Universal relaxational dynamics near two-dimensional quantum critical points

Sachdev, Subir

doi:10.1103/physrevb.59.14054

Cited by 104 publications

(162 citation statements)

References 37 publications

(65 reference statements)

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“…b) there is an analog of the singular nature of χ for zero temperature phases with a broken continuous symmetry. In two dimensions and with Goldstone modes whose frequencies ω = cq vanish linearly with momentum q = | q|, the dynamical susceptibility has a singular contribution (q 2 − ω 2 /c 2 ) −1/2 as noted first by Sachdev on the basis of a 1/N expansion [5]. The dynamic structure factor of 2D quantum antiferromagnets thus exhibits a critical continuum above the standard δ-function spin wave peak.…”

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confidence: 92%

“…The continuum results from the decay of a normally massive amplitude mode with momentum p into a pair of spin waves with momenta q and p − q, which is possible for any ω > cq, with a singular cross section because of the large phase space. The amplitude mode is thus completely overdamped in two dimensions [5]. The relative weight of these fluctuations compared to the dominant transverse contribution is determined by the dimensionless parameter qξ J , with ξ J =hc/ρ s the Josephson correlation length.…”

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Anomalous Fluctuations in Phases with a Broken Continuous Symmetry

Zwerger

2004

Phys. Rev. Lett.

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It is shown that the Goldstone modes associated with a broken continuous symmetry lead to anomalously large fluctuations of the zero field order parameter at any temperature below Tc. In dimensions 2 < d < 4, the variance of the extensive spontaneous magnetization scales as L 4 with the system size L, independent of the order parameter dynamics. The anomalous scaling is a consequence of the 1/q 4−d divergence of the longitudinal susceptibility. For ground states in two dimensions with Goldstone modes vanishing linearly with momentum, the dynamical susceptibility contains a singular contribution (q 2 − ω 2 /c 2 ) −1/2 . The dynamic structure factor thus exhibits a critical continuum above the undamped spin wave pole, which may be detected by neutron scattering in the Néel-phase of 2D quantum antiferromagnets.It is one of the basic properties of any thermodynamic system that the fluctuations VarÂ = Â 2 − Â 2 ∼ V of an extensive variableÂ scale linearly with the system volume V . This property guarantees that the intensive variablesÂ/V are self-averaging, with rms fluctuations vanishing proportional to V −1/2 . Physically, the extensive nature of VarÂ∼ V is due to the existence of a finite, microscopic correlation length ξ. A large system can thus be partitioned into an extensive number of V /ξ d subvolumes which are statistically independent. Since the variance in each subvolume is finite, the central limit theorem then quite generally implies a Gaussian distribution forÂ with a variance of order V , in agreement with the standard Einstein theory of fluctuations in macroscopic thermodynamics.The above argument indicates that the linear scaling VarÂ ∼ V may break down only at a critical point of a continuous phase transition where the correlation length ξ diverges. In fact, from the standard relation VarM = V · T χ between the fluctuations of the total 'magnetization'M and the corresponding linear susceptibility, the finite size scaling χ(T c , L) ∼ L 2−η of the susceptibilty right at T c [1] implies a nontrivial dependence VarM ∼ L d+2−η on system size L at the critical point. Below, it will be shown that anomalous fluctuations of the order parameter are present not only at T c but in fact at any temperature below T c provided the broken symmetry is continuous. This is a result of the presence of Goldstone modes, which imply that correlations decay algebraically below T c with exponents that are independent of temperature. A specific example of recent interest are fluctuations of the condensate numberN 0 in a Bose Einstein condensate. Using a weak coupling Bogoliubov approach, Giorgini et.al have shown [2] that VarN 0 ∼ T 2 L 4 scale anomalously at low temperatures. In fact, this result also applies to strongly interacting superfluids, being essentially a consequence of the long wavelength phase fluctuations [3]. In the present Letter, we will show, that the anomalous fluctuations in a BoseEinstein condensate are just one particular example of a rather general phenomenon which appears in any phase with a broken co...

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Anomalous Fluctuations in Phases with a Broken Continuous Symmetry

Zwerger

2004

Phys. Rev. Lett.

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“…(1). Beyond perturbation, calculations based on the renormalized classical Ginzburg-Landau theory [21,22] at finite temperature yield the result G = 2πg 2π+g [23]. A recent nonperturbative renormalization-group (NPRG) calculation also provides complete thermodynamic calculations.…”

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Strongly Interacting Two-Dimensional Bose Gases

Hung

Zhang

et al. 2013

Phys. Rev. Lett.

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We prepare and study strongly interacting two-dimensional Bose gases in the superfluid, the classical Berezinskii-Kosterlitz-Thouless (BKT) transition, and the vacuum-to-superfluid quantum critical regimes. A wide range of the two-body interaction strength 0.05 < g < 3 is covered by tuning the scattering length and by loading the sample into an optical lattice. Based on the equations of state measurements, we extract the coupling constants as well as critical thermodynamic quantities in different regimes. In the superfluid and the BKT transition regimes, the extracted coupling constants show significant down-shifts from the mean-field and perturbation calculations when g approaches or exceeds one. In the BKT and the quantum critical regimes, all measured thermodynamic quantities show logarithmic dependence on the interaction strength, a tendency confirmed by the extended classical-field and renormalization calculations.PACS numbers: 51.30.+i, 67.25.dj, 64.70.Tg, 37.10.Jk Two-dimensional (2D) Bose gases are an intriguing system to study the interplay between quantum statistics, fluctuations, and interaction. For noninteracting bosons in 2D, fluctuations prevail at finite temperatures and Bose-Einstein condensation occurs only at zero temperature. The presence of interaction can drastically change the picture. With repulsive interactions, fluctuations are reduced and superfluidity emerges at finite temperature via the Berezenskii-Kosterliz-Thouless (BKT) mechanism [1,2]. Interacting Bose gases in two dimensions and BKT physics have been actively investigated in many condensed matter experiments [3][4][5][6][7]. In cold atoms, the BKT transition and the suppression of fluctuations are observed based on 2D gases in the weak interaction regimes [8][9][10][11].Extensive theoretical research on 2D Bose systems addresses the role of interactions in the superfluid phase [12][13][14][15][16][17] and near the BKT critical point [18,19]. In the weak interaction regime, the classical φ 4 field theory [18,19] predicts the logarithmic corrections to the critical chemical potential µ c = k B T (g/π) ln(13.2/g) and the critical density n c = λ −2 dB ln(380/g) for small two-body interaction strength g < 0.2. Here k B T is the thermal energy and λ dB is the thermal de Broglie wavelength. The classical-field predictions are consistent with weakly interacting 2D gas experiments [9][10][11]20]. The upper blue shaded area is the superfluid regime, and the red boundary corresponds to the BKT transition regime. The black dashed lineμ = 0 indicates where we evaluate the density and pressure for a vacuum-to-superfuid quantum critical gas. The inset compares the equations of state of a 2D gas and a 2D lattice gas with an almost identical g ≈ 0.4. µ MF = 2 gn/m logarithmically [13]. Here, m is the mass of the boson, n is the density, and 2π is the Planck constant. Defining the superfluid coupling constant as G = m/( 2 κ), where κ = ∂n/∂µ is the compressibility, we can summarize the perturbation expansion result of G as [12]

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“…where C is a constant (30), Ӎ (380 Ϯ 3)͞2 according to numerical simulation (31)(32)(33)(34). We write the interparticle coupling constant in two dimensions in the form g ϭ 2 ␣͞m, where the parameter ␣ is dimensionless.…”

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Superfluid transition of homogeneous and trapped two-dimensional Bose gases

Holzmann

Baym

Blaizot

et al. 2007

Proc. Natl. Acad. Sci. U.S.A.

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Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.two dimensions ͉ phase transitions ͉ trapped atoms T he ability to produce two-dimensional atomic gases trapped in optical potentials has stimulated considerable interest in the Kosterlitz-Thouless-Berezinskii (KTB) transition in such systems (1-7); recently, the transition has been observed in a quasi two-dimensional system of trapped rubidium atoms (8, 9). A homogeneous Bose gas in two dimensions undergoes BoseEinstein condensation (BEC) only at zero temperature, since long wavelength phase fluctuations destroy long range order (10-13); nonetheless, interparticle interactions drive a phase transition to a superfluid state at finite temperature, as first pointed out by Berezinskii (14,15) and by Kosterlitz and Thouless (16,17). The phase transition is characterized by an algebraic decay of the off-diagonal one-body density matrix (or single particle Green's function) in real space below the transition temperature, T KT . Furthermore, the superfluid mass density, s , jumps with falling temperature, T, from 0 just above T KT to a universal value s ϭ 2m 2 T KT ͞ just below (18), where m is the atomic mass. (We use units ϭ k B ϭ 1.)A noninteracting homogeneous Bose gas in two dimensions does not undergo Bose-Einstein condensation at finite temperature. In such a system, the density is given in terms of the chemical potential bywhere ␤ ϭ 1͞T, and ϭ (2 ͞mT) 1/2 is the thermal wavelength.As approaches zero at fixed temperature, the density grows arbitrarily, implying the absence of condensation. On the other hand, a trapped noninteracting gas does undergo a BoseEinstein condensation at a finite temperature (19). In the semiclassical limit, the total number of particles is given by N(␤ ) ϭ g 2 (Ϫ␤ )(T͞ ) 2 , w...

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Universal relaxational dynamics near two-dimensional quantum critical points

Cited by 104 publications

References 37 publications

Anomalous Fluctuations in Phases with a Broken Continuous Symmetry

Anomalous Fluctuations in Phases with a Broken Continuous Symmetry

Strongly Interacting Two-Dimensional Bose Gases

Superfluid transition of homogeneous and trapped two-dimensional Bose gases

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