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arXiv:1805.01300 · Physical review. B./Physical review. B

Klein bottle entropy of compactified boson conformal field theory

Wei Tang, X. C. Xie, Lei Wang, Hong-Hao Tu

Published Mar 5, 2019• 16 citations

Abstract

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 $lng$ in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius $R$, $lng=lnR$. 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-$1/2$ 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 $S=1$ XXZ chain that cannot be exactly solved, our numerical results provide an accurate numerical determination of the Luttinger parameter in this model.

DOI: 10.1103/physrevb.99.115105

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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

lng

in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius

R

,

lng=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-

1/2

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=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.

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.

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

q

-state Potts model are compared with the CFT predictions.

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr ρ A n for any integer n is calculated as the four-point function of twist fields of a particular type and the final result is expressed in a compact form in terms of the Riemann–Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.

Abstract We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class. While much insight has been gained in homogeneous systems by making use of conformal field theory techniques, our focus is on systems for which the Luttinger parameter K depends on position, and conformal invariance is broken. An important point of our analysis is that contributions stemming from the UV cutoff have to be treated very carefully, since they now depend on position. We show that such terms can be removed either by considering regularized entropies specifically designed to do so, or by tabulating numerically the cutoff, and reconstructing its contribution to the entropy through the local density approximation. We check our method numerically in the spin-1/2 XXZ spin chain in a spatially varying magnetic field, and find excellent agreement.

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.