By analogy with the Landau-Ginzburg theory of ordinary zero-form symmetries, we introduce and develop a Landau-Ginzburg theory of one-form global symmetries, which we call mean string field theory. The basic dynamical variable is a string field – defined on the space of closed loops – that can be used to describe the creation, annihilation, and condensation of effective strings. Like its zero-form cousin, the mean string field theory provides a useful picture of the phase diagram of broken and unbroken phases. We provide a transparent derivation of the area law for charged line operators in the unbroken phase and describe the dynamics of gapless Goldstone modes in the broken phase. The framework also provides a theory of topological defects of the broken phase and a description of the phase transition that should be valid above an upper critical dimension, which we discuss. We also discuss general consequences of emergent one-form symmetries at zero and finite temperature.
By analogy with the Landau-Ginzburg theory of ordinary zero-form symmetries, we introduce and develop a Landau-Ginzburg theory of one-form global symmetries, which we call mean string field theory. The basic dynamical variable is a string field – defined on the space of closed loops – that can be used to describe the creation, annihilation, and condensation of effective strings. Like its zero-form cousin, the mean string field theory provides a useful picture of the phase diagram of broken and unbroken phases. We provide a transparent derivation of the area law for charged line operators in the unbroken phase and describe the dynamics of gapless Goldstone modes in the broken phase. The framework also provides a theory of topological defects of the broken phase and a description of the phase transition that should be valid above an upper critical dimension, which we discuss. We also discuss general consequences of emergent one-form symmetries at zero and finite temperature.
Abstract Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.
Recently, a generalized Landau paradigm has been proposed, in which a broad class of "unconventional" phase transitions lying beyond the original symmetry-breaking framework of Landau are understood as the symmetry-breaking transitions of generalized symmetries. In particular, in (1+1)D, such "topological" or "deconfined" transitions can be mapped to the symmetry-breaking transitions of non-invertible symmetries. How to extract the universal dynamical properties of the transitions from this understanding? In this paper, we follow the Ginzburg-Landau philosophy and develop a field-theoretic description of such critical points in terms of the fluctuations of local order parameters, now of non-invertible symmetries. The Symmetry Topological Field Theory formalism plays a central role in our analysis, allowing us to identify the algebra of local order parameters from the Lagrangian algebra of a (2+1)D topological order. We illustrate the key ideas and the full steps of this generalized Ginzburg-Landau procedure using the transitions between gapped phases with a simple non-invertible symmetry --- the Rep(S3) symmetry.
Symmetries are important guiding principle for phase transitions. We systematically construct field theory models with local quantum fields that exhibit the following phase transitions: (1) different symmetry protected topological (SPT) phases with generalized symmetries; (2) different symmetry enriched topological (SET) phases with generalized symmetries differ by symmetry fractionalizations; (3) spontaneously broken generalized symmetries, where the unbroken phases can have nontrivial SPT or SET. The models are ordinary gauge theories with bosons or fermions in 3+1d and 2+1d. We focus on one-form symmetries and symmetries generated by condensation defects, which do not act on local operators. The phase transitions are protected from local operator perturbations which do not change the asymptotic phases. In particular, we show that continuous gauge theories in 3+1d can have different phases distinguished by fractionalizations of unbroken one-form symmetries.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.
In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking have been accumulating, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative tests except for certain solvable models, therefore limiting their usage in the more generic quantum many-body systems. In this work, we study Z2 higher-form symmetry in a quantum Ising model, which is dual to the global (zero-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a nonlocal order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in (2+1)d. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic or ordered phase. At the Ising critical point, we find that the disorder operator also obeys a ``perimeter'' law scaling with possibly multiplicative power-law corrections. We discuss examples where both the global zero-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide nontrivial working examples of higher-form symmetry operators, including the direct computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.