arXiv:1708.06985 · Acta Physica Polonica A

Order in Quantum Compass and Orbital egModels

Piotr Czarnik, Jacek Dziarmaga, Andrzej M. Oleś

Published Mar 1, 2018• 0 citations

Abstract

We investigate thermodynamic phase transitions in the compass model and in eg orbital model on an infinite square lattice by variational tensor network renormalization (VTNR) in imaginary time.The onset of nematic order in the quantum compass model is estimated at Tc{J 0.0606p4q.For the eg orbital model one finds: (i) a very accurate estimate of Tc{J 0.3566 0.0001 and (ii) the critical exponents in the Ising universality class.Remarkably large difference in frustration results in so distinct values of Tc, while entanglement influences the quality of Tc estimation.

DOI: 10.12693/aphyspola.133.359

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arXiv:1708.06985 · Acta Physica Polonica A

Order in Quantum Compass and Orbital egModels

Piotr Czarnik, Jacek Dziarmaga, Andrzej M. Oleś

Published Mar 1, 2018• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.12693/aphyspola.133.359

Related papers

Variational tensor network renormalization in imaginary time: Two-dimensional quantum compass model at finite temperature

Piotr Czarnik, Jacek Dziarmaga, Andrzej M. Oleś

Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension $D$ . The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at ${\mathcal{T}}_{c}=0.0606(4)J$ , where $J$ is the isotropic coupling constant between $S=\frac{1}{2}$ pseudospins.Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension $D$. The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at ${\mathcal{T}}_{c}=0.0606(4)J$, where $J$ is the isotropic coupling constant between $S=\frac{1}{2}$ pseudospins.

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Nov 12, 20203 citations

H

can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension

D

. The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at

{\mathcal{T}}_{c}=0.0606(4)J

, where

J

is the isotropic coupling constant between

S=\frac{1}{2}

pseudospins.

{e}_{g}

model with spatially highly anisotropic orbital interactions. Coarse graining of the tensor network along the inverse temperature $\ensuremath{\beta}$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension

D

---limiting the amount of entanglement---is a natural refinement parameter. Increasing

D

we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature

{T}_{c}

, provide a numerically exact estimate of

{T}_{c}

, and give the critical exponents within

1%

of the 2D Ising universality class.

C_4

and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two

z

-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.

\beta^*\approx 0.02

for one of our topological order parameters, which we relate to disorder parameters in the 3D Ising model. This shows that our topological order parameters can provide additional means to characterize the universal data at topological phase transitions, and altogether demonstrates the power of this framework to microscopically study topological phase transitions and identify the universal data underlying the transition.