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arXiv:1401.1304

Quantum critical dynamics of the boson system in the Ginsburg-Landau model

Mikhail Gennag'evich Vasin

Published Jan 7, 2014• 0 citations
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Abstract

The quantum critical dynamics of the quantum phase transitions is considered. In the framework of the unified theory, based on the Keldysh technique, we consider the crossover from the classical to the quantum description of the boson many-particle system dynamics close to the second order quantum phase transition. It is shown that in this case the upper critical space dimension of this model is $d_c^{+}=2$, therefore the quantum critical dynamics approach is useful in case of $d<2$. In the one-dimension system the phase coherence time does diverge at the quantum critical point, $g_c$, and has the form of $τ\propto -\ln |g-g_c|/(g-g_c) $, the correlation radius diverges as $r_c\propto |g-g_c|^{-ν} (ν= 0.6)$.

DOI
10.48550/arxiv.1401.1304

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The quantum critical dynamics of the quantum phase transitions is considered. In the framework of the unified theory, based on the Keldysh technique, we consider the crossover from the classical to the quantum description of the boson many-particle system dynamics close to the second order quantum phase transition. It is shown that in this case the upper critical space dimension of this model is dc+=2d_c^{+}=2dc+​=2, therefore the quantum critical dynamics approach is useful in case of d<2d<2d<2. In the one-dimension system the phase coherence time does diverge at the quantum critical point, gcg_cgc​, and has the form of τ∝−ln⁡∣g−gc∣/(g−gc)τ\propto -\ln |g-g_c|/(g-g_c) τ∝−ln∣g−gc​∣/(g−gc​), the correlation radius diverges as rc∝∣g−gc∣−ν(ν=0.6)r_c\propto |g-g_c|^{-ν} (ν= 0.6)rc​∝∣g−gc​∣−ν(ν=0.6).
The non-equilibrium dynamics of the critical behaviour of the Boson system undergoing the second order quantum phase transition is discussed. The analysis is carried out using the Keldysh technique of the non-equilibrium dynamics description. The critical behaviour close to the quantum critical point has been shown to be multi-critical. Crossover among three different critical regimes (modes) is possible: adiabatic quantum mode (AQM), dissipative classical mode (classical critical dynamics mode (CCDM) and dissipative quantum critical mode (QCDM). As a result, the observed critical behavior can essentially depend on the conditions of approaching the critical point.
We analyze the quantum-classical crossover in the vicinity of the continuous quantum critical point (QCP) of a Boson system. The analysis is based on the Keldysh approach for the description of of the non-equilibrium quantum dynamics. The critical behavior close to QCP has three different regimes (modes): adiabatic quantum mode (AQM), dissipative classical mode (classical critical dynamics mode (CCDM) and dissipative quantum critical mode (QCDM). Crossover among these regimes (modes) is possible: it is shown that the experimentally observed changing of the critical exponents close to QCP is the dynamical effect accompanying the crossover from CCDM where the thermal fluctuations dominate to QCDM where the quantum fluctuations determine the critical behavior. In this case the effective dimension of the ddd-dimensional system continuously changes from Deff=dD_{eff}=dDeff​=d to Deff=d+2D_{eff}=d+2Deff​=d+2 while the universality class of the system does not change.
Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension ddd and for MMM complex fields. We concentrate on nonrelativistic bosons and an action with a linear time derivative. The ordered phase can be associated with a nonzero density of (quasi)particles nnn. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum kph{k}_{\mathit{ph}}kph​, which is characteristic for a given experiment. For the dilute regime ${k}_{\mathit{ph}}\ensuremath{\gtrsim}{n}^{1∕d}$, the quantum phase transition is simple, with the same ``mean field'' critical exponents for all ddd and MMM. On the other hand, the dense regime kph⪡n1∕d{k}_{\mathit{ph}}⪡{n}^{1∕d}kph​⪡n1∕d reveals a rather rich spectrum of features, depending on ddd and MMM. In this regime, one observes for $d\ensuremath{\leqslant}3$ a crossover to a relativistic action with second time derivatives. This admits order for d>1d>1d>1, whereas d=1d=1d=1 shows a behavior similar to the low temperature phase of the classical two-dimensional O(2M)\mathrm{O}(2M)O(2M) models.
We describe the quantum phase transitions in the ferromagnetic Dicke-Ising model using a Landau theory approach. The theory quantitatively captures the change from a second- to a first-order transition between the normal and superradiant phases through a tricritical point. We identify virtual nearest-neighbor double spin-flip processes as the crucial mechanism responsible for this behavior. The tricritical point constitutes a quantum phase transition above the upper critical dimension. We discuss the modifications to finite-size scaling required for the correct interpretation of numerical data at the tricritical point. Our results emphasize the need for adapted finite-size scaling forms in all-to-all interacting quantum systems and establish the ferromagnetic Dicke-Ising model as a paradigmatic platform for quantum phase transitions above the upper critical dimension, encompassing both standard φ4φ^4φ4 criticality and beyond.
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.
A nonconventional renormalization-group (RG) treatment close to and below four dimensions is used to explore, in a unified and systematic way, the low-temperature properties of a wide class of systems in the influence domain of their quantum critical point. The approach consists in a preliminary averaging over quantum degrees of freedom and a successive employment of the Wilsonian RG transformation to treat the resulting effective classical Ginzburg-Landau free energy functional. This allows us to perform a detailed study of criticality of the quantum systems under study. The emergent physics agrees, in many aspects, with the known quantum critical scenario. However, a richer structure of the phase diagram appears with additional crossovers which are not captured by the traditional RG studies. In addition, in spite of the intrinsically static nature of our theory, predictions about the dynamical critical exponent, which parametrizes the link between statics and dynamics close to a continuous phase transition, are consistently derived from our static results.