Tensor renormalization group (TRG) method is a real space renormalization group approach. It has been successfully applied to both classical and quantum systems. In this paper, we study a disordered and frustrated system, the two-dimensional Edwards-Anderson model, by a new topological invariant TRG scheme. We propose an approach to calculate the local magnetizations and nearest pair correlations simultaneously. The Nishimori multicritical point predicted by the topological invariant TRG agrees well with the recent Monte Carlo results. The TRG schemes outperform the mean-field methods on the calculation of the partition function. We notice that it might obtain a negative partition function at sufficiently low temperatures. However, the negative contribution can be neglected if the system is large enough. This topological invariant TRG can also be used to study three-dimensional spin glass systems.
Tensor renormalization group method (TRG) is a real space renormalization group approach. It has been successfully applied to both classical and quantum systems. In this paper, we study a disordered and frustrated system, the two-dimensional Edward-Anderson model, by a new topological invariant TRG scheme. We propose an approach to calculate the local magnetizations and nearest pair correlations simultaneously. The Nishimori multi-critical point predicted by the topological invariant TRG agrees well with the recent Monte-Carlo results. The TRG schemes outperform the mean field methods on the calculation of the partition function. We notice that it maybe obtain a negative partition function at sufficiently low temperatures. However, the negative contribution can be neglected if the systems is large enough. This topological invariant TRG can also be used to study three-dimensional spin glass systems.
A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ2 and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ2 Edwards-Anderson model in d=4. Good estimates of the critical index ν and of Tc are obtained, and an RG flow diagram is sketched for the first time.
The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length $\ensuremath{\xi}$, such that for system sizes $L\ensuremath{\gg}\ensuremath{\xi}$ finite-size calculations actually simulate the thermodynamic limit. In this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width
W=10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site ${u}_{g}^{*}\ensuremath{\simeq}\ensuremath{-}0.6694(4)$ and a spontaneous magnetization ${m}_{S}^{*}\ensuremath{\simeq}0.30(1)$ already consistent with the ground-state properties, which is obtained from extrapolated low-
T thermal data on $W\ensuremath{\le}8$ cylinders and $W\ensuremath{\le}10$ open strips, respectively. We extract an exponential divergence versus
T of the structure factor
S(M), as well as the correlation length $\ensuremath{\xi}$, at the ordering wave vector $M=(\ensuremath{\pi},\ensuremath{\pi})$, which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature
Tc.