arXiv:0906.4469 · Journal of Physics Conference Series

Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

Published Jan 1, 2010• 8 citations

Abstract

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.

DOI: 10.1088/1742-6596/200/1/012017

Related papers

Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

Nov 27, 20128 citations

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 proceedingsJan 1, 20102 citations

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

Wojciech Brzezicki, Andrzej M. Oleś

arXiv:1306.0791Physical Review BJun 21, 201328 citations

Exact solution for a quantum compass ladder

Wojciech Brzezicki, Andrzej M. Oleś

arXiv:0906.4462Physical Review BJul 2, 200930 citations

Topological phase transition in the extended cluster compass ladder

Rouhollah Jafari, S. Mahdavifar

arXiv:1310.5501Progress of Theoretical and Experimental PhysicsApr 14, 20148 citations

Ground-State Behavior of the Quantum Compass Model in an External Field

Kewei Sun, Qing‐Hu Chen

Ground-state (GS) properties of the two-dimensional (2D) quantum compass model in an external field on a square 5 × 5 lattice are investigated by using the exact diagonalization (ED) method. We obtain the GS energy and evaluate quantities such as its correlation functions, nearest-neighbor entanglement and local order parameter. As the external field is presented, the first-order quantum phase point is absent and the system exhibits the behaviors of the second-order phase transition.

Chinese Physics LettersSep 1, 20110 citations

arXiv:0906.4469 · Journal of Physics Conference Series

Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

Published Jan 1, 2010• 8 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1742-6596/200/1/012017

Related papers

Compass model on a ladder and square clusters

Wojciech Brzezicki, Andrzej M. Oleś

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. ThisAbstract. 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

Nov 27, 20128 citations

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

Wojciech Brzezicki, Adolfo Avella, Ferdinando Mancini

AIP conference proceedingsJan 1, 20102 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.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:1306.0791Physical Review BJun 21, 201328 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.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.

arXiv:0906.4462Physical Review BJul 2, 200930 citations

Topological phase transition in the extended cluster compass ladder

Rouhollah Jafari, S. Mahdavifar

We have studied the exact solution of the extended cluster compass ladder, which is equivalent to the extended quantum compass model with cluster interaction between next-nearest-neighbor spins, by using the Jordan–Wigner transformation. We show that this model is always gapful except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical couplings which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and topological ordered on the legs, in addition to the phase with antiparallel ordering of the spin | $y$ | component on the rungs. We also study the universality and scaling properties of the three point correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis. On the other hand, we have replaced the cluster interaction with the usual form and using the Lanczos method a numerical experiment is done. Analyzing the numerical results, we show that the effect of the cluster interaction between next-nearest-neighbor spins is completely different from the usual form.We have studied the exact solution of the extended cluster compass ladder, which is equivalent to the extended quantum compass model with cluster interaction between next-nearest-neighbor spins, by using the Jordan–Wigner transformation. We show that this model is always gapful except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical couplings which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and topological ordered on the legs, in addition to the phase with antiparallel ordering of the spin |$y$| component on the rungs. We also study the universality and scaling properties of the three point correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis. On the other hand, we have replaced the cluster interaction with the usual form and using the Lanczos method a numerical experiment is done. Analyzing the numerical results, we show that the effect of the cluster interaction between next-nearest-neighbor spins is completely different from the usual form.

arXiv:1310.5501Progress of Theoretical and Experimental PhysicsApr 14, 20148 citations

Ground-State Behavior of the Quantum Compass Model in an External Field

Kewei Sun, Qing‐Hu Chen

Chinese Physics LettersSep 1, 20110 citations

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.

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.

y

| component on the rungs. We also study the universality and scaling properties of the three point correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis. On the other hand, we have replaced the cluster interaction with the usual form and using the Lanczos method a numerical experiment is done. Analyzing the numerical results, we show that the effect of the cluster interaction between next-nearest-neighbor spins is completely different from the usual form.