Physical Review B

Quantum phase transition in the one-dimensional compass model

Wojciech Brzezicki, Jacek Dziarmaga, Andrzej M. Oleś

Published Apr 16, 2007• 99 citations

Abstract

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

DOI: 10.1103/physrevb.75.134415

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Physical Review B

Quantum phase transition in the one-dimensional compass model

Wojciech Brzezicki, Jacek Dziarmaga, Andrzej M. Oleś

Published Apr 16, 2007• 99 citations

Publisher

Abstract

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}

DOI: 10.1103/physrevb.75.134415

Related papers

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 $N\to\infty$ .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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Wojciech Brzezicki, Andrzej M. Oleś

We introduce a one-dimensional (1D) pseudospin model on a ladder where the Ising interactions along the legs and along the rungs alternate between ${X}_{i}{X}_{i+1}$ and ${Z}_{i}{Z}_{i+1}$ for even/odd bond (rung). We include also the next-nearest-neighbor Ising interactions on plaquettes' diagonals that alternate in such a way that a model where only leg interactions are switched on is equivalent to the one when only the diagonal ones are present. Thus in the absence of rung interactions the model can interpolate between two 1D compass models. The model possesses local symmetries which are the parities within each $2\ifmmode\times\else\texttimes\fi{}2$ cell (plaquette) of the ladder. We find that for different values of the interaction it can realize ground states that differ by the patterns formed by these local parities. By exact diagonalization we derive detailed phase diagrams for small systems of $L=4$ , 6, and 8 plaquettes, and use next $L=12$ to identify generic phases that appear in larger systems as well. Among them we find a nematic phase with macroscopic degeneracy when the leg and diagonal interactions are equal and the rung interactions are larger than a critical value. By performing a perturbative expansion around this phase we find indeed a very complex competition around the nematic phase which has to do with releasing frustration in this range of parameters. The nematic phase is similar to the one found in the two-dimensional compass model. For particular parameters the low-energy sector of the present plaquette model reduces to a 1D compass model with spins $S=1$ which suggests that it realizes peculiar crossovers within the class of compass models. Finally, we show that the model can realize phases with broken translation invariance which can be either dimerized, trimerized, etc., or completely disordered and highly entangled in a well identified window of the phase diagram.We introduce a one-dimensional (1D) pseudospin model on a ladder where the Ising interactions along the legs and along the rungs alternate between ${X}_{i}{X}_{i+1}$ and ${Z}_{i}{Z}_{i+1}$ for even/odd bond (rung). We include also the next-nearest-neighbor Ising interactions on plaquettes' diagonals that alternate in such a way that a model where only leg interactions are switched on is equivalent to the one when only the diagonal ones are present. Thus in the absence of rung interactions the model can interpolate between two 1D compass models. The model possesses local symmetries which are the parities within each $2\ifmmode\times\else\texttimes\fi{}2$ cell (plaquette) of the ladder. We find that for different values of the interaction it can realize ground states that differ by the patterns formed by these local parities. By exact diagonalization we derive detailed phase diagrams for small systems of $L=4$, 6, and 8 plaquettes, and use next $L=12$ to identify generic phases that appear in larger systems as well. Among them we find a nematic phase with macroscopic degeneracy when the leg and diagonal interactions are equal and the rung interactions are larger than a critical value. By performing a perturbative expansion around this phase we find indeed a very complex competition around the nematic phase which has to do with releasing frustration in this range of parameters. The nematic phase is similar to the one found in the two-dimensional compass model. For particular parameters the low-energy sector of the present plaquette model reduces to a 1D compass model with spins $S=1$ which suggests that it realizes peculiar crossovers within the class of compass models. Finally, we show that the model can realize phases with broken translation invariance which can be either dimerized, trimerized, etc., or completely disordered and highly entangled in a well identified window of the phase diagram.

arXiv:1605.08178Physical review. B./Physical review. BJun 1, 20162 citations

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}

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

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.

We introduce a one-dimensional (1D) pseudospin model on a ladder where the Ising interactions along the legs and along the rungs alternate between

{X}_{i}{X}_{i+1}

and

{Z}_{i}{Z}_{i+1}

for even/odd bond (rung). We include also the next-nearest-neighbor Ising interactions on plaquettes' diagonals that alternate in such a way that a model where only leg interactions are switched on is equivalent to the one when only the diagonal ones are present. Thus in the absence of rung interactions the model can interpolate between two 1D compass models. The model possesses local symmetries which are the parities within each $2\ifmmode\times\else\texttimes\fi{}2$ cell (plaquette) of the ladder. We find that for different values of the interaction it can realize ground states that differ by the patterns formed by these local parities. By exact diagonalization we derive detailed phase diagrams for small systems of

L=4

, 6, and 8 plaquettes, and use next

L=12

to identify generic phases that appear in larger systems as well. Among them we find a nematic phase with macroscopic degeneracy when the leg and diagonal interactions are equal and the rung interactions are larger than a critical value. By performing a perturbative expansion around this phase we find indeed a very complex competition around the nematic phase which has to do with releasing frustration in this range of parameters. The nematic phase is similar to the one found in the two-dimensional compass model. For particular parameters the low-energy sector of the present plaquette model reduces to a 1D compass model with spins

S=1

which suggests that it realizes peculiar crossovers within the class of compass models. Finally, we show that the model can realize phases with broken translation invariance which can be either dimerized, trimerized, etc., or completely disordered and highly entangled in a well identified window of the phase diagram.