This presentation is an analysis of the role of symmetry in second-order quantum phase transitions. It seeks to explain why transitions between phases of systems, associated with different symmetries, exhibit critical phenomena. It transpires that a system in one phase, tends to hold onto the symmetry associated with that phase until a breaking point is reached at which a rapid transition occurs to a new phase associated with a different symmetry. Understanding what is happening presents the challenge of explaining how a system "holds on" to a symmetry in the face of strong symmetry-breaking interactions. The explanation is of fundamental interest in physics, for understanding why models with built in symmetries often work extremely well even when the models ignore large symmetry-breaking interactions that are known to be present. When this phenomenon occurs, we say that the system has a quasidynamical symmetry. This concept is of interest in mathematics because it turns out that quasidyndamical symmetries are the physical realizations of a new concept in group theory, which we refer to as an embedded representation. PACS Nos.: 21.60.Fw, 21.60.Ev, 64.6–.Ht, 68.18.Jk
This presentation is an analysis of the role of symmetry in second-order quantum phase transitions. It seeks to explain why transitions between phases of systems, associated with different symmetries, exhibit critical phenomena. It transpires that a system in one phase, tends to hold onto the symmetry associated with that phase until a breaking point is reached at which a rapid transition occurs to a new phase associated with a different symmetry. Understanding what is happening presents the challenge of explaining how a system "holds on" to a symmetry in the face of strong symmetry-breaking interactions. The explanation is of fundamental interest in physics, for understanding why models with built in symmetries often work extremely well even when the models ignore large symmetry-breaking interactions that are known to be present. When this phenomenon occurs, we say that the system has a quasidynamical symmetry. This concept is of interest in mathematics because it turns out that quasidyndamical symmetries are the physical realizations of a new concept in group theory, which we refer to as an embedded representation. PACS Nos.: 21.60.Fw, 21.60.Ev, 64.6–.Ht, 68.18.Jk
The apparent persistence of symmetry in the face of strong symmetry-breaking interactions is examined in a many-boson model. The model exhibits a transition between two phases associated with U(5) and O(6) symmetries, respectively, as the value of a control parameter progresses from 0 to 1. The remarkable fact is that, in spite of strong mixing of the symmetries for intermediate values of the control parameter, the model continues to exhibit the characteristics of its closest symmetry limit for all but a relatively narrow transition region that becomes progressively narrower as the boson number increases. This phenomenon is explained in terms of quasidynamical symmetry.
Quantum phase transitions are strongly relevant in a number of fields, ranging from condensed matter to cold atom physics and quantum field theory. This book, now in its second edition, approaches the problem of quantum phase transitions from a new and unifying perspective. Topics addressed include the concepts of scale and time invariance and their significance for quantum criticality, as well as brand new chapters on superfluid and superconductor quantum critical points, and quantum first order transitions. The renormalisation group in real and momentum space is also established as the proper language to describe the behaviour of systems close to a quantum phase transition. These phenomena introduce a number of theoretical challenges which are of major importance for driving new experiments. Being strongly motivated and oriented towards understanding experimental results, this is an excellent text for graduates, as well as theorists, experimentalists and those with an interest in quantum criticality.
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 work deals with quantum phase transitions in Dirac systems where the symmetries involved play a key role in the nature of the transition and are enlarged at criticality. By using functional renormalization group (FRG) methods, we show that an emergent relativistic symmetry not present at the bare level is a common feature of several different kinds of phase transitions. In three different projects, the interplay of emergent symmetries and universality in critical phenomena are explored through FRG techniques. In the first project, we studied the relation between discrete symmetry breaking and the emergence of two diverging length scales with their respective critical exponents. A second project explored the relation between multicritical points where two adjacent symmetry-breaking phases are compatible and the possibility of a continuous, order-to-order transition. The third and final project was concerned with the persistence of chiral symmetry in three-dimensional quantum electrodynamics - here seen as an effective description of different condensed matter systems - for the lowest possible number of fermion flavours, as well as with a dual description that has the potential to provide exact results for this strongly correlated theory.
The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century--there is a great theoretical challenge in solving special statistical mechanical models. Additionally, beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions. Although a few spectacular successes in the exact solution of simple models have occurred, interesting systems have mostly proven to be intractable from the analytic perspective. Because many physically disparate systems show identical critical behavior, identifying and understanding general principles of universality and their relationship to microscopic symmetries are of fundamental importance
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.