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arXiv:1708.04034 · Physical review. B./Physical review. B

Conformal thermal tensor network and universal entropy on topological manifolds

Lei Chen, Haoxin Wang, Lei Wang, Wei Li

Published Nov 21, 2017• 15 citations
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Abstract

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 ${S}_{\mathcal{K}}=lnk$ 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 $lng$ $(g$ reflects noninteger ground-state degeneracy), ${S}_{\mathcal{K}}$ has $no$ 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 ${S}_{\mathcal{M}}=\frac{1}{2}(lng+lnk)$ in CFT, where $\frac{1}{2}lng$ is associated with the only open edge of the M\"obius strip and $\frac{1}{2}lnk$ 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.

DOI
10.1103/physrevb.96.174429

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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 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.
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, RP2{\mathbb{RP}}^{2}RP2). Through a cut-and-sew process, RP2{\mathbb{RP}}^{2}RP2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term 12lnk\frac{1}{2}lnk21​lnk related to a nonorientable genus, with kkk 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 ccc 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.
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.
We show that the Klein bottle entropy [H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.261603] for conformal field theories perturbed by a relevant operator is a universal function of the dimensionless coupling constant. The universal scaling of the Klein bottle entropy near criticality provides an efficient approach to extract the scaling dimension of lattice operators via data collapse. As paradigmatic examples, we validate the universal scaling of the Klein bottle entropy for Ising and Z_{3} parafermion conformal field theories with various perturbations using numerical simulation with continuous matrix product operator approach.
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.
A bstract Tensor networks offer a sign-problem-free approach to study lattice gauge theories, but extracting precise universal information associated with the deconfinement transition remains challenging. In this work, we study the deconfinement transition of (2+1)-dimensional ℤ N lattice gauge theories at finite temperature using a thermal tensor network approach, where the partition functions at finite temperature are formulated as three-dimensional tensor networks. These tensor networks are first contracted in the temporal direction, and the subsequent coarse-graining in the spatial directions yields a renormalized transfer matrix, the spectrum of which directly encodes the universal conformal field theory data. In particular, by numerically extracting the central charge and scaling dimensions, we verify that the universality class of the thermal deconfinement transition matches the prediction of the Svetitsky-Yaffe conjecture for N = 2, 3, 5. Moreover, we show that the ℤ 5 theory at finite temperature exhibits an intermediate phase with an emergent U(1) symmetry. Critical couplings are determined via Gu-Wen ratios and agree with existing Monte Carlo simulations. Finally, extrapolating these critical couplings at finite temperature enables us to determine the deconfinement transition points for N = 2, 3 at zero temperature.