Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Phase Transition in the One-Dimensional XZ Model

Wojciech Brzezicki, Andrzej M. Oleś

Published Jan 1, 2008• 34 citations
OpenAlex

Abstract

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 fluctuations

Related papers

Quantum Phase Transition in the One-Dimensional XZ Model

Wojciech Brzezicki, Andrzej M. Oleś

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.

arXiv:0805.3904•Acta Physica Polonica A•Jan 1, 2009•0 citations

Quantum phase transition in the one-dimensional compass model

Wojciech Brzezicki, Jacek Dziarmaga, Andrzej M. Oleś

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 $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

Physical Review B•Apr 16, 2007•99 citations

Quantum phase transition in the one-dimensional compass model

Wojciech Brzezicki, Jacek Dziarmaga, Andrzej M. Oleś

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 $N\to\infty$.

arXiv:cond-mat/0702369•Feb 15, 2007•0 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

Quantum phase transition in the one-dimensional period-two and uniform compass model

Kewei Sun, Yuyu Zhang, Qing‐Hu Chen

Quantum phase transitions in the one-dimensional period-two and uniform quantum compass model are studied by using the pseudospin transformation method and the trace map method. The exact solutions are presented, the fidelity, the nearest-neighbor pseudospin entanglement, spin and pseudospin-correlation functions are then calculated. At the critical point, the fidelity and its susceptibility change substantially, the gap of pseudospin concurrence is observed, which scales as $1/N$ ($N$ 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.

arXiv:0809.3569•Physical Review B•Mar 25, 2009•30 citations

Quantum phase transition in the one-dimensional extended quantum compass model in a transverse field

Rouhollah Jafari

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:1101.3673•Physical Review B•Jul 21, 2011•27 citations
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 fluctuations
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.
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 σizσi+1zσ_i^zσ_{i+1}^zσiz​σi+1z​ and σixσi+1xσ_i^xσ_{i+1}^xσix​σi+1x​, 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 2×2N/22\times 2^{N/2}2×2N/2 in the limit of N→∞N\to\inftyN→∞.
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.
Quantum phase transitions in the one-dimensional period-two and uniform quantum compass model are studied by using the pseudospin transformation method and the trace map method. The exact solutions are presented, the fidelity, the nearest-neighbor pseudospin entanglement, spin and pseudospin-correlation functions are then calculated. At the critical point, the fidelity and its susceptibility change substantially, the gap of pseudospin concurrence is observed, which scales as 1/N1/N1/N (NNN 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.
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 yyy 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.