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arXiv:cond-mat/9810399 · Physical review. B, Condensed matter

Universal relaxational dynamics near two-dimensional quantum critical points

Subir Sachdev A

Published Jun 1, 1999• 103 citations
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Abstract

We describe the nonzero temperature $(T)$, low frequency $(\ensuremath{\omega})$ dynamics of the order parameter near quantum critical points in two spatial dimensions $(d)$, with a special focus on the regime $\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T$. For the case of a ``relativistic,'' $\mathrm{O}(n)$-symmetric, bosonic quantum field theory we show that, for small $\ensuremath{\epsilon}=3\ensuremath{-}d$, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in $d=2$. We describe the crossover from the finite frequency, ``amplitude fluctuation,'' gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency ``phase'' $(n>~2)$ or ``domain wall'' $(n=1)$ relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For $n=2$, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For $n=3$, our computations of the dynamic structure factor relate to neutron scattering experiments on ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{Cu}\mathrm{}\mathrm{O}}_{4}$, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in $d=2$. Although computations in appendixes do rely upon technical results on the $\ensuremath{\epsilon}$-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.

DOI
10.1103/physrevb.59.14054

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We describe the nonzero temperature (T)(T)(T), low frequency (\ensuremathω)(\ensuremath{\omega})(\ensuremathω) dynamics of the order parameter near quantum critical points in two spatial dimensions (d)(d)(d), with a special focus on the regime \ensuremath\Elzxh\ensuremathω\ensuremath≪kBT\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T\ensuremath\Elzxh\ensuremathω\ensuremath≪kB​T. For the case of a relativistic,'' $\mathrm{O}(n)$-symmetric, bosonic quantum field theory we show that, for small $\ensuremath{\epsilon}=3\ensuremath{-}d$, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in $d=2$. We describe the crossover from the finite frequency, amplitude fluctuation,'' gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency phase'' $(n>~2)$ or domain wall'' (n=1)(n=1)(n=1) relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For n=2n=2n=2, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For n=3n=3n=3, our computations of the dynamic structure factor relate to neutron scattering experiments on La1.85Sr0.15CuO4{\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{Cu}\mathrm{}\mathrm{O}}_{4}La1.85​Sr0.15​CuO4​, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in d=2d=2d=2. Although computations in appendixes do rely upon technical results on the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.
This paper presents a unified perspective on the results of two recent works (C. Buragohain and S. Sachdev cond-mat/9811083 and S. Sachdev cond-mat/9810399) along with additional background. We describe the low frequency, non-zero temperature, order parameter relaxational dynamics of a number of systems in the vicinity of a quantum critical point. The dynamical correlations are properties of the high temperature limit of renormalizable quantum field theories in spatial dimensions d=1,2. We study, as a function of d and the number of order parameter components, n, the crossover from the finite frequency, "amplitude fluctuation", gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency "phase" (n>=2) or "domain wall" (n=1) relaxation mode near the ordered phase. Implications for dynamic measurements on the high temperature superconductors and antiferromagnetic spin chains are noted.
A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=Tc{\mathrm{T}}_{\mathrm{c}}Tc​(t), t0, with Tc{\mathrm{T}}_{\mathrm{c}}Tc​(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-Tc{\mathrm{T}}_{\mathrm{c}}Tc​(t)|\ensuremath{\ll}Tc{\mathrm{T}}_{\mathrm{c}}Tc​(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}kB{\mathrm{k}}_{\mathrm{B}}kB​T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$kB{\mathrm{k}}_{\mathrm{B}}kB​T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.
The dilute, two-dimensional Bose gas exhibits a novel regime of relaxational dynamics in the regime ${k}_{B}T\ensuremath{\gtrsim}\ensuremath{\mid}\ensuremath{\mu}\ensuremath{\mid}$ where TTT is the absolute temperature and $\ensuremath{\mu}$ is the chemical potential. This may also be interpreted as the quantum criticality of the zero density quantum critical point at $\ensuremath{\mu}=0$. We present a theory for this dynamic, to leading order in $1∕\mathrm{ln}[\ensuremath{\Lambda}∕({k}_{B}T)]$, where $\ensuremath{\Lambda}$ is a high energy cutoff. Although pairwise interactions between the bosons are weak at low energy scales, the collective dynamics are strongly coupled even when $\mathrm{ln}(\ensuremath{\Lambda}∕T)$ is large. We argue that the strong-coupling effects can be isolated in an effective classical model, which is then solved numerically. Applications to experiments on the gap-closing transition of spin-gap antiferromagnets in an applied field are presented.
Following on from our previous work [M. J. Bhaseen et al., Phys. Rev. Lett. 98, 166801 (2007)] we examine the finite temperature magnetothermoelectric response in the vicinity of a quantum critical point (QCP). We begin with general scaling considerations relevant to an arbitrary QCP, either with or without Lorentz invariance, and in arbitrary dimension. In view of the broad connections to high-temperature superconductivity and cold atomic gases, we focus on the quantum critical fluctuations of the relativistic Landau-Ginzburg theory. This paradigmatic model arises in many contexts and describes the (particle-hole symmetric) superfluid-Mott insulator quantum phase transition in the Bose-Hubbard model. The application of a magnetic field opens up a wide range of physical observables, and we present a detailed overview of the charge and thermal transport and thermodynamic response. We combine several different approaches including the epsilon expansion and associated quantum Boltzmann equation, entropy drift, and arguments based on Lorentz invariance. The results differ markedly from the zero-field case, and we include an extended discussion of the finite thermal conductivity which emerges in the presence of a magnetic field. We derive an integral equation that governs its response and explore the crossover upon changing the magnetic field. This equation may be interpreted as a projection equation in the low-field limit, and clearly highlights the important role of collision invariants (or zero modes) in the hydrodynamic regime. Using an epsilon expansion around three dimensions, our analytic and numerical results interpolate between our previously published value and the exact limit of two-dimensional relativistic magnetohydrodynamics.
We study the thermodynamics of the relativistic quantum O(N)(N)(N) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N({T}^{3}/{c}^{2}){\mathcal{F}}_{N}(\ensuremath{\Delta}/T)$, where ccc is the velocity of the excitations at the QCP and $|\ensuremath{\Delta}|$ a characteristic zero-temperature energy scale. Using both a large-NNN approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function FN{\mathcal{F}}_{N}FN​. For small values of NNN ($N\ensuremath{\lesssim}10$) we find that FN(x){\mathcal{F}}_{N}(x)FN​(x) is nonmonotonic in the quantum critical regime ($|x|\ensuremath{\lesssim}1$) with a maximum near x=0x=0x=0. The large-NNN approach---if properly interpreted---is a good approximation both in the renormalized classical ($x\ensuremath{\lesssim}\ensuremath{-}1$) and quantum disordered ($x\ensuremath{\gtrsim}1$) regimes, but fails to describe the nonmonotonic behavior of FN{\mathcal{F}}_{N}FN​ in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio ${T}_{\mathrm{BKT}}/{\ensuremath{\rho}}_{s}(0)$ is very close to $\ensuremath{\pi}/2$, implying that the stiffness ${\ensuremath{\rho}}_{s}({T}_{\mathrm{BKT}}^{\ensuremath{-}})$ at the transition is only slightly reduced with respect to the zero-temperature stiffness ${\ensuremath{\rho}}_{s}(0)$. Finally, we briefly discuss the experimental determination of the universal function F2{\mathcal{F}}_{2}F2​ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.
We study the relativistic O (N) model, a quantum generalization of the φ⁴ theory used in statistical physics to study some phase transitions. This model describes quantum phase transitions such as the Mott insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson’s renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which define the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. Then, we study the excitation spectrum in the zero-temperature ordered phase. For N = 2 and 3, we establish the existence of an amplitude mode, also called “Higgs mode” by analogy with the Higgs mechanism in high-energy physics. For N = 1, we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and confirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a “superuniversal” quantity depending neither on the distance to the critical point nor on N.