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arXiv:1605.05460 · Physical review. B./Physical review. B

Nonequilibrium quantum mechanics: A “hot quantum soup” of paramagnons

Harley D. Scammell, Oleg P. Sushkov

Published Jan 20, 2017• 14 citations
arXivHTMLPDF

Abstract

Motivated by recent measurements of the lifetime (decay width) of paramagnons in quantum antiferromagnet ${\mathrm{TlCuCl}}_{3}$, we investigate paramagnon decay in a heat bath and formulate an appropriate quantum theory. Our formulation can be split into two regimes: (i) a nonperturbative, ``hot quantum soup'' regime where the paramagnon width is comparable to its energy; (ii) a usual perturbative regime where the paramagnon width is significantly lower than its energy. Close to the Neel temperature, the paramagnon width becomes comparable to its energy and falls into the hot quantum soup regime. To describe this regime, we develop a new finite frequency, finite temperature technique for a nonlinear quantum field theory; the ``golden rule of quantum kinetics.'' The formulation is generic and applicable to any three-dimensional quantum antiferromagnet in the vicinity of a quantum critical point. Specifically, we apply our results to ${\mathrm{TlCuCl}}_{3}$ and find agreement with experimental data. Additionally, we show that logarithmic running of the coupling constant in the upper critical dimension changes the commonly accepted picture of the quantum disordered and quantum critical regimes.

DOI
10.1103/physrevb.95.024420

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Motivated by recent measurements of the lifetime (decay width) of paramagnons in quantum antiferromagnet TlCuCl3{\mathrm{TlCuCl}}_{3}TlCuCl3​, we investigate paramagnon decay in a heat bath and formulate an appropriate quantum theory. Our formulation can be split into two regimes: (i) a nonperturbative, hot quantum soup'' regime where the paramagnon width is comparable to its energy; (ii) a usual perturbative regime where the paramagnon width is significantly lower than its energy. Close to the Neel temperature, the paramagnon width becomes comparable to its energy and falls into the hot quantum soup regime. To describe this regime, we develop a new finite frequency, finite temperature technique for a nonlinear quantum field theory; the golden rule of quantum kinetics.'' The formulation is generic and applicable to any three-dimensional quantum antiferromagnet in the vicinity of a quantum critical point. Specifically, we apply our results to TlCuCl3{\mathrm{TlCuCl}}_{3}TlCuCl3​ and find agreement with experimental data. Additionally, we show that logarithmic running of the coupling constant in the upper critical dimension changes the commonly accepted picture of the quantum disordered and quantum critical regimes.
We study the magnetic behavior of an itinerant, strongly interacting fermion system for varying values of the coupling constant and for arbitrary dimensionalities 1<d< 31<d<~31<d< 3. We show that, at zero temperature, due to quantum effects, the critical exponents are the mean-field ones below two dimensions as well as above. We then focus on the critical behavior of the two-dimensional case, which, within some assumptions that are discussed, exhibits a ferromagnetic rather than antiferromagnetic character. We also compute the effects of critical fluctuations on the lowtemperature properties of such systems.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures – the so-called ‘hot entanglement’ – has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit ‘hot entanglement’ to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillators in a shared bath depicted by a finite-temperature scalar field . From the Langevin equations we derived for the normal modes and the entanglement measure constructed from the covariance matrix we examine the interplay between direct coupling, field-induced interaction and finite separation on the structure of late-time entanglement. We show that the coupling between oscillators plays a crucial role in sustaining entanglement at intermediate temperatures and over finite separations. In contrast, the field-induced interaction between the oscillators which is a non-Markovian effect becomes very ineffective at high temperature. We determine the critical temperature above which entanglement disappears to be bounded in the leading order by the inverse frequency of the center-of-mass mode of the reduced oscillator system, a result not unexpected, which rules out hot entanglement in such settings.
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.
It is shown that an analytic approach which includes vertex corrections in a paramagnon-like self-energy can quantitatively explain the two-dimensional Hubbard model in the weak-to-intermediate coupling regime. All parameters are determined self-consistently. This approach clearly shows that in two dimensions Fermi-liquid quasiparticles disappear in the finite-temperature paramagnetic state when the antiferromagnetic correlation length becomes larger than the electronic thermal de Broglie wavelength. Quantum Monte Carlo results are used to compare the accuracy of this approach with others.