The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought new concepts that revolutionized the way we understand many-body systems. Recently, through the discovery of symmetry protected topological phases, it is realized that quantum phase transition between states with the same symmetry but different topology can also occur continuously. Here we ask "what distinguishes these two types of phase transitions".
The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought new concepts that revolutionized the way we understand many-body systems. Recently, through the discovery of symmetry protected topological phases, it is realized that quantum phase transition between states with the same symmetry but different topology can also occur continuously. Here we ask "what distinguishes these two types of phase transitions".
The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. The subject of this dissertation is the phase transition between these symmetry protected topological states (SPTs). There are two main parts in this dissertation.In the first part we consider spatial dimension d and symmetry group G so that the cohomology group, Hd+1(G,U(1)), contains at least one Z2n or Z factor. We show that the phase transition between the trivial SPT and the root states that generate the $ Z_{2n} $ or Z groups can be induced on the boundary of a d+1 dimensional GtimesZ2T-symmetric SPT by a Z2T symmetry breaking field. Moreover we show these boundary phase transitions can be ``transplanted'' to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase. In the second part, we study the phase transition between bosonic topological phases protected by ZntimesZn in 1+1 dimensions. We find a direct transition occurs when $ n=2,3,4 $ and in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint($ c\\geq1 $) on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
The Landau paradigm is a central dogma for understanding phase and phase transitions in condensed matter systems, yet for decades it has been known that a variety of quantum phases exist beyond the framework. Is there a more general framework that provides a systematic understanding of phase and phase transitions in quantum many-body systems? Recent developments of the notion of generalized symmetry and generalized gauging seem to point to a way to generalize the Landau Paradigm. In this essay, we discuss how `beyond Landau' phases and phase transitions can be captured by a generalized Landau paradigm in terms of the breaking of generalized symmetries, often induced by the generalized gauging procedure facilitated through the topological holography formalism. We also discuss what needs to be understood to make the generalized Landau paradigm useful in the study of quantum phase and phase transitions.
The Landau Paradigm The usual paradigm to describe phase transitions is that of Landau. It is based on an expansion of a free energy functional or an action in terms of an order parameter. Close to the phase transition the order parameter is small, and only terms dictated by symmetry are allowed in this expansion. Associated with this paradigm are many important ideas, such as that of broken symmetry and Goldstone modes. The extension of this paradigm to include quantum phase transitions certainly brought new ideas and progress, as we have seen in this book: the inextricability between the dynamics and static properties, the modified hyperscaling relation with the notion of an effective dimensionality, the role of the uncertainty principle linking energy and temporal fluctuations and so on. The underlying principle here is always that of symmetry. However, as we will see in this chapter, this paradigm is not sufficient to describe certain classes of phenomena that do not involve symmetry changes but are nonetheless still associated with diverging lengths, singularities and critical exponents: features that are clearly related to critical phenomena and phase transitions. We have already met similar behaviour when studying density-driven or Lifshitz transitions. We will see in this chapter that these transitions are much more common and interesting that initially thought. The keyword here is topology, which contains principles and constraints that protect quasi-particle excitations and are still more powerful than those of symmetry. Topological Quantum Phase Transitions We will refer to topological quantum phase transitions (TQPT) as those that separate two phases where at least one of them is topologically non-trivial, as characterised by some topological parameter like a winding number or a Berry phase. In some cases these transitions either give rise to singularities in a thermodynamic quantity or are accompanied by a diverging length. However, differently from Landau-type phase transitions they are not necessarily associated with a symmetry change. Also there is no identifiable order parameter to distinguish between the different phases. Topological quantities are not useful to play the role of an order parameter. They are in general discontinuous at topological transitions and this does not necessarily reflect the nature of the transition. On the other hand, we will show that the renormalisation group is a useful tool to describe TQPT.
Condensed matter is the area of physics which deals with the phases of matter in condensed form with an aim to understand the properties of these phases and the transition between them based on the known laws of physics. This is usually done by figuring out some characteristic properties of the phase which sets it apart from the others, e.g., symmetry, topological order, entanglement, etc. The most common way of characterizing the transition between these phases is the broken symmetry. In this classification method, one finds the symmetry which phase breaks spontaneously during the transition and develops a local long-range order. This is the famous Landau’s symmetry breaking paradigm [ 1 ]. Landau’s symmetry breaking theory is a phenomenological theory which describes the transition from a symmetric phase (high-temperature phases) to a less symmetric/broken symmetry phase (phases at low temperature). A zero-temperature phase transition is also possible (by tuning the parameters of the Hamiltonian) and is known as quantum phase transition [ 2 ]. Although there were other paradigms of phase transition available, like Anderson localization [ 3 ], Berezinsky—Kosterlitz—Thouless transition [ 4 ] (broken symmetry but no long-range order), etc., Landau’s spontaneous symmetry breaking dominated over the others.
Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.