arXiv:0805.3904 · Acta Physica Polonica A

Quantum Phase Transition in the One-Dimensional XZ Model

Wojciech Brzezicki, Andrzej M. Oleś

Published Jan 1, 2009• 0 citations

Abstract

We introduce a one-dimensional XZ model with alternating σ z i σ z i+1 and σ x i σ x i+1 interactions on even/odd bonds, interpolating between the Ising model and the quantum compass model.We present two ways of its exact solution by: (i) mapping to the quantum Ising models, and (ii) using fermions with spin 1/2.In certain cases the nearest neighbor pseudospin correlations change discontinuously at the quantum phase transition, where one finds highly degenerate ground state of the 1D compass model.

DOI: 10.12693/aphyspola.115.162

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arXiv:0805.3904 · Acta Physica Polonica A

Quantum Phase Transition in the One-Dimensional XZ Model

Wojciech Brzezicki, Andrzej M. Oleś

Published Jan 1, 2009• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.12693/aphyspola.115.162

Related papers

Quantum Phase Transition in the One-Dimensional XZ Model

Wojciech Brzezicki, Andrzej M. Oleś

We introduce a one-dimensional (1D) XZ model with alternating σ z i σ z i+1 and σ x i σ x i+1 interactions on even/odd bonds, interpolating between the Ising model and the quantum compass model. We present two ways of its exact solution by: (i) mapping to the quantum Ising models, and (ii) using fermions with spin 1/2. In certain cases the nearest neighbor pseudospin correlations change discontinuously at the quantum phase transition, where one finds highly degenerate ground state of the 1D compass model. PACS numbers: 75.10.Jm, 03.67.-a, 05.70.Fh, 63.70.Tg Introduction — Models of magnetism with exotic interactions are motivated by rather complex orbital superexchange in Mott insulators. In certain case the degeneracy of 3d orbitals is only partly lifted and the remaining orbital degrees of freedom are frequently described as 1/2 spins. They also arise from spin-orbital superexchange, with rich dynamics leading to enhanced quantum fluctuationsWe introduce a one-dimensional (1D) XZ model with alternating σ z i σ z i+1 and σ x i σ x i+1 interactions on even/odd bonds, interpolating between the Ising model and the quantum compass model. We present two ways of its exact solution by: (i) mapping to the quantum Ising models, and (ii) using fermions with spin 1/2. In certain cases the nearest neighbor pseudospin correlations change discontinuously at the quantum phase transition, where one finds highly degenerate ground state of the 1D compass model. PACS numbers: 75.10.Jm, 03.67.-a, 05.70.Fh, 63.70.Tg Introduction — Models of magnetism with exotic interactions are motivated by rather complex orbital superexchange in Mott insulators. In certain case the degeneracy of 3d orbitals is only partly lifted and the remaining orbital degrees of freedom are frequently described as 1/2 spins. They also arise from spin-orbital superexchange, with rich dynamics leading to enhanced quantum fluctuations

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Quantum phase transition in the one-dimensional compass model

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We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions $σ_i^zσ_{i+1}^z$ and $σ_i^xσ_{i+1}^x$ , alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly and indicate divergent correlation length at the first order quantum phase transition. At this transition one finds the disordered ground state of the compass model with high degeneracy $2\times 2^{N/2}$ in the limit of $N\to\infty$ .We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions $σ_i^zσ_{i+1}^z$ and $σ_i^xσ_{i+1}^x$, alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly and indicate divergent correlation length at the first order quantum phase transition. At this transition one finds the disordered ground state of the compass model with high degeneracy $2\times 2^{N/2}$ in the limit of $N\to\infty$.

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We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions ${\ensuremath{\sigma}}_{i}^{z}{\ensuremath{\sigma}}_{i+1}^{z}$ and ${\ensuremath{\sigma}}_{i}^{x}{\ensuremath{\sigma}}_{i+1}^{x}$, alternating between even and/or odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest-neighbor pseudospin correlations change discontinuously and indicate divergent correlation length at the first-order quantum phase transition. At this transition, one finds the disordered ground state of the compass model with high degeneracy $2\ifmmode\times\else\texttimes\fi{}{2}^{N∕2}$ in the limit of $N\ensuremath{\rightarrow}\ensuremath{\infty}$.We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions ${\ensuremath{\sigma}}_{i}^{z}{\ensuremath{\sigma}}_{i+1}^{z}$ and ${\ensuremath{\sigma}}_{i}^{x}{\ensuremath{\sigma}}_{i+1}^{x}$, alternating between even and/or odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest-neighbor pseudospin correlations change discontinuously and indicate divergent correlation length at the first-order quantum phase transition. At this transition, one finds the disordered ground state of the compass model with high degeneracy $2\ifmmode\times\else\texttimes\fi{}{2}^{N∕2}$ in the limit of $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

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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.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.

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Quantum phase transitions in the one-dimensional extended quantum compass model in a transverse field are studied by using the Jordan-Wigner transformation. This model is always gapful, except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical fields which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and saturated ferromagnetic phases in addition to the phase with antiparallel ordering of the $y$ spin component on odd bonds. We also study also the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis.Quantum phase transitions in the one-dimensional extended quantum compass model in a transverse field are studied by using the Jordan-Wigner transformation. This model is always gapful, except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical fields which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and saturated ferromagnetic phases in addition to the phase with antiparallel ordering of the $y$ spin component on odd bonds. We also study also the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis.

arXiv:1101.3673Physical Review BJul 21, 201127 citations

We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions

σ_i^zσ_{i+1}^z

and

σ_i^xσ_{i+1}^x

, alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly and indicate divergent correlation length at the first order quantum phase transition. At this transition one finds the disordered ground state of the compass model with high degeneracy

2\times 2^{N/2}

in the limit of

N\to\infty

1/N

(

N

is the system size). The spin-correlation functions show smooth behavior around the critical point. In the period-two chain, the pseudospin-correlation functions exhibit an oscillating behavior, which is absent in the uniform chain. The divergent correlation length at the critical point is demonstrated in the general trend for both cases.

{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.

y

spin component on odd bonds. We also study also the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis.