Conformal field theories and critical phenomena

Al.B. Zamolodchikov, Al.B. Zamolodchikov

Norio Kawakami, Sung-Kil Yang

Recent progress in conformal field theory enables us to precisely treat the critical phenomena in (1 +I)-dimension. We review how conformal field theory applies to correlated electron systems in one dimension and explain the way of exact calculation of the correlation exponents. We demonstrate that conformal field theory provides the microscopic foundation of the Luttinger liquid which describes the universal behavior of lD correlated electron systems.

Deniz N. Bozkurt

Conformal field theories (CFTs) and supersymmetric field theories constitute special classes of quantum field theories with applications ranging from critical phenomena and confinement to quantum gravity via holography. Their large symmetry groups make them ideal testing grounds for theoretical developments, enabling the use of powerful and elegant methods. Among these, conformal bootstrap and integrability have proven particularly successful. In this thesis, we take steps toward extending these methods beyond their conventional domains by breaking conformal symmetry through finite temperature effects and reducing supersymmetry via orbifolding. For spin chains arising from $\mathcal{N} = 2$ superconformal field theories obtained by orbifolding $\mathcal{N} = 4$ super Yang–Mills theory in four dimensions, we generalize the coordinate Bethe ansatz by incorporating long-range corrections.On the conformal bootstrap side, after breaking conformality by placing the CFT on a d-dimensional cylinder, we develop an analytic framework based on dispersion relations. We apply this method to several theories, including large-N CFTs in four dimensions. Furthermore, we combine this analytic approach with existing numerical bootstrap techniques at finite temperature to compute new thermal correlation functions in the three-dimensional Ising CFT.

Yevgeny Kats

Malte Henkel

Yifan Liu, Natalia Chepiga, Yoshiki Fukusumi +1 more

We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the $\mathbb{Z}_2$-orbifold of the $c=1$ compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying $\mathrm{SU}(2)$ symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with $D_4$ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.