Critical points and their properties are one of the most active topics in condensed-matter physics.The activity in this field can be partially attributed to the abundance of theoretically and experimentally accessible phenomena in the vicinity of continuous phase transitions.In recent years quantum critical points have attracted a particular amount of interest.Although quantum phase transitions are observed only at absolute zero temperature, they also affect certain properties at nonzero temperatures, which can be studied experimentally.One remarkable feature of critical points is that in their vicinity macroscopic properties do not depend on the microscopic details of the sample and the underlying Hamiltonian.Therefore, the various transitions can be assigned to different universality classes, where all representatives of a class show exactly the same properties.Furthermore, the so-called scaling hypothesis implies that close to the critical point the observable quantities follow a universal scaling form.As a result, by properly rescaling all system-dependent parameters it is possible to collapse all data of a particular quantity onto a single curve, given by the respective universal scaling function.
Critical points and their properties are one of the most active topics in condensed-matter physics.The activity in this field can be partially attributed to the abundance of theoretically and experimentally accessible phenomena in the vicinity of continuous phase transitions.In recent years quantum critical points have attracted a particular amount of interest.Although quantum phase transitions are observed only at absolute zero temperature, they also affect certain properties at nonzero temperatures, which can be studied experimentally.One remarkable feature of critical points is that in their vicinity macroscopic properties do not depend on the microscopic details of the sample and the underlying Hamiltonian.Therefore, the various transitions can be assigned to different universality classes, where all representatives of a class show exactly the same properties.Furthermore, the so-called scaling hypothesis implies that close to the critical point the observable quantities follow a universal scaling form.As a result, by properly rescaling all system-dependent parameters it is possible to collapse all data of a particular quantity onto a single curve, given by the respective universal scaling function.
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 .
Critical phenomena, characterized by abrupt changes in the macroscopic properties of a system, exhibit a profound degree of universality, meaning that vastly different physical systems can display identical behavior near their critical points. This universality is typically understood through the lens of scaling laws and the Renormalization Group theory, which classify critical points into distinct universality classes. This paper proposes and explores the "Entropic Law of Critical Universality," an overarching principle positing that not only do thermodynamic quantities scale universally, but the fundamental information-theoretic measures of a system, particularly entropy in its various forms (thermodynamic entropy, entanglement entropy, Renyi entropy), also conform to universal scaling behaviors that uniquely characterize these universality classes. We review how entanglement entropy exhibits universal logarithmic corrections at quantum critical points, linked to the central charge of underlying conformal field theories. We also discuss how specific entropy behaves non-monotonically near quantum chromodynamics critical points and how entropic measures can identify phase transitions and their classes in various condensed matter systems, including graphene and spin models. The paper synthesizes theoretical frameworks and empirical evidence to demonstrate that entropy serves as a powerful and often more direct probe of the universal properties underlying critical phenomena, offering a comprehensive perspective on the shared information landscape of diverse critical systems.
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.
Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.