Correlated Quantum Walks In Optical Lattices, Holographic Quantum Error-Correcting Codes, Valence-Bond States And Weak Nonlinearities, Entangled Microwave Photon States, Photon Emission From A Cavity-Coupled Double Quantum Dot, Classical Simulation Of Qua
Properties of systems near quantum critical points (QCPs) have been studied extensively. A QCP is a point across which the symmetry of (e) the ground state of a quantum system changes (e&eb) in a fundamental way; such a point can be accessed by (e) changing some parameter, say λ, in the Hamiltonian governing the system. The change in the ground state across a QCP is mediated by (e&eb) quantum fluctuations. Unlike conventional thermal critical points, thermal fluctuations do not (e) play a crucial role in such transitions. Similar to its thermal counterparts, the low-energy physics near a QCP is associated with (e&*eb) a number of critical exponents which characterize (eb) the universality class of such a transition. Among these exponents,....... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-414213-418614.pdf
Properties of systems near quantum critical points (QCPs) have been studied extensively. A QCP is a point across which the symmetry of (e) the ground state of a quantum system changes (e&eb) in a fundamental way; such a point can be accessed by (e) changing some parameter, say λ, in the Hamiltonian governing the system. The change in the ground state across a QCP is mediated by (e&eb) quantum fluctuations. Unlike conventional thermal critical points, thermal fluctuations do not (e) play a crucial role in such transitions. Similar to its thermal counterparts, the low-energy physics near a QCP is associated with (e&*eb) a number of critical exponents which characterize (eb) the universality class of such a transition. Among these exponents,....... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-414213-418614.pdf
Properties of systems near quantum critical points (QCPs) have been studied extensively. A QCP is a point across which the symmetry of (e) the ground state of a quantum system changes (ee such a point can be accessed by (e) changing some parameter, say λ, in the Hamiltonian governing the system. The change in the ground state across a QCP is mediated by (e&eb) quantum fluctuations. Unlike conventional thermal critical points, thermal fluctuations do not (e) play a crucial role in such transitions The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-409888-414212.pdf
Anatoli Polkovnikov Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, U.S.A. Vladimir Gritsev Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland Continuous quantum phase transitions (QPTs) have been a subject of intense theoretical research in recent decades (see, e.g., Refs. [1-3] for overview). Unlike usual phase transitions driven by temperature, QPTs are driven entirely by quantum fluctuations. They are believed to occur in many situations as described later in this book. Quite recently a second order QPT was observed in a cold atom system of interacting bosons in an optical lattice. There a system of interacting bosons was driven in real time from the superfluid to the insulating phase [4], confirming an earlier theoretical prediction [5]. Up till now, Ref. [4] has provided probably the cleanest experimental confirmation of a QPT. The unifying property of all continuous (second order) phase transitions is the emergent universality and scale invariance of the long-distance low energy properties of the system near the quantum critical point (QCP) [1]. This universality implies that low-energy properties of the system can be described by very few parameters, like the correlation length or the gap, which typically have power-law scaling with the tuning parameter characterized by critical exponents. These exponents are not sensitive to the microscopic details of the Hamiltonian describing the system, but rather depend only on the universality class to which a given phase transition belongs [1].
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.
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 d-dimensional system continuously changes from Deff=d to Deff=d+2 while the universality class of the system does not change.
A critical point is associated with a diverging relaxation time which always makes the dynamics across a critical point fascinating. For example, when the temperature of a classical system undergoing a classical phase transition (CPT) at a finite temperature T c is suddenly changed from a value higher than the critical temperature to a lower value across the critical point, the system does not equilibrate instantaneously. Domains of ordered regions are formed which grow following a phase ordering dynamics which leads to a dynamical scaling [104]. In this section, we will discuss the recent studies of non-equilibrium dynamics of quantum systems driven across QCPs where the dynamics is unitary unlike the dynamics across a finite temperature critical point. The non-equilibrium dynamics of a transverse XY spin chain was first investigated in a series of papers [48, 46, 47] where the time evolution of the model was studied in the presence of various time-dependent magnetic fields and the nonergodic behavior of the magnetization was pointed out. A similar result was also obtained in [505]. There is a recent upsurge in studies of non-equilibrium dynamics of a quantum system swept across a QCP. These studies are important for exploring the universality associated with quantum critical dynamics. Moreover, recent experiments with ultracold atomic gases [343, 663, 478, 87] have stimulated numerous theoretical studies. The main properties of these atomic gases are low dissipation rates and phase coherence over a long time so that the dynamics is well described by the usual quantum evolution of a closed system. In the subsequent sections, we shall discuss that when a quantum system initially prepared in its ground state is driven across a QCP, the dynamics fails to be adiabatic however slow the rate of change in the parameters of the Hamiltonian may be. This is due to the divergence of the characteristic time scale of the quantum system, namely, the relaxation time close to the QCP. This non-adiabaticity results in the occurrence of defects in the final state of the quantum Hamiltonian.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .