We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-
21 models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number
K⪢1 and weak dimensionless coupling
g⪡1. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ensuremath{\ge}{K}_{c}(g)$, where the critical temperature
Tc of the ordering transition vanishes. We find that the local probability distribution
P(B) of the order parameter
B has a long power-law tail in the region where
B is much larger than its typical value
B0. Near the quantum-critical point, at $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale
Ninh of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. In the disordered regime, realized at
K<Kc(g) we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<{K}^{\ensuremath{\ast}}(g)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.
Zero-temperature or quantum phase transitions in itinerant electronic systems both with and without quenched disordered are discussed. Phase transitions considered include, the ferromagnetic transition, the antiferromagnetic transition, the superconductor-metal transition, and various metal-insulator transitions. Emphasis is placed on how to determine the universal properties that characterize these quantum phase transitions. For the first three of the phase transitions listed above, one of the main physical ideas established is that in zero-temperature systems there are soft or slow modes that exist in addition to the soft order parameter fluctuations, and that these modes can couple to the critical modes. These extra soft modes are shown to have a profound effect on the quantum critical properties. For quantum phase transitions involving zero wavenumber order parameters, i.e., the ferromagnetic and superconductor-metal transitions, these extra modes effectively lead to long-ranged effective interactions between order parameter fluctuations, which in turn lead to exactly soluble critical behaviors. For the antiferromagnetic case, we argue that while in low enough dimensions disorder fluctuation effects tend to destroy long-range order, quantum fluctuations counteract this effect and in some parameter regions manage to re-establish antiferromagnetic long-range order. For the metal-insulator transition, some recent new ideas are reviewed. In particular, it is pointed out that for interacting disordered electrons, one expects that in high dimensions the metal-insulator transition is related to the phase transition that occurs in random-field magnets in high dimensions. If the analogy also holds in three dimensions this suggests that the metal-insulator transition might have glassy characteristics.
Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.
The interplay of interactions and disorder in low-dimensional superconductors supports the formation of multiple quantum phases as possible instabilities of the superconductor-insulator transition (SIT) at a singular quantum critical point. We explore a one-dimensional model which exhibits such a variety of phases in the strongly quantum fluctuations regime. Specifically, we study the effect of weak disorder on a two-leg Josephson ladder with comparable Josephson and charging energies $({E}_{J}\ensuremath{\sim}{E}_{C})$. An additional key feature of our model is the requirement of perfect
Z2 symmetry, respected by all parameters including the disorder. Using a perturbative renormalization-group (RG) analysis, we derive the phase diagram and identify at least one intermediate phase between a full-fledged superconductor and a disorder-dominated insulator. Most prominently, for repulsive interactions on the rungs we identify two distinct mixed phases: In both of them the longitudinal charge mode is a gapless superconductor, however one phase exhibits a dipolar charge density order on the rungs, while the other is disordered. This latter phase is characterized by coexisting superconducting (phase-locked) and charge-ordered rungs, and encompasses the potential of evolving into a Griffith's phase characteristic of the random-field Ising model in the strong disorder limit.
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.
This article reviews recent studies of mean-field and one dimensional quantum disordered spin systems coupled to different types of dissipative environments. The main issues discussed are: (i) The real-time dynamics in the glassy phase and how they compare to the behaviour of the same models in their classical limit. (ii) The phase transition separating the ordered -- glassy -- phase from the disordered phase that, for some long-range interactions, is of second order at high temperatures and of first order close to the quantum critical point (similarly to what has been observed in random dipolar magnets). (iii) The static properties of the Griffiths phase in random Ising chains. (iv) The dependence of all these properties on the environment. The analytic and numeric techniques used to derive these results are briefly mentioned.