Discontinuous quantum phase transitions besides their general interest are clearly relevant to the study of heavy fermions and magnetic transition metal compounds. Recent results show that in many systems belonging to these classes of materials the magnetic transition changes from second order to first order as they approach the quantum critical point (QCP). We investigate here some mechanisms that may be responsible for this change, specifically the coupling of the order parameter to soft modes and the competition between different types of order near the QCP. For weak first order quantum phase transitions general results are obtained. In particular, we describe the thermodynamic behaviour at this transition when it is approached from finite temperatures. This is the discontinuous equivalent of the non-Fermi-liquid trajectory close to a conventional QCP in a heavy-fermion material.
Discontinuous quantum phase transitions besides their general interest are clearly relevant to the study of heavy fermions and magnetic transition metal compounds. Recent results show that in many systems belonging to these classes of materials the magnetic transition changes from second order to first order as they approach the quantum critical point (QCP). We investigate here some mechanisms that may be responsible for this change, specifically the coupling of the order parameter to soft modes and the competition between different types of order near the QCP. For weak first order quantum phase transitions general results are obtained. In particular, we describe the thermodynamic behaviour at this transition when it is approached from finite temperatures. This is the discontinuous equivalent of the non-Fermi-liquid trajectory close to a conventional QCP in a heavy-fermion material.
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.
Quantum phase transition arises in general as second order phase transition at zero temperature, tuned by a non-thermal parameter such as pressure, doping or a magnetic field. The point in the phase diagram of the material in which different phases meet is called a quantum critical point (QCP). Physics around QCPs are of extensive current interest because the critical quantum fluctuations influence the physical properties in a wide temperature range (quantum criticality), and are believed to be responsible for many emergent physical properties such as non-Fermi liquids and unconventional superconductivity. In this research we explore dynamics and thermodynamics near QCPs via investigating three classes of models, which all have real material correspondence. Specifically first, we study local dynamics in a perturbed quantum critical Ising chain with E8 symmetry, where we show the local dynamical spin susceptibility has a singular dependence on frequency, but differs from the diffusion form. The nuclear magnetic resonance (NMR) relaxation rate at low temperatures depends exponentially on the inverse temperature, whose prefactor we also determine. We propose NMR experiments as a means to further test the applicability of the E8 description for CoNb2O6. Second, we investigate the thermodynamic properties of itinerant ferromagnets near quantum critical points, described by the quantum Landau-Ginzburg theory. We provide a regularized perturbative renormalization group procedure to calculate the free energy. We further carry out numerical calculations on thermodynamic quantities, capturing not only the leading critical behaviors, but also the subleading and nonsingular contributions. We demonstrate various thermodynamic signatures of quantum criticality, including the entropy accumulation effect and the divergence of the specific heat coefficient. A detailed comparison to the recent experimental results on an itinerant ferromagnet Sr3Ru2O7 is also presented. Third, we explore Ising-nematic and magnetic phases and their transitions in iso-electronically doped iron pnictides by carrying out a large-N study of an effective low-energy Ginzburg-Landau model for these systems. We demonstrate that the magnetic and Ising orders transitions are concurrent at zero temperature, and both transitions are weakly first-order, which is consistent with RG-based prediction and experimental observations.
Competing scenarios for quantum critical points (QCPs) of strongly interacting Fermi systems signaled by a divergent density of states at zero temperature are contrasted. The conventional scenario, which enlists critical fluctuations of a collective mode and attributes the divergence to a coincident vanishing of the quasi-particle strength z , is shown to be incompatible with identities arising from conservation laws prevailing in the fermionic medium. An alternative scenario, in which the topology of the Fermi surface is altered at the QCP, is found to explain the non-Fermi-liquid thermodynamic behavior observed experimentally in Yb-based compounds close to the QCP. It is suggested that combination of the topological scenario with the theory of quantum phase transitions will provide a proper foundation for analysis of the extended QCP region.
Competing scenarios for quantum critical points (QCPs) of strongly interacting Fermi systems signaled by a divergent density of states at zero temperature are contrasted. The conventional scenario, which enlists critical fluctuations of a collective mode and attributes the divergence to a coincident vanishing of the quasiparticle strength z, is shown to be incompatible with identities arising from conservation laws prevailing in the fermionic medium. An alternative scenario, in which the topology of the Fermi surface is altered at the QCP, is found to explain the non-Fermi-liquid thermodynamic behavior observed experimentally in Yb-based compounds close to the QCP. It is suggested that combination of the topological scenario with the theory of quantum phase transitions will provide a proper foundation for analysis of the extended QCP region.
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.
This article is aimed at a pedagogical introduction to the physics of quantum phase transitions that is unique to metallic systems. It has been recognized for some time that quantum criticality can result in a breakdown of Landau's Fermi liquid theory. Its converse, however, has not been appreciated until very recently: non-Fermi liquid behavior can in turn lead to new classes of quantum phase transition. A concrete example is provided by ``local quantum critical points''. I summarize the theoretical reasoning and experimental evidence for local quantum criticality, in the context of heavy fermion metals. The underlying physics is likely to be relevant to other correlated electron systems including doped Mott insulators.