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arXiv:1210.4166 · Physical Review B

Quasinormal modes of quantum criticality

William Witczak‐Krempa, Subir Sachdev A

Published Dec 12, 2012• 66 citations
arXivHTMLPDF

Abstract

We study charge transport of quantum critical points described by conformal field theories in 2+1 space-time dimensions. The transport is described by an effective field theory on an asymptotically anti-de Sitter space-time, expanded to fourth order in spatial and temporal gradients. The presence of a horizon at nonzero temperatures implies that this theory has quasinormal modes with complex frequencies. The quasinormal modes determine the poles and zeros of the conductivity in the complex frequency plane, and so fully determine its behavior on the real frequency axis, at frequencies both smaller and larger than the absolute temperature. We describe the role of particle-vortex or S duality on the conductivity, specifically how it maps poles to zeros and vice versa. These analyses motivate two sum rules obeyed by the quantum critical conductivity: the holographic computations are the first to satisfy both sum rules, while earlier Boltzmann-theory computations satisfy only one of them. Finally, we compare our results with the analytic structure of the $O(N)$ model in the large-$N$ limit, and other CFTs.

DOI
10.1103/physrevb.86.235115

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We study charge transport of quantum critical points described by conformal field theories in 2+1 space-time dimensions. The transport is described by an effective field theory on an asymptotically anti-de Sitter space-time, expanded to fourth order in spatial and temporal gradients. The presence of a horizon at nonzero temperatures implies that this theory has quasinormal modes with complex frequencies. The quasinormal modes determine the poles and zeros of the conductivity in the complex frequency plane, and so fully determine its behavior on the real frequency axis, at frequencies both smaller and larger than the absolute temperature. We describe the role of particle-vortex or S duality on the conductivity, specifically how it maps poles to zeros and vice versa. These analyses motivate two sum rules obeyed by the quantum critical conductivity: the holographic computations are the first to satisfy both sum rules, while earlier Boltzmann-theory computations satisfy only one of them. Finally, we compare our results with the analytic structure of the O(N)O(N)O(N) model in the large-NNN limit, and other CFTs.
Understanding the real time dynamics of quantum systems without quasiparticles constitutes an important yet challenging problem. We study the superfluid-insulator quantum-critical point of bosons on a two-dimensional lattice, a system whose excitations cannot be described in a quasiparticle basis. We present detailed quantum Monte Carlo results for two separate lattice realizations: their low-frequency conductivities are found to have the same universal dependence on imaginary frequency and temperature. We then use the structure of the real time dynamics of conformal field theories described by the holographic gauge/gravity duality to make progress on the dicult problem of analytically continuing the Monte Carlo data to real time. Our method yields quantitative and experimentally testable results on the frequency-dependent conductivity near the quantum critical point, and on the spectrum of quasinormal modes in the vicinity of the superfluid-insulator quantum phase transition. Extensions to other observables and universality classes are discussed.
Many condensed matter experiments explore the finite temperature dynamics of systems near quantum critical points. Often, there are no well-defined quasiparticle excitations, and so quantum kinetic equations do not describe the transport properties completely. The theory shows that the transport coefficients are not proportional to a mean free scattering time (as is the case in the Boltzmann theory of quasiparticles), but are completely determined by the absolute temperature and by equilibrium thermodynamic observables. Recently, explicit solutions of this quantum critical dynamics have become possible via the anti-de Sitter/conformal field theory duality discovered in string theory. This shows that the quantum critical theory provides a holographic description of the quantum theory of black holes in a negatively curved anti-de Sitter space, and relates its transport coefficients to properties of the Hawking radiation from the black hole. We review how insights from this connection have led to new results for experimental systems: (i) the vicinity of the superfluid-insulator transition in the presence of an applied magnetic field, and its possible application to measurements of the Nernst effect in the cuprates, (ii) the magnetohydrodynamics of the plasma of Dirac electrons in graphene and the prediction of a hydrodynamic cyclotron resonance.
We compute three-point correlators between the stress-energy tensor and the conserved currents of conformal field theories (CFTs) in 2+1 dimensions. We first compute the correlators in the large-flavor-number expansion of conformal gauge theories and then perform the computation using holography. In the holographic approach, the correlators are computed from an effective action on (3+1)-dimensional anti-de Sitter space (AdS4{\mathrm{AdS}}_{4}AdS4​) and depend upon the coefficient $\ensuremath{\gamma}$ of a four-derivative term in the action. We find a precise match between the CFT and the holographic results, thus, fixing the values of $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of U(1)U(1)U(1) gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike transport near quantum phase transitions of interest in condensed-matter physics. This paper includes Appendices reviewing key features of the AdS-CFT correspondence for condensed-matter physicists.
Strongly correlated quantum critical points with conformal symmetry in two spatial dimensions describe many phase transitions in condensed matter systems, as for example that from superfluid to insulator. Combining field theory methods, Monte Carlo simulations, and the holographic duality from string theory, the authors use an operator product expansion to derive non-perturbative results for these systems, including the leading terms in the frequency dependence of the nonzero temperature conductivity and sum rules obeyed by the conductivity. These methods and results are applicable to a wide class of conformal field theories for condensed matter systems.
We compute the universal conductivity of the (2+1)-dimensional XY universality class, which is realized for a superfluid-to-Mott insulator quantum phase transition at constant density. Based on large-scale Monte Carlo simulations of the classical (2+1)-dimensional J-current model and the two-dimensional Bose-Hubbard model, we can precisely determine the conductivity on the quantum critical plateau, σ(∞) = 0.359(4)σQ with σQ the conductivity quantum. The universal conductivity curve is the standard example with the lowest number of components where the bottoms-up AdS/CFT correspondence from string theory can be tested and made to use [R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011)]. For the first time, the shape of the σ(iω(n)) - σ(∞) function in the Matsubara representation is accurate enough for a conclusive comparison and establishes the particlelike nature of charge transport. We find that the holographic gauge-gravity duality theory for transport properties can be made compatible with the data if temperature of the horizon of the black brane is different from the temperature of the conformal field theory. The requirements for measuring the universal conductivity in a cold gas experiment are also determined by our calculation.
In an earlier work, Damle and the author [Phys. Rev. B 56, 8714 (1997)] demonstrated the central role played by incoherent, inelastic processes in transport near two-dimensional quantum critical points. This paper extends these results to the case of a quantum transition between a fractional quantized Hall state and an insulator, induced by varying the strength of an external periodic potential. We use the quantum field theory for this transition introduced by Chen, Fisher, and Wu [Phys. Rev. B 48, 13 749 (1993)]. The longitudinal and Hall conductivities at the critical point are both e2/h{e}^{2}/he2/h times nontrivial, fully universal functions of $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}T$ (\ensuremath{\omega} is the measuring frequency). These functions are computed using a combination of perturbation theory on the Kubo formula, and the solution of a quantum Boltzmann equation for the anyonic quasiparticles and quasiholes. The results include the values of the dc conductivities $(\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}\stackrel{\ensuremath{\rightarrow}}{T}0);$ earlier work was restricted strictly to T=0,T=0,T=0, and therefore computed only the high frequency ac conductivities with $\ensuremath{\Elzxh}\ensuremath{\omega}{/k}_{B}\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty}.$