Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe an ...
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.
The quantum phase of a physical system is defined as its ground state. As examples of quantum phases we mention super uids, superconductors, Bose- Einstein condensates and quantum Hall states. Different phases of a system originate from the different ways a systems' constituents can be organised or, in other words, the different orders of the system. Landau had the insight that the difference between two phases can be attributed to the corresponding orders having different symmetries [1]. Also, the transition between phases can be associated with the breaking of symmetries. This insight led to Ginzburg and Landau describing orders using local order parameters and developing a general theory of phase transitions based on symmetry breaking. This theory was successful in describing a large number of observed phases of matter. However, phases of matter have been discovered which cannot be described using a local order parameter and Landau's theory of symmetry breaking known as topological phases.
Exotic phases of matter represent a paradigm shift in condensed matter physics, transcending classical thermodynamic states and embracing quantum mechanical principles. These phases, including topological insulators, quantum spin liquids, and time crystals, exhibit emergent properties driven by symmetry breaking, entanglement, and quantum coherence. This paper explores the fundamental principles governing these novel phases, recent breakthroughs in their realization, and their implications for quantum computing and materials science. Emphasis is placed on theoretical models such as the Kitaev honeycomb lattice and experimental advances in ultracold atom traps and strongly correlated electron systems. By reviewing current challenges and prospects, this study outlines a roadmap for navigating this new frontier in physics.
Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe and explain the nature of any phase transition. However, since the late 1980s, it has become apparent that it is necessary to introduce some new kind of order, named topological order, that transcends the traditional symmetry description. In this thesis we will study the Kitaev model, which is a Hamiltonian lattice model that allows one to incorporate the concept of topological order, as well as the corresponding operators and algebras. First, we consider the model on an infinite lattice, and show how to relate local and global degrees of freedom of the anyons/quasi-particles living on sites to the ribbon operators. Afterwards, we introduce holes and an external boundary to the lattice, and examine the ramifications of this generalization in terms of the ground state degeneracy. Lastly, we verify that the algebra formed by boundary site operators has the structure of a quasi-Hopf algebra.
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.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.