Conformal Thermal Tensor Network and Universal Entropy on Topological Manifolds

Abstract

Partition functions of quantum critical systems, expressed as conformal thermal tensor networks, are defined on various manifolds which can give rise to universal entropy corrections. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy $S_{\mathcal{K}}=\ln k$ on the Klein bottle, where $k$ relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy $\ln g$ ($g$ reflects non-integer groundstate degeneracy), $S_{\mathcal{K}}$ has \textit{no} boundary dependence or surface energy terms accompanied, and can be very conveniently extracted from thermal data. On the Möbius-strip manifold, we uncover an entropy $S_{\mathcal{M}}=\frac{1}{2}(\ln g+\ln k)$ in CFT, where $\frac{1}{2}\ln g$ is associated with the only open edge of the Möbius strip, and $\frac{1}{2}\ln k$ with the non-orientable topology. We employ $S_{\mathcal{K}}$ to accurately pinpoint the quantum phase transitions, even for those without local order parameters.

arXiv comment4 pages + references, 5 figures, supplementary material; minor revisions

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• Exponential Thermal Tensor Network Approach for Quantum Lattice Models
• Critical properties of the two-dimensional $q$-state clock model
• Automatic Differentiation for Second Renormalization of Tensor Networks
• Tensor network simulation for the frustrated $J_1$-$J_2$ Ising model on the square lattice
• Universal entropy of conformal critical theories on a Klein bottle
• Continuous matrix product operator approach to finite temperature quantum states
• Klein bottle entropy of compactified boson conformal field theory
• Universal scaling of Klein bottle entropy near conformal critical points
• Extracting the Luttinger parameter from a single wave function
• Topological and Geometric Universal Thermodynamics in Conformal Field Theory
• Numerical extraction of crosscap coefficients in microscopic models for (2+1)D conformal field theory
• On the topological and crosscap entropies in non-oriented RCFTs
• Crosscap states and duality of Ising field theory in two dimensions
• Tensor network simulations for nonorientable surfaces