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arXiv:2211.09745 · Physical Review Letters

Universal Scaling of Klein Bottle Entropy near Conformal Critical Points

Yueshui Zhang, Anton Hulsch, Hua-Chen Zhang, Wei Tang, Lei Wang, Hong-Hao Tu

Published Apr 14, 2023• 10 citations
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Abstract

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.

DOI
10.1103/physrevlett.130.151602

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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.
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.
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.
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.
Abstract We compare the time evolution of entanglement measures after local operator excitation in the critical Ising model with predictions from conformal field theory. For the spin operator and its descendants we find that Rényi entropies of a block of spins increase by a constant that matches the logarithm of the quantum dimension of the conformal family. However, for the energy operator we find a small constant contribution that differs from the conformal field theory answer equal to zero. We argue that the mismatch is caused by the subtleties in the identification between the local operators in conformal field theory and their lattice counterpart. Our results indicate that evolution of entanglement measures in locally excited states not only constraints this identification, but also can be used to extract non-trivial data about the conformal field theory that governs the critical point. We generalize our analysis to the Ising model away from the critical point, states with multiple local excitations, as well as the evolution of the relative entropy after local operator excitation and discuss universal features that emerge from numerics.
Measuring universal data in the strongly correlated regime of quantum critical points remains a fundamental objective for quantum simulators. In foundational work, Calabrese and Cardy demonstrated how these data govern the dynamics of certain global quenches to 1+1-dimensional conformal field theories. While the quasiparticle picture they introduce has been widely successful in both theory and experiment, their seminal prediction that the critical exponents are simply encoded in the relaxation rates of local observables is challenging to investigate experimentally. In this Letter, we examine the critical quench dynamics of local observables from two types of readily accessible initial conditions: ground states and finite-temperature ensembles. We identify universal scaling collapses and scaling functions, utilizing a combination of conformal perturbation theory and tensor network numerics. For the finite-temperature quenches, we determine a regime in which the conformal field theory results are recovered, thereby allowing universal quantum critical data to be extracted from realistic quenches.