Abstract The study of the competition or coexistence of different ground states in many-body systems is an exciting and actual topic of research, both experimentally and theoretically. Quantum fluctuations of a given phase can suppress or enhance another phase depending on the nature of the coupling between the order parameters, their dynamics and the dimensionality of the system. The zero temperature phase diagrams of systems with competing scalar order parameters with quartic and bilinear coupling terms have been previously studied for the cases of a zero temperature bicritical point and of coexisting orders. In this work, we apply the Matsubara summation technique from finite temperature quantum field theory to introduce the effects of thermal fluctuations on the effective potential of these systems. This is essential to make contact with experiments. We consider two and three-dimensional materials characterized by a Lorentz invariant quantum critical theory, i.e., with dynamic critical exponent z = 1, such that time and space scale in the same way. We obtain that in both cases, thermal fluctuations lead to weak first-order temperature phase transitions, at which coexisting phases arising from quantum corrections become unstable. We show that above this critical temperature ( T c ), the system presents scaling behavior consistent with that approaching a quantum critical point. Below the transition the specific heat has a thermally activated contribution with a gap related to the size of the domains of the ordered phases. We obtain that T c decreases as a function of the distance to the zero temperature classical bicritical point (ZTCBP) in the coexistence region, implying that in our approach, the system attains the highest T c above the fine tuned value of this ZTCBP.
Abstract The study of the competition or coexistence of different ground states in many-body systems is an exciting and actual topic of research, both experimentally and theoretically. Quantum fluctuations of a given phase can suppress or enhance another phase depending on the nature of the coupling between the order parameters, their dynamics and the dimensionality of the system. The zero temperature phase diagrams of systems with competing scalar order parameters with quartic and bilinear coupling terms have been previously studied for the cases of a zero temperature bicritical point and of coexisting orders. In this work, we apply the Matsubara summation technique from finite temperature quantum field theory to introduce the effects of thermal fluctuations on the effective potential of these systems. This is essential to make contact with experiments. We consider two and three-dimensional materials characterized by a Lorentz invariant quantum critical theory, i.e., with dynamic critical exponent z = 1, such that time and space scale in the same way. We obtain that in both cases, thermal fluctuations lead to weak first-order temperature phase transitions, at which coexisting phases arising from quantum corrections become unstable. We show that above this critical temperature ( T c ), the system presents scaling behavior consistent with that approaching a quantum critical point. Below the transition the specific heat has a thermally activated contribution with a gap related to the size of the domains of the ordered phases. We obtain that T c decreases as a function of the distance to the zero temperature classical bicritical point (ZTCBP) in the coexistence region, implying that in our approach, the system attains the highest T c above the fine tuned value of this ZTCBP.
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.
The dynamical critical exponent is introduced. The phase diagram and the phase transitions in the Bose–Hubbard model of interacting bosons on a lattice are determined. The concept of quantum fluctuations is introduced on the example of an interacting superfluid, and finally the special scaling of conductivity is discussed. Dynamical critical exponent The finite temperature phase transitions studied in previous chapters are the result of the competition between the entropy and the energy terms in the free energy: the weight of entropy increases with temperature, and ultimately destroys the order that may be existing in the system. A sharp phase transition between two phases exhibiting qualitatively different correlations may, however, occur even at zero temperature, by varying a coupling constant in the Hamiltonian. The transition then corresponds to a non-analyticity of the energy of the ground state. A simple example is provided by the interacting bosons, where the superfluid transition may be brought about by tuning the chemical potential at T = 0. Such T = 0 phase transitions are called quantum phase transitions , and will be the subject of the present chapter. In general, a quantum phase transition may lie at the end of a line of thermal phase transitions, as in the bosonic example mentioned above. It is possible, however, that the system may not even have an ordered state at finite temperatures, but still exhibits a quantum critical point. Two different situations are depicted in Fig. 8.1. Examples of such phase diagrams are provided by the system of interacting bosons which will be discussed in Sections 8.2 and 8.3.
Introduction The quantum phase transitions considered so far in this book are mostly continuous, second order. They are associated with the divergence of a correlation length and of a characteristic time that renders the quantum critical point scale invariant. However, there are many important phase transitions at which the order parameter goes abruptly to zero and, more important, at which the characteristics of length and critical time do not diverge. In general there is also a latent heat , or work required, to transform one phase into another. In this and the next chapter we study these discontinuous quantum phase transitions. We proceed by considering initially some important cases of what is known as fluctuation induced quantum phase transitions to next treat a more general case of a first-order quantum phase transition. We will show through the study of these problems that the concepts of critical exponents and scaling remain useful, even though these systems do not exhibit scale invariance at their critical point. In Chapter 15 we present the scaling theory of first-order quantum phase transitions and use its results to treat some general problems. One of the most interesting cases of fluctuation-induced transitions occurs in superconductors due to the coupling of the charged carriers to the electromagnetic field. We consider initially a wider class of systems with macroscopic wave functions, the superfluids, of which superconductors are the charged version. We adopt a macroscopic approach to this problem and describe the superfluid by a complex order parameter φ , such that | φ | 2 is proportional to the density of condensed particles. The zero temperature phase diagram is expressed in terms of a control parameter, m 2 , such that, at m 2 = 0, the system presents a quantum phase transition from a superfluid state, for m 2 < 0, φ ≠ 0, to a normal state for m 2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2 e , we will include the coupling of these particles to the electromagnetic field and study its effect in the phase diagram of these systems. We show that in the neutral superfluid, the zero temperature transition to the normal state is continuous and occurs at a quantum critical point.
We face the problem of detecting and featuring footprints of quantum criticality in the finite-temperature behavior of quantum many-body systems. Our strategy is that of comparing the phase diagram of a system displaying a T=0 quantum phase transition with that of its classical limit, in order to single out the genuinely quantum effects. To this aim, we consider the one-dimensional Ising model in a transverse field: while the quantum S=1∕2 Ising chain is exactly solvable and extensively studied, results for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ of such model are lacking, and we supply them here. They are obtained numerically, via the transfer-matrix method, and their asymptotic low-temperature behavior is also derived analytically by self-consistent spin-wave theory. We draw the classical phase diagram according to the same procedure followed in the quantum analysis, and the two phase diagrams are found unexpectedly similar: Three regimes are detected also in the classical case, each characterized by a functional dependence of the correlation length on temperature and field analogous to that of the quantum model. What discriminates the classical from the quantum case are the different values of the exponents entering such dependencies, a consequence of the different nature of zero-temperature quantum fluctuations with respect to the thermal ones.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantumphase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.