arXiv:2101.00268 · Physical review. B./Physical review. B

Stripping the planar quantum compass model to its basics

Soumya Sur, M. S. Laad, Syed R. Hassan

Published Apr 12, 2021• 1 citation

Abstract

We introduce a novel mean-field theory (MFT) around the exactly soluble two-leg ladder limit for the planar quantum compass model (QCM). In contrast to usual MFT, our construction respects the stringent constraints imposed by emergent, lower (here $d=1$) dimensional symmetries of the QCM. Specializing our construction to the QCM on a periodic four-leg ladder, we find that a first-order transition separates two mutually dual Ising nematic phases, in good accord with state-of-the-art numerics for the planar QCM. One pseudo-spin-flip excitation in the ordered phase turns out to be two (Jordan-Wigner) fermion bound states, reminiscent of spin waves in spin-$1/2$ Heisenberg chains. We discuss the novel implications of our work on (1) the emergence of coupled orbital and magnetic ordered and liquidlike disordered phases and (2) a rare instance of orbital-spin separation in $d>1$, in the context of a Kugel-Khomskii view of multiorbital Mott insulators.

DOI: 10.1103/physrevb.103.144419

Related papers

A Tale of Two Quantum Compass Models

Soumya Sur, M. S. Laad, Arya Subramonian +1 more

arXiv:2206.15199Jun 30, 20220 citations

Exact ground states and phase diagram of the quantum compass model under an in-plane field

A. D. S. Richards, Erik Schwartz Sørensen

arXiv:2310.01384Physical review. B./Physical review. BJun 26, 20243 citations

Quantum phase transitions in exactly solvable one-dimensional compass models

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

arXiv:1403.5455Physical Review BMar 31, 201429 citations

Mechanism of hole propagation in the orbital compass models

Wojciech Brzezicki, Maria Daghofer, Andrzej M. Oleś

arXiv:1311.4796Physical Review BJan 27, 201410 citations

Exact ground state of a spin ladder with a quantum phase transition

Wojciech Brzezicki, Andrzej M. Oleś

The European Physical Journal BDec 1, 200810 citations

Exact solution for a quantum compass ladder

Wojciech Brzezicki, Andrzej M. Oleś

arXiv:0906.4462Physical Review BJul 2, 200930 citations

arXiv:2101.00268 · Physical review. B./Physical review. B

Stripping the planar quantum compass model to its basics

Soumya Sur, M. S. Laad, Syed R. Hassan

Published Apr 12, 2021• 1 citation

arXiv HTML PDF

Abstract

d=1

) dimensional symmetries of the QCM. Specializing our construction to the QCM on a periodic four-leg ladder, we find that a first-order transition separates two mutually dual Ising nematic phases, in good accord with state-of-the-art numerics for the planar QCM. One pseudo-spin-flip excitation in the ordered phase turns out to be two (Jordan-Wigner) fermion bound states, reminiscent of spin waves in spin-

1/2

Heisenberg chains. We discuss the novel implications of our work on (1) the emergence of coupled orbital and magnetic ordered and liquidlike disordered phases and (2) a rare instance of orbital-spin separation in

d>1

, in the context of a Kugel-Khomskii view of multiorbital Mott insulators.

DOI: 10.1103/physrevb.103.144419

Related papers

A Tale of Two Quantum Compass Models

Soumya Sur, M. S. Laad, Arya Subramonian +1 more

We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a $XX$ - to $ZZ$ -ordered phase in the $3d$ -Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a $p$ -wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature ( $T$ )-dependent dimensional crossover from a high- $T$ quasi- $1d$ insulator-like state, via an incoherent bad-metal-like state at intermediate $T$ , to a $2d$ symmetry-broken insulator at low $T$ , well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a $XX$- to $ZZ$-ordered phase in the $3d$-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a $p$-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature ($T$)-dependent dimensional crossover from a high-$T$ quasi-$1d$ insulator-like state, via an incoherent bad-metal-like state at intermediate $T$, to a $2d$ symmetry-broken insulator at low $T$, well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.

arXiv:2206.15199Jun 30, 20220 citations

Exact ground states and phase diagram of the quantum compass model under an in-plane field

A. D. S. Richards, Erik Schwartz Sørensen

We consider the square lattice $S=1/2$ quantum compass model (QCM) parametrized by ${J}_{x},{J}_{z}$ , under a field, $\mathbf{h}$ , in the $x\text{\ensuremath{-}}z$ plane. At the special field value $({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})$ , we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at $({h}_{x}^{★},{h}_{z}^{★})$ and states obtained for fields within the vicinity of $({h}_{x}^{★},{h}_{z}^{★})$ . This gapped phase occupies most of the in-plane field phase diagram. For some values of ${h}_{x}>{h}_{z}$ and ${h}_{z}>{h}_{x}$ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.We consider the square lattice $S=1/2$ quantum compass model (QCM) parametrized by ${J}_{x},{J}_{z}$, under a field, $\mathbf{h}$, in the $x\text{\ensuremath{-}}z$ plane. At the special field value $({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})$, we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at $({h}_{x}^{★},{h}_{z}^{★})$ and states obtained for fields within the vicinity of $({h}_{x}^{★},{h}_{z}^{★})$. This gapped phase occupies most of the in-plane field phase diagram. For some values of ${h}_{x}>{h}_{z}$ and ${h}_{z}>{h}_{x}$ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.

arXiv:2310.01384Physical review. B./Physical review. BJun 26, 20243 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.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:1403.5455Physical Review BMar 31, 201429 citations

Mechanism of hole propagation in the orbital compass models

Wojciech Brzezicki, Maria Daghofer, Andrzej M. Oleś

We explore the propagation of a single hole in the quantum compass model, whose nematic ground state is given by mutually decoupled antiferromagnetic chains. The compass model can be seen as the strong-coupling limit of a spinless two-band Hubbard model, which we study here using mean-field theory and the variational cluster approach. Due to the symmetries of the compass model, the inherent disorder along one lattice direction turns out not to affect hole motion, and doping a hole consequently does not lift the subextensive degeneracy of the nematic phase. To broaden and deepen understanding, we derive a generalized itinerant model and address the transition to two-dimensional Ising order. We observe coherent hole motion in both the nematic and the antiferromagnetic phases, also in the presence of quantum fluctuations away from pure Ising exchange. In addition to quantum fluctuations and interorbital hopping, three-site hopping is found to play an important role and to dominate propagation in the two-dimensional Ising limit as well as along the antiferromagnetic chains in the nematic order which forms in the compass model.We explore the propagation of a single hole in the quantum compass model, whose nematic ground state is given by mutually decoupled antiferromagnetic chains. The compass model can be seen as the strong-coupling limit of a spinless two-band Hubbard model, which we study here using mean-field theory and the variational cluster approach. Due to the symmetries of the compass model, the inherent disorder along one lattice direction turns out not to affect hole motion, and doping a hole consequently does not lift the subextensive degeneracy of the nematic phase. To broaden and deepen understanding, we derive a generalized itinerant model and address the transition to two-dimensional Ising order. We observe coherent hole motion in both the nematic and the antiferromagnetic phases, also in the presence of quantum fluctuations away from pure Ising exchange. In addition to quantum fluctuations and interorbital hopping, three-site hopping is found to play an important role and to dominate propagation in the two-dimensional Ising limit as well as along the antiferromagnetic chains in the nematic order which forms in the compass model.

arXiv:1311.4796Physical Review BJan 27, 201410 citations

Exact ground state of a spin ladder with a quantum phase transition

Wojciech Brzezicki, Andrzej M. Oleś

We introduce a spin ladder with Ising interactions along the legs and intrinsically frustrated Heisenberg-like ferromagnetic interactions on the rungs. The model is solved exactly in the subspaces relevant for the ground state by mapping to the quantum Ising model, and we show that a first order quantum phase transition separates the classical from quantum regime, with the spin correlations on the rungs being either ferromagnetic or antiferromagnetic, and different spin excitations in both regimes. The present case resembles the quantum phase transition found in the compass model in one and two dimensions.We introduce a spin ladder with Ising interactions along the legs and intrinsically frustrated Heisenberg-like ferromagnetic interactions on the rungs. The model is solved exactly in the subspaces relevant for the ground state by mapping to the quantum Ising model, and we show that a first order quantum phase transition separates the classical from quantum regime, with the spin correlations on the rungs being either ferromagnetic or antiferromagnetic, and different spin excitations in both regimes. The present case resembles the quantum phase transition found in the compass model in one and two dimensions.

The European Physical Journal BDec 1, 200810 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

d=1

1/2

d>1

, in the context of a Kugel-Khomskii view of multiorbital Mott insulators.

We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a

XX

- to

ZZ

-ordered phase in the

3d

-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a

p

-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature (

T

)-dependent dimensional crossover from a high-

T

quasi-

1d

insulator-like state, via an incoherent bad-metal-like state at intermediate

T

, to a

2d

symmetry-broken insulator at low

T

, well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.

We consider the square lattice

S=1/2

quantum compass model (QCM) parametrized by

{J}_{x},{J}_{z}

, under a field,

\mathbf{h}

, in the $x\text{\ensuremath{-}}z$ plane. At the special field value

({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})

, we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at

({h}_{x}^{★},{h}_{z}^{★})

and states obtained for fields within the vicinity of

({h}_{x}^{★},{h}_{z}^{★})

. This gapped phase occupies most of the in-plane field phase diagram. For some values of

{h}_{x}>{h}_{z}

and

{h}_{z}>{h}_{x}

at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.

{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 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.