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arXiv:1707.05812 · Physical Review Letters

Universal Entropy of Conformal Critical Theories on a Klein Bottle

Hong-Hao Tu

Published Dec 28, 2017• 33 citations
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Abstract

We show that rational conformal field theories in 1+1 dimensions on a Klein bottle, with a length L and width β, satisfying L≫β, have a universal entropy. This universal entropy depends on the quantum dimensions of the primary fields and can be accurately extracted by taking a proper ratio between the Klein bottle and torus partition functions, enabling the characterization of conformal critical theories. The result is checked against exact calculations in quantum spin-1/2 XY and Ising chains.

DOI
10.1103/physrevlett.119.261603

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We show that rational conformal field theories in 1+1 dimensions on a Klein bottle, with a length L and width β, satisfying L≫β, have a universal entropy. This universal entropy depends on the quantum dimensions of the primary fields and can be accurately extracted by taking a proper ratio between the Klein bottle and torus partition functions, enabling the characterization of conformal critical theories. The result is checked against exact calculations in quantum spin-1/2 XY and Ising chains.
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We extend the scope of the Klein bottle entropy, originally introduced by Tu [Phys. Rev. Lett. 119, 261603 (2017)] in the rational conformal field theory (CFT), to the compactified boson CFT, which are relevant to the studies of Luttinger liquids. We first review the Klein bottle entropy in rational CFT and discuss details of how to extract the Klein bottle entropy from lattice models using the example of the transverse field Ising model. We then go beyond the scope of rational CFT and study the Klein bottle entropy lnglnglng in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius RRR, lng=lnRlng=lnRlng=lnR. This relation indicates a convenient and efficient method to extract the Luttinger parameter from lattice model calculations. Our numerical results on the Klein bottle entropy in the spin-1/21/21/2 XXZ chain show excellent agreement with the CFT predictions, up to some small deviations near the isotropic point, which we attribute to the marginally irrelevant terms. For the S=1S=1S=1 XXZ chain that cannot be exactly solved, our numerical results provide an accurate numerical determination of the Luttinger parameter in this model.
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 addition to the celebrated Affleck-Ludwig entropy originating from the open boundaries of the path-integral manifold, recent research has shown that the entropy correction on nonorientable manifolds such as the Klein bottle is also a universal characterization of critical systems with an emergent conformal field theory (CFT). Here, the authors show that the Klein bottle entropy can be interpreted as a boundary effect by transforming the Klein bottle into an orientable manifold, whereby the Klein bottle entropy bears a resemblance to the Affleck-Ludwig entropy. The authors then propose a generic scheme to extract these universal boundary entropies from lattice models. Numerical results for the qqq-state Potts model are compared with the CFT predictions.
We study the relative entropy and the trace square distance, both of which measure the distance between reduced density matrices of two excited states in two dimensional conformal field theories. We find a general formula for the relative entropy between two primary states with the same conformal dimension in the limit of a single small interval and find that in this case the relative entropy is proportional to the trace square distance. We check our general formulae by calculating the relative entropy between two generalized free fields and the trace square distance between the spin and disorder operators of the critical Ising model. We also give the leading term of the relative entropy in the small interval expansion when the two operators have different conformal dimensions. This turns out to be universal when the CFT has no primaires lighter than the stress tensor. The result reproduces the previously known special cases.
We study the scaling of the R\'enyi and entanglement entropy of two disjoint blocks of critical Ising models as function of their sizes and separations. We present analytic results based on conformal field theory that are quantitatively checked in numerical simulations of both the quantum spin chain and the classical two-dimensional Ising model. Theoretical results match the ones obtained from numerical simulations only after taking properly into account the corrections induced by the finite length of the blocks to their leading scaling behavior.