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arXiv:1801.07635 · Physical review. B./Physical review. B

Topological and geometric universal thermodynamics in conformal field theory

Haoxin Wang, Lei Chen, Hai Lin, Wei Li

Published Jun 28, 2018• 7 citations
arXivHTMLPDF

Abstract

Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, ${\mathbb{RP}}^{2}$). Through a cut-and-sew process, ${\mathbb{RP}}^{2}$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{1}{2}lnk$ related to a nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and $c$ the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.

DOI
10.1103/physrevb.97.220407

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Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy SK=lnk{S}_{\mathcal{K}}=lnkSK​=lnk on the Klein bottle, where kkk relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy lnglnglng (g(g(g reflects noninteger ground-state degeneracy), SK{S}_{\mathcal{K}}SK​ has nonono boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the M\"obius-strip manifold, we uncover an entropy SM=12(lng+lnk){S}_{\mathcal{M}}=\frac{1}{2}(lng+lnk)SM​=21​(lng+lnk) in CFT, where 12lng\frac{1}{2}lng21​lng is associated with the only open edge of the M\"obius strip and 12lnk\frac{1}{2}lnk21​lnk is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.
In this study, we explore the geometric construction of the Klein bottle and the real projective plane (RP^{2}) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on RP^{2} with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.
We propose two distinct crosscap states for the two-dimensional (2D) Ising field theory. These two crosscap states, identifying Ising spins or dual spins (domain walls) at antipodal points, are shown to be related via the Kramers-Wannier duality transformation. We derive their Majorana free field representations and extend bosonization techniques to calculate correlation functions of the 2D Ising conformal field theory (CFT) with different crosscap boundaries. Away from criticality, we develop a conformal perturbation theory to calculate the Klein bottle entropy (norm-square of the crosscap overlap) as a universal scaling function []. For the Ising field theory, our analytical results support the conjectured monotonicity of the Klein bottle entropy under relevant perturbations. The formalism provides a general framework for studying perturbed 2D CFTs on nonorientable manifolds.
A bstract Two-dimensional conformal field theories (CFTs) defined on non-orientable Riemann surfaces obey consistency Cardy conditions analogous to those in the orientable case. We revisit those conditions for irrational theories with central charge c > 1 in the context of two-point functions of primaries on the Real Projective plane $$ {\mathbbm{RP}}^2 $$ RP 2 and the partition function on the Klein bottle $$ {\mathbbm{K}}^2 $$ K 2 . Using the irrational versions of the Virasoro fusion and modular kernels we derive universal expressions for the non-orientable CFT data at large conformal dimension, assuming a gap in the spectrum of scalar primaries. In particular, we derive asymptotic formulas at finite central charge for the averaged Light-Light-Heavy product C LLH × Γ H of OPE coefficients with the $$ {\mathbbm{RP}}^2 $$ RP 2 one-point function normalizations, as well as for the parity-weighted density of heavy scalar primaries (or equivalently the density of heavy ΓH2 {\Gamma}_H^2 ΓH2​ Γ H 2 ). We discuss the gravitational interpretation of the results.
Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T → ∞, and a universal spectrum in the Cardy regime, Δ → ∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all Δ≥c6 \Delta \ge \frac{c}{6} Δ≥6c​ . The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal ‘enigma’ saddlepoints in the thermodynamics of 3d gravity.
Abstract In Can et al (2016 Phys. Rev. Lett. 117 ), quantum Hall states on singular surfaces were shown to possess an emergent conformal symmetry. In this paper, we develop this idea further and flesh out details on the emergent conformal symmetry in holomorphic adiabatic states, which we define in the paper. We highlight the connection between the universal features of geometric transport of quantum Hall states and holomorphic dimension of primary fields in conformal field theory. In parallel we compute the universal finite-size corrections to the free energy of a critical system on a hyperbolic sphere with conical and cusp singularities, thus extending the result of Cardy and Peschel for critical systems on a flat cone (Cardy and Peschel 1988 Nucl. Phys . B 300 377–92), and the known results for critical systems on polyhedra and flat branched Riemann surfaces.