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arXiv:1111.0629 · Physical Review B

Quantum criticality in a dissipative (2+1)-dimensional XY model of circulating currents in high- Tc cuprates

Iver Bakken Sperstad, Einar B. Stiansen, Asle Sudbø

Published Nov 8, 2011• 6 citations
arXivHTMLPDF

Abstract

We present large-scale Monte Carlo results for the dynamical critical exponent $z$ and the spatio-temporal two-point correlation function of a (2+1)-dimensional quantum $XY$ model with bond dissipation, proposed to describe a quantum critical point in high-${T}_{c}$ cuprates near optimal doping. The phase variables of the model, originating with a parametrization of circulating currents within the CuO${}_{2}$ 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.

DOI
10.1103/physrevb.84.180503

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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, \ensuremathθr,\ensuremathτ\ensuremath∈[\ensuremath−\ensuremathπ,\ensuremathπ\ensuremath⟩{{\ensuremath{\theta}}_{\mathbf{r},\ensuremath{\tau}}}\ensuremath{\in}[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}\ensuremath{\rangle}\ensuremathθr,\ensuremathτ​\ensuremath∈[\ensuremath−\ensuremathπ,\ensuremathπ\ensuremath⟩. The dynamical critical exponent is found to be z\ensuremath≈1z\ensuremath{\approx}1z\ensuremath≈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.
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.
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.
This thesis presents seven research papers on topics in condensed-matter theory. Five of the papers report on Monte Carlo studies of quantum phase transitions in various (d+1)-dimensional statistical mechanics models featuring Caldeira-Leggettlike dissipation. The principal motivation for these studies was to investigate a particular bond-dissipative (2+1)-dimensional XY model of circulating currents in cuprate high-temperature superconductors. It has been proposed that quantum critical fluctuations associated with a local quantum critical point described by this model can explain the marginal-Fermi-liquid behaviour of the normal state of these compounds. We present simulation results for this model for both compact and noncompact phase variables and show unambiguously that the quantum critical point in the compact case is not local. If the phases are taken to be noncompact variables, the model is also a model of resistively shunted Josephson junction arrays. The results in this case reveal a more complicated phase diagram, but we have not been able to establish critical behaviour consistent with the scenario of local quantum criticality.\n\nThe study of extended quantum dissipative models is also motivated by the general effect on condensed-matter systems of the coupling to environmental degrees of freedom. Their influence on quantum critical phenomena is characterized by the dynamical critical exponent z, a measure of spatiotemporal anisotropy, the value of which can be estimated by naive scaling arguments. We confirm by numerical means that such scaling estimates give correct results to a good approximations (with a few reservations), irrespective of system dimensionality, order parameter symmetry, or whether the variables are compact or noncompact. Corrections to the naive scaling estimates have to be invoked for strongly super-Ohmic dissipation for d = 1 due to relatively large values of the anomalous scaling dimension η.\n\nThe two last research papers are concerned with the superconducting pairing state of the recently discovered class of iron-based high-temperature superconductors. Here, we calculate possible signatures of the proposed s±-wave pairing state in conductance-spectroscopy and Josephson-effect experiments.
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.
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.
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.