Universal postquench coarsening and aging at a quantum critical point

Pia Gagel, Peter P. Orth, and Jörg Schmalian

Phys. Rev. B 92, 115121 – Published 9 September 2015

Abstract

The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an $N$ -component $φ^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large- $N$ limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.

Received 23 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.115121

Authors & Affiliations

Pia Gagel¹, Peter P. Orth^1,2, and Jörg Schmalian^1,3

¹Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
²School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
³Institute for Solid State Physics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

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Vol. 92, Iss. 11 — 15 September 2015

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Images

Figure 1
(a) Schematic description of the $φ^{4}$ model coupled to an external bath. The order parameter $ϕ$ experiences a potential landscape that depends on a set of parameters $R$ . If the initial parameters $R_{i}$ are such that the system is prepared in the symmetry-broken state with finite order parameter $ϕ_{i}$ , we consider a sudden change of the parameters from $R_{i}$ to $R_{f}$ that brings the system to the quantum critical point, where $ϕ = 0$ in equilibrium. (b) One experimental realization of the $φ^{4}$ model for $N = 3$ is a quantum dimer model in contact with phonons. The quench can be performed by rapidly changing the pressure.
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Figure 2
(a) Schematic equilibrium phase diagram of the $φ^{4}$ model at $T = 0$ including the different quench paths that we consider in this article. We either approach the quantum critical point $C$ from the symmetric side (path $A \to C$ ) or from the symmetry-broken side (path $B \to C$ ). We also consider a quench in the presence of a magnetic field $h_{i}$ that induces a finite initial order parameter $ϕ_{i}$ (path $B^{'} \to C$ ). (b) Schematic plot of the quench protocol. We consider fast quenches where the switching time $τ_{s}$ from the initial parameter set $R_{i}$ to the final one $R_{f}$ occurs on a microscopic time scale.
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Figure 3
(a) Keldysh three-time contour. (b) Bath spectral function for different power-law exponents corresponding to the sub-Ohmic case $α < 1$ , the Ohmic case $α = 1$ , and the super-Ohmic case $α > 1$ .
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Figure 4
Prethermalization exponent $θ$ as a function of dynamic critical exponents $z$ . Here, $ε = 4 - d - z$ and $z = 2 / α$ is determined by the form of the bath spectral function at low energies $Im η (ω) \propto {| ω |}^{α}$ . For $θ < 0$ one observes an underdamped approach of the free Keldysh Green's function $G_{0}^{K} (t, t)$ towards equilibrium. The transition from overdamped to underdamped behavior occurs close the location of the sign change.
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Figure 5
Visualization of the correction to the distribution function $δ n (t_{a}, ω)$ as a function of dimensionless variable $ω γ / q^{2}$ [see Eq. (110)]. Plot shows $2 n (t_{a}, ω) + 1 = coth (\frac{ω}{2 T}) [1 + 2 r (t_{a}) Re G_{eq}^{R}]$ [see Eq. (107)] as a function of the dimensionless variable $x = γ ω / q^{2}$ for $z = 2, γ T / q^{2} = 1.7$ , and $| θ | γ / (q^{2} t_{a}^{2 / z}) = 1$ .
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Figure 6
Time evolution of the order parameter $ϕ (t)$ following a quench from an initial state $ϕ_{i} \neq 0$ . The two panels show the cases of (a) $θ > 0$ , and (b) $θ < 0$ . First, the order parameter collapses due to the quench. After a microscopic time scale $t_{γ}$ (see also Appendix pp1), the order parameter recovers for a while $t_{γ} < t < t^{*}$ according to $ϕ (t) \propto t^{θ}$ for $θ > 0$ , before it eventually relaxes to zero for $t > t^{*}$ via a power law described by equilibrium exponents. For $θ < 0$ the order parameter always decays in a power-law fashion but exhibits two different exponents for $t < t^{*}$ and $t > t^{*}$ . The inset in (b) shows the time dependence of the correlation length which diverges after an initial collapse in a light-cone fashion $ξ (t) \propto t^{1 / z}$ .
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