In this paper, we perform a comprehensive study of the renormalization group (RG) method on thermal tensor networks (TTN). By Trotter-Suzuki decomposition, one obtains the 1+1D TTN representing the partition function of 1D quantum lattice models, and then employs efficient RG contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN in an efficient and accurate way, is briefly reviewed. In addition, the single-layer LTRG can be generalized to a bilayer form, dubbed as LTRG++, in both finite-and infinite-size systems, with accuracies significantly improved. We provide the details of LTRG++ in finite-size system, comparing its accuracy with single-layer algorithm, and elaborate the infinite-size LTRG++ in the context of fermion chain model. We show that the LTRG++ algorithm in infinite-size system is in essence equivalent to transfer-matrix renormalization group method, while expressed in a tensor network language. LTRG++ is then applied to study an extended Hubbard model, where the phase separation phenomenon, groundstate phase diagram, as well as quantum criticality-enhanced magnetocaloric effects under external fields, are investigated.

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 SK = 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), SK has 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 SM = ln k with the non-orientable topology. We employ SK to accurately pinpoint the quantum phase transitions, even for those without local order parameters.Introduction.-Quantum critical point (QCP) expands to a finite regime at T > 0, where the thermal properties exhibit intriguing universal features [1]. For d dimensional quantum critical systems, there emerges conformal symmetry in the partition function defined on d + 1 manifold (Euclidean worldsheet), described by the conformal field theory (CFT) [2,3]. The partition functions can be expressed as thermal tensor networks (TTNs) [4][5][6], which provides a powerful tool exploring finite-temperature properties of exotic thermal matter near QCPs, e.g., extracting universal data to identify the corresponding CFTs.According to 2D CFT, for the quantum critical chain with a torus worldsheet, i.e., periodic boundary condition (PBC) on both L and β directions [ Figs. 1(a,b)], the (logarithmic) partition function takes the universal form (when L vβ) ln Z T = L[− 0 β +

Recently, entropy corrections on nonorientable manifolds such as the Klein bottle are proposed as a universal characterization of critical systems with an emergent conformal field theory (CFT). We show that entropy correction on the Klein bottle can be interpreted as a boundary effect via transforming the Klein bottle into an orientable manifold with nonlocal boundary interactions. The interpretation reveals the conceptual connection of the Klein bottle entropy with the celebrated Affleck-Ludwig entropy in boundary CFT. We propose a generic scheme to extract these universal boundary entropies from quantum Monte Carlo calculation of partition function ratios in lattice models. Our numerical results on the Affleck-Ludwig entropy and Klein bottle entropy for the q-state quantum Potts chains with q = 2, 3 show excellent agreement with the CFT predictions. For the quantum Potts chain with q = 4, the Klein bottle entropy slightly deviates from the CFT prediction, which is possibly due to marginally irrelevant terms in the low-energy effective theory.

There have been extensive recent developments on kagome metals, such as TmXn (T= Fe, Co and X= Sn, Ge) and AV3Sb5 (A = Cs, K, Rb). An emerging issue is the nature of correlated phases when topologically non-trivial bands cross the Fermi level. Here, we consider an extended Hubbard model on the kagome lattice in the presence of spin-orbit couplings, involving a Kramer's pair of bands that have opposite Chern numbers and are isolated in the band structure. We construct an effective model in a time-reversal (T) symmetric lattice description. We determine the correlated phases of this model and identify a density-wave order in the phase diagram. We show that this order is T-breaking, which originates from the Wannier orbitals lacking a common Wannier center -a fingerprint of the underlying Z2 topology. Implications of our results for the correlation physics of the kagome metals are discussed.

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, RP 2 ). Through a cut-and-sew process, RP 2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term 1 2 ln k related to nonorientable genus, with k a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term c 4 ln β, with β the manifold size and c 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. arXiv:1801.07635v3 [cond-mat.str-el]