitions in the two-dimensional dissipative quantum XY model with and without fourfold anisotropy. Without anisotropy, the model describes the superconductor-to-insulator transition in two-dimensional dirty superconductors. With anisotropy, the model represents the loop-current order observed in the underdoped cuprates and its fluctuations, as well as the fluctuations near the ordering vector in simple models of two-dimensional itinerant ferromagnets and itinerant antiferromagnets. These calculations test an analytic solution of the model which reexpressed it in terms of topological excitations: the vortices with interactions only in space but none in time, and warps with leading interactions only in time but none in space, as well as subleading interactions which are both space and time dependent. For parameters where the proliferation of warps dominates the phase transition, the critical fluctuations as functions of the deviation of the dissipation parameter $\ensuremath{\alpha}$ on the disordered side from its critical value ${\ensuremath{\alpha}}_{c}$ are scale invariant in imaginary time $\ensuremath{\tau}$ as the correlation length in time ${\ensuremath{\xi}}_{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}{e}^{{[{\ensuremath{\alpha}}_{c}/({\ensuremath{\alpha}}_{c}\ensuremath{-}\ensuremath{\alpha})]}^{1/2}}$ diverges, where ${\ensuremath{\tau}}_{c}$ is a short-time cutoff. On the other hand, the spatial correlations develop with a correlation length ${\ensuremath{\xi}}_{x}\ensuremath{\approx}{\ensuremath{\xi}}_{0}ln({\ensuremath{\xi}}_{\ensuremath{\tau}})$, with ${\ensuremath{\xi}}_{0}$ of the order of a lattice constant. The dynamic correlation exponent zzz is therefore $\ensuremath{\infty}$. The Monte Carlo calculations also directly show warps and vortices. Their densities and correlations across the various transitions in the model are calculated and related to those of the order-parameter correlations in the dissipative quantum XY model.
An important problem in contemporary physics concerns quantum-critical fluctuations in metals. A scaling function for the momentum, frequency, temperature, and magnetic field dependence of the correlation function near a 2D-ferromagnetic quantum-critical point (QCP) is constructed, and its singularities are determined by comparing to the recent calculations of the correlation functions of the dissipative quantum XY model (DQXY). The calculations are motivated by the measured properties of the metallic compound YFe2Al10{\mathrm{YFe}}_{2}{\mathrm{Al}}_{10}YFe2​Al10​, which is a realization of the DQXY model in 2D. The frequency, temperature, and magnetic field dependence of the scaling function as well as the singularities measured in the experiments are given by the theory without adjustable exponents. The same model is applicable to the superconductor-insulator transitions, classes of metallic AFM-QCPs, and as fluctuations of the loop-current ordered state in hole-doped cuprates. The results presented here lend credence to the solution found for the 2D-DQXY model and its applications in understanding quantum-critical properties of diverse systems.
We present large-scale Monte Carlo results for the dynamical critical exponent zzz and the spatio-temporal two-point correlation function of a (2+1)-dimensional quantum XYXYXY model with bond dissipation, proposed to describe a quantum critical point in high-Tc{T}_{c}Tc​ cuprates near optimal doping. The phase variables of the model, originating with a parametrization of circulating currents within the CuO2{}_{2}2​ unit cells in cuprates, are compact, ${{\ensuremath{\theta}}_{\mathbf{r},\ensuremath{\tau}}}\ensuremath{\in}[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}\ensuremath{\rangle}$. The dynamical critical exponent is found to be $z\ensuremath{\approx}1$, and the spatio-temporal correlation functions are explicitly demonstrated to be isotropic in space-imaginary time. The model thus has a fluctuation spectrum where momentum and frequency enter on equal footing, rather than having the essentially momentum-independent marginal Fermi-liquid-like fluctuation spectrum previously reported for the same model.
Earlier Monte Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY transition, i.e., one with a dynamical critical exponent z=1z=1z=1. The transition changes from z=1z=1z=1 to the class of criticality with $z\ensuremath{\rightarrow}\ensuremath{\infty}$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.
In a recent paper [V. Aji and C. M. Varma, Phys. Rev. Lett. 99, 067003 (2007)] we have shown that the dissipation driven quantum phase transition of the two-dimensional XYXYXY model represents a universality class where the correlations at criticality are local in space and power law in time. Here we provide a detailed analysis of the model. The local criticality is brought about by the decoupling of infrared singularities in space and time. The former leads to a Kosterlitz-Thouless transition whereby the excitations of the transverse component of the velocity field (vortices) unbind in space. The latter, on the other hand, leads to a transition among excitations (warps) in the longitudinal component of the velocity field, which unbind in time. The quantum Ashkin-Teller model, with which the observed loop order in the cuprates is described, maps in the critical regime to the quantum XYXYXY model. We also discuss other models which are expected to have similar properties.