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arXiv:1209.3823 · Physical Review B

Mapping between finite temperature classical and zero temperature quantum systems: Quantum critical jamming and quantum dynamical heterogeneities

Zohar Nussinov, Patrick Johnson, Matthias J. Graf, Alexander V. Balatsky

Published May 15, 2013• 19 citations
arXivHTMLPDF

Abstract

Many electronic systems (e.g., the cuprate superconductors and heavy fermions) exhibit striking features in their dynamical response over a prominent range of experimental parameters. While there are some empirical suggestions of particular increasing length scales that accompany such transitions in some cases, this identification is not universal and in numerous instances no large correlation length is evident. To better understand, as a matter of principle, such behavior in quantum systems, we extend a known mapping (earlier studied in stochastic or supersymmetric quantum mechanics) between finite temperature classical Fokker-Planck systems and related quantum systems at zero temperature to include general nonequilibrium dynamics. Unlike Feynman mappings or stochastic quantization methods in field theories (as well as more recent holographic type dualities), the classical systems that we consider and their quantum duals reside in the same number of space-time dimensions. The upshot of our very broad and rigorous result is that a Wick rotation exactly relates (i) the dynamics in general finite temperature classical dissipative systems to (ii) zero temperature dynamics in the corresponding dual many-body quantum systems. Using this correspondence, we illustrate that, even in the absence of imposed disorder, many continuum quantum fluid systems (and possible lattice counterparts) may exhibit a zero-point ``quantum dynamical heterogeneity'' wherein the dynamics, at a given instant, is spatially nonuniform. While the static length scales accompanying this phenomenon do not seem to exhibit a clear divergence in standard correlation functions, the length scale of the dynamical heterogeneities can increase dramatically. We further study ``quantum jamming'' and illustrate how a hard-core bosonic system can undergo a zero temperature quantum critical metal-to-insulator-type transition with an extremely large effective dynamical exponent $z>4$ that is consistent with length scales that increase far more slowly than the relaxation time as a putative critical transition is approached. Similar results may hold for spin-liquid-type as well as interacting electronic systems. We suggest ways to analyze experimental data in order to adduce such phenomena. Our approach may be used to analyze other quenched quantum systems.

DOI
10.1103/physrevb.87.184202

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A viewpoint article on the very interesting work of Biroli, Chamon, and Zamponi on superglasses. I further suggest how additional new superglass and "spin-superglass" phases of matter (the latter phases contain quenched disorder) and general characteristics may be proven as a theoretical proof of concept in various electronic systems. The new phases include: (1) superglasses of Cooper pairs, i.e., glassy superconductors, (2) superglass phases of quantum spins, and (3) superglasses of the electronic orbitals. New general features which may be derived by the same construct include (a) quantum dynamical heterogeneities- a low temperature quantum analogue of dynamical heterogeneities known to exist in classical glasses and spin-glasses wherein the local dynamics and temporal correlations are spatially non-uniform. I also discuss on a new class of quantum critical systems. In particular, I outline (b) the derivation of the quantum analogue of the zero temperature jamming transition that has a non-trivial dynamical exponent. We very briefly comment on (c) quantum liquid crystals.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8){(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)(h/J)c​=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256L=32--256L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7)3.044\phantom{\rule{0.16em}{0ex}}51(7)3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24L=24L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.
In this thesis I investigate several aspects of local quantum criticality, a concept of key importance in a number of physical contexts ranging from critical heavy fermion compounds to quantum dot systems. Quantum critical points are associated with second order phase transitions at zero temperature. In contrast to their finite-temperature counterparts, the zero-point motion cannot be neglected near a quantum critical point. As a result, the incorporation of quantum dynamics leads to an effective dimension larger than the spatial dimension of the system for the order parameter fluctuations within the Ginzburg-Landau-Wilson treatment of criticality. This so-called quantum-to-classical mapping works well for the critical properties in insulating systems but apparently fails in systems containing gapless fermions. This has been experimentally most clearly been demonstrated within a particular class of intermetallic compounds called heavy fermions. A particular way in which the Ginzburg-Landau-Wilson paradigm fails is for critical Kondo destruction that seems to underlie the unconventional quantum criticality seen in the heavy fermions. I focus on studying the properties of critical Kondo destruction and the emergence of energy-over-temperature-scaling in systems without spatial degrees of freedom, i.e., so-called quantum impurity systems. In particular, I employ large-N techniques to address critical properties of this class of quantum phase transitions in and out of equilibrium. As quantum critical systems are characterized by a scale-invariant spectrum with many low-lying excitations, it may appear that any perturbation can lead to a response beyond the linear response regime. Understanding what governs the non-linear response regime near quantum criticality is an interesting area. Here, I first present a path integral version of the Schrieffer-Wolff transformation which relates the functional integral form of the partition function of the Anderson model to that of its effective low-energy model. The equivalence between the low-energy sector of the Anderson model in the Kondo regime and the spin-isotropic Kondo model is usually established via a canonical transformation performed on the Hamiltonian, followed by a projection. The resulting functional integral assumes the form of a spin path integral and includes a geometric phase factor, i.e. a Berry phase. The approach stresses the underlying symmetries of the model and allows for a straightforward generalization of the transformation to more involved models. As an example of the efficiency of the approach I apply it to a single electron transistor attached to ferromagnetic leads and derive the effective low-energy model of such a magnetic transistor. As Kondo screening is a local phenomenon, it and its criticality can be studied using the appropriate impurity model. A general impurity model to study critical Kondo destruction is the pseudogap Bose-Fermi Kondo model. Here, I concentrate on the multi-channel version of the model using the dynamical large-N study. This model allows to study the non-trivial interplay between two different mechanisms of critical Kondo destruction. The interplay of two processes that can each by itself lead to critical Kondo destruction. The zero-temperature residual entropy at various fixed points for the model is also discussed. The two channel Anderson model exhibits several continuous quantum phase transitions between weak- and strong-coupling phases. The non-crossing approximation (NCA) is believed to give reliable results for the standard two-channel Anderson model of a magnetic impurity in a metal. I revisit the reliability of the NCA for the standard two channel Anderson model (constant conduction electron density of states) and investigate its reliability for the two-channel pseudogap Anderson model. This is done by comparing finite-temperature, finite-frequency solutions of the NCA equations and asymptotically exact zero-temperature NCA solutions with numerical renormalization-group calculations. The phase diagram of this model is well established. The focus here will be on the dynamical scaling properties obtained within the NCA. Finally, I study the thermal and non-thermal steady state scaling functions and the steady-state dynamics of the pseudogap Kondo model. This model allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points and compare the out-of-equilibrium scaling properties of critical Kondo destruction to those of the traditional spin-density wave (SDW) scenario. The differences I identify can be experimentally probed. This may be helpful in identifying the nature of the quantum critical points observed in certain heavy fermion compounds.
Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition.Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the λ\lambdaλ-transition in 4^{4}4He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario.For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent z=1.8±0.05z=1.8 \pm 0.05z=1.8±0.05 contradicting the long standing result z=dz=dz=d.Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role.Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations.
Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition.Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the λ\lambdaλ-transition in 4^{4}4He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario.For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent z=1.8±0.05z=1.8 \pm 0.05z=1.8±0.05 contradicting the long standing result z=dz=dz=d.Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role.Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations.