A bstract We study quantum field theories which have quantum groups as global internal symmetries. We show that in such theories operators are generically non-local, and should be thought as living at the ends of topological lines. We describe the general constraints of the quantum group symmetry, given by Ward identities, that correlation functions of the theory should satisfy. We also show that generators of the symmetry can be represented by topological lines with some novel properties. We then discuss a particular example of U q ( sl 2 ) symmetric CFT, which we solve using bootstrap techniques and relying on the symmetry. We finally show strong evidence that for a special value of q a subsector of this theory reproduces the fermionic formulation of the Ising model. This suggests that a quantum group can act on local operators as well, however, it generically transforms them into non-local ones.
This thesis studies Quantum Field Theories (QFTs) and many-body systems that have quantum groups as global internal symmetries. We establish that quantum groups can act as genuine global symmetries, constraining the spectrum, correlation functions and dynamics. Operators that transform naturally under this symmetry are shown to be non-local, living at the ends of topological lines that carry some novel properties. These properties are illustrated by explicit examples including spin chains and their field-theoretic continuum limit. We solve the continuum theory in two ways: first, by using constraints coming from quantum group symmetry together with conformal bootstrap techniques; second, by applying the Coulomb gas formalism. We further show that this symmetry is not an artifact of integrability: it survives as a global symmetry even when integrability is broken and the system becomes chaotic. Finally, we observe that it governs late-time hydrodynamic transport, as manifest in the slow decay of the transverse spin autocorrelation function. Altogether, these results position quantum groups as a new and robust class of symmetry governing the physics of QFTs and many-body systems.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of
n point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
In this paper we study the ground-state properties of a ladder Hamiltonian with chiral
SU(2)-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of
SU(2) symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a
c=1 conformal field theory (CFT) in
(1+1) dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-
21 Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an
SU(2)1 Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for
SU(2)-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.
Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic “topological symmetries” that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.
This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to
SO(d+1,1), and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra
[Lm,Ln]=(m−n)Lm+n+12c(m3−m)δm+n,0 from infinitesimal generators
ℓn=−zn+1∂z, computing the central charge explicitly for free theories (boson
c=1, Majorana fermion
c=1/2,
bc-ghosts
c=1−3(2λ−1)2) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states
∣h,hˉ⟩, deriving the transformation law
T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}] that reveals the geometric origin of the conformal anomaly.