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Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

Published Nov 27, 2012• 8 citations
OpenAlex

Abstract

Abstract. We obtained exact heat capacities of the quantum compass model on the square L×L clusters with L = 2, 3,4, 5 using Kernel Polynomial Method and compare them with heat capacity of a large compass ladder. Intersite correlations found in the ground state for these systems demonstrate that the quantum compass model differs from its classical version. Quantum compass model originating from the orbital (Kugel-Khomskii) superexchange in transition metal oxides has been recently studied both using analytical [1] and numerical [2, 3] approach. In spite of remarkable progress in numerical methods for two–dimensional (2D) Ising– like models [4, 5], exact solutions are necessary to investigate the structure of excited states. The one–dimensional (1D) compass model is exactly solvable by mapping to quantum Ising model [6] and exhibits interesting properties while approaching to the quantum critical point at zero temperature [7]. In addition, its ladder version, first considered by Douçot et al. [1] is solvable in a similar way and its partition function [8] can be obtained exactly in case of a large (but finite) system. There is no exact solution for the 2D compass model but the latest Monte Carlo data [3] prove that the model exhibits a phase transition at finite temperature both in quantum and classical version with symmetry breaking between x and z part of the Hamiltonian. This

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Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

We obtained exact specific heat of the quantum compass model on the square L × L clusters with L = 2, 3, 4, 5 using Kernel Polynomial Method and compare them with the specific heat of a large compass ladder. Intersite correlations found in the ground state for these systems demonstrate that the quantum compass model differs from its classical version.

arXiv:0906.4469•Journal of Physics Conference Series•Jan 1, 2010•8 citations

Exact solution for a quantum compass ladder

Wojciech Brzezicki, Andrzej M. Oleś

We present a spin ladder with antiferromagnetic Ising $ZZ$ interactions along the legs and interactions on the rungs which interpolate between the Ising ladder and the quantum compass ladder. We show that the entire energy spectrum of the ladder may be determined exactly for finite number of spins $2N$ by mapping to the quantum Ising chain and using Jordan-Wigner transformation in invariant subspaces. We also demonstrate that subspaces with spin defects lead to excited states using finite-size scaling, and the ground state corresponds to the quantum Ising model without defects. At the quantum phase transition to maximally frustrated interactions of the compass ladder, the $ZZ$ spin-correlation function on the rungs collapses to zero and the ground-state degeneracy increases by two. We formulate a systematic method to calculate the partition function for a mesoscopic system and employ it to demonstrate that fragmentation of the compass ladder by kink defects increases with increasing temperature. The obtained heat capacity of a large compass ladder consisting of $2N=104$ spins reveals two relevant energy scales and has a broad maximum due to dense energy spectrum. The present exact results elucidate the nature of the quantum phase transition from ordered to disordered ground state found in the compass model in two dimensions.

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Quantum compass model on a chain, ladder and finite square clusters

Wojciech Brzezicki, Adolfo Avella, Ferdinando Mancini

I show the exact solutions for the quantum compass model on a chain and ladder which are based on mapping to quantum Ising models in certain subspaces. I discuss the ground state and thermodynamic properties of both models. I show the spin transformation that maps L×L quantum compass model to (L−1)×(L−1) spin model and its applications to compass clusters of the sizes 4×4 and 5×5. I discuss the ground‐state and thermodynamic properties of these models.

AIP conference proceedings•Jan 1, 2010•2 citations

Symmetry properties and spectra of the two-dimensional quantum compass model

Wojciech Brzezicki, Andrzej M. Oleś

We use exact symmetry properties of the two-dimensional quantum compass model to derive nonequivalent invariant subspaces in the energy spectra of $L\ifmmode\times\else\texttimes\fi{}L$ clusters up to $L=6$. The symmetry allows one to reduce the original $L\ifmmode\times\else\texttimes\fi{}L$ compass cluster to the $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ one with modified interactions. This step is crucial and enables (i) exact diagonalization of the $6\ifmmode\times\else\texttimes\fi{}6$ quantum compass cluster and (ii) finding the specific heat for clusters up to $L=6$, with two characteristic energy scales. We investigate the properties of the ground state and the first excited states and present extrapolation of the excitation energy with increasing system size. Our analysis provides physical insights into the nature of nematic order realized in the quantum compass model at finite temperature. We suggest that the quantum phase transition at the isotropic interaction point is second order with some admixture of the discontinuous transition, as indicated by the entropy, the overlap between two types of nematic order (on horizontal and vertical bonds), and the existence of the critical exponent. Extrapolation of the specific heat to the $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit suggests the classical nature of the quantum compass model and high degeneracy of the ground state with nematic order.

arXiv:1306.0791•Physical Review B•Jun 21, 2013•28 citations

Quantum phase transitions in exactly solvable one-dimensional compass models

Wen‐Long You, Peter Horsch, Andrzej M. Oleś

We present an exact solution for a class of one-dimensional compass models which stand for interacting orbital degrees of freedom in a Mott insulator. By employing the Jordan-Wigner transformation we map these models on noninteracting fermions and discuss how spin correlations, high degeneracy of the ground state, and ${Z}_{2}$ symmetry in the quantum compass model are visible in the fermionic language. Considering a zigzag chain of ions with singly occupied ${e}_{g}$ orbitals (${e}_{g}$ orbital model) we demonstrate that the orbital excitations change qualitatively with increasing transverse field, and that the excitation gap closes at the quantum phase transition to a polarized state. This phase transition disappears in the quantum compass model with maximally frustrated orbital interactions which resembles the Kitaev model. Here we find that the finite transverse field destabilizes the orbital-liquid ground state with macroscopic degeneracy, and leads to peculiar behavior of the specific heat and orbital susceptibility at finite temperature. We show that the entropy and the cooling rate at finite temperature exhibit quite different behavior near the critical point for these two models.

arXiv:1403.5455•Physical Review B•Mar 31, 2014•29 citations

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model

S. C. Wenzel, Wolfhard Janke E

A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion, and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a preinvestigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed-loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as ${T}_{c}=0.1464(2)J$ and ${T}_{c}=0.055(1)J$ for the classical and quantum versions, respectively, and our data support a transition in the 2D Ising universality class for both cases.

arXiv:0804.2972•Physical Review B•Aug 5, 2008•70 citations
Abstract. We obtained exact heat capacities of the quantum compass model on the square L×L clusters with L = 2, 3,4, 5 using Kernel Polynomial Method and compare them with heat capacity of a large compass ladder. Intersite correlations found in the ground state for these systems demonstrate that the quantum compass model differs from its classical version. Quantum compass model originating from the orbital (Kugel-Khomskii) superexchange in transition metal oxides has been recently studied both using analytical [1] and numerical [2, 3] approach. In spite of remarkable progress in numerical methods for two–dimensional (2D) Ising– like models [4, 5], exact solutions are necessary to investigate the structure of excited states. The one–dimensional (1D) compass model is exactly solvable by mapping to quantum Ising model [6] and exhibits interesting properties while approaching to the quantum critical point at zero temperature [7]. In addition, its ladder version, first considered by Douçot et al. [1] is solvable in a similar way and its partition function [8] can be obtained exactly in case of a large (but finite) system. There is no exact solution for the 2D compass model but the latest Monte Carlo data [3] prove that the model exhibits a phase transition at finite temperature both in quantum and classical version with symmetry breaking between x and z part of the Hamiltonian. This
We present a spin ladder with antiferromagnetic Ising ZZZZZZ interactions along the legs and interactions on the rungs which interpolate between the Ising ladder and the quantum compass ladder. We show that the entire energy spectrum of the ladder may be determined exactly for finite number of spins 2N2N2N by mapping to the quantum Ising chain and using Jordan-Wigner transformation in invariant subspaces. We also demonstrate that subspaces with spin defects lead to excited states using finite-size scaling, and the ground state corresponds to the quantum Ising model without defects. At the quantum phase transition to maximally frustrated interactions of the compass ladder, the ZZZZZZ spin-correlation function on the rungs collapses to zero and the ground-state degeneracy increases by two. We formulate a systematic method to calculate the partition function for a mesoscopic system and employ it to demonstrate that fragmentation of the compass ladder by kink defects increases with increasing temperature. The obtained heat capacity of a large compass ladder consisting of 2N=1042N=1042N=104 spins reveals two relevant energy scales and has a broad maximum due to dense energy spectrum. The present exact results elucidate the nature of the quantum phase transition from ordered to disordered ground state found in the compass model in two dimensions.
We use exact symmetry properties of the two-dimensional quantum compass model to derive nonequivalent invariant subspaces in the energy spectra of $L\ifmmode\times\else\texttimes\fi{}L$ clusters up to L=6L=6L=6. The symmetry allows one to reduce the original $L\ifmmode\times\else\texttimes\fi{}L$ compass cluster to the $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ one with modified interactions. This step is crucial and enables (i) exact diagonalization of the $6\ifmmode\times\else\texttimes\fi{}6$ quantum compass cluster and (ii) finding the specific heat for clusters up to L=6L=6L=6, with two characteristic energy scales. We investigate the properties of the ground state and the first excited states and present extrapolation of the excitation energy with increasing system size. Our analysis provides physical insights into the nature of nematic order realized in the quantum compass model at finite temperature. We suggest that the quantum phase transition at the isotropic interaction point is second order with some admixture of the discontinuous transition, as indicated by the entropy, the overlap between two types of nematic order (on horizontal and vertical bonds), and the existence of the critical exponent. Extrapolation of the specific heat to the $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit suggests the classical nature of the quantum compass model and high degeneracy of the ground state with nematic order.
We present an exact solution for a class of one-dimensional compass models which stand for interacting orbital degrees of freedom in a Mott insulator. By employing the Jordan-Wigner transformation we map these models on noninteracting fermions and discuss how spin correlations, high degeneracy of the ground state, and Z2{Z}_{2}Z2​ symmetry in the quantum compass model are visible in the fermionic language. Considering a zigzag chain of ions with singly occupied eg{e}_{g}eg​ orbitals (eg{e}_{g}eg​ orbital model) we demonstrate that the orbital excitations change qualitatively with increasing transverse field, and that the excitation gap closes at the quantum phase transition to a polarized state. This phase transition disappears in the quantum compass model with maximally frustrated orbital interactions which resembles the Kitaev model. Here we find that the finite transverse field destabilizes the orbital-liquid ground state with macroscopic degeneracy, and leads to peculiar behavior of the specific heat and orbital susceptibility at finite temperature. We show that the entropy and the cooling rate at finite temperature exhibit quite different behavior near the critical point for these two models.
A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion, and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a preinvestigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed-loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as Tc=0.1464(2)J{T}_{c}=0.1464(2)JTc​=0.1464(2)J and Tc=0.055(1)J{T}_{c}=0.055(1)JTc​=0.055(1)J for the classical and quantum versions, respectively, and our data support a transition in the 2D Ising universality class for both cases.