Physical review. B, Condensed matter

Majorana representation of the quantum O(N) Heisenberg antiferromagnet: Its success at N=∞ and the reason for its failure at N=3

D. Foerster, François Triozon

Published Oct 1, 1997• 4 citations

Abstract

The Majorana representation of spin $1/2$ could be useful in the theory of strongly correlated electrons because it respects isotropy in spin space and has no constraints. To judge its reliability, we apply it to the $O(N)$ quantum Heisenberg antiferromagnet for general $N.$ At $N\ensuremath{\rightarrow}\ensuremath{\infty}$ we find a resonant valence bond-like spin-liquid ground state in precise agreement with previous results from an SU(\ensuremath{\infty}) extrapolation. At $N=3,$ however, the representation fails and gives an incorrect gap in the $O(3)$ Heisenberg chain. We trace this failure at $N=3$ to (i) a redundancy of states and (ii) the presence of a local gauge invariance both of which are intrinsic properties of the Majorana representation of spins at any $N$ but irrelevant at $N=\ensuremath{\infty}.$ Specifically at $N=3,$ there is an unphysical ``longitudinal'' mode. A correct treatment of these difficulties must precede applications of the Majorana representation of spin-$1/2$ operators to systems of physical interest.

DOI: 10.1103/physrevb.56.8069

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Bond operator theory of isotropic spin-1 antiferromagnetic chains

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Physical review. B, Condensed matterDec 1, 199729 citations

Can Majorana Fermions correctly describe the ordered state of an antiferromagnetic Heisenberg chain?

D. Foerster

To check the reliability of the Majorana representation of quantum spin systems, we use it to compute the ground state energy of an antiferromagnetic spin 1/2 chain to one loop order. We find a very small one loop correction of the mean field energy and a discrepancy of >100% compared to the Bethe Ansatz result. We conclude that a careful handling of the gauge degrees of freedom of this representation is crucial to get correct results.

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Physics of Resonating Valence Bond Spin Liquids

Julia Wildeboer

Jan 1, 20130 citations

Physical review. B, Condensed matter

Majorana representation of the quantum O(N) Heisenberg antiferromagnet: Its success at N=∞ and the reason for its failure at N=3

D. Foerster, François Triozon

Published Oct 1, 1997• 4 citations

Publisher

Abstract

The Majorana representation of spin

1/2

could be useful in the theory of strongly correlated electrons because it respects isotropy in spin space and has no constraints. To judge its reliability, we apply it to the

O(N)

quantum Heisenberg antiferromagnet for general

N.

N\ensuremath{\rightarrow}\ensuremath{\infty}

we find a resonant valence bond-like spin-liquid ground state in precise agreement with previous results from an SU(\ensuremath{\infty}) extrapolation. At

N=3,

however, the representation fails and gives an incorrect gap in the

O(3)

Heisenberg chain. We trace this failure at

N=3

to (i) a redundancy of states and (ii) the presence of a local gauge invariance both of which are intrinsic properties of the Majorana representation of spins at any

N

but irrelevant at

N=\ensuremath{\infty}.

Specifically at

N=3,

there is an unphysical ``longitudinal'' mode. A correct treatment of these difficulties must precede applications of the Majorana representation of spin-

1/2

operators to systems of physical interest.

DOI: 10.1103/physrevb.56.8069

Related papers

Bond operator theory of isotropic spin-1 antiferromagnetic chains

Hanting Wang, Jue-Lian Shen, Zhao-Bin Su

The bond operator representation of quantum $S=\frac{1}{2}$ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.The bond operator representation of quantum $S=\frac{1}{2}$ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.

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Alexei M. Tsvelik

A novel approach to S=1/2 antiferromagnets with strong quantum fluctuations based on the representation of spin-1/2 operators as bilinear forms of real (Majorana) fermions is suggested. This representation does not contain unphysical states and thus does not require the imposition of constraints on the fermionic Hilbert space. This property allows one to construct a simple and effective mean field theory of the spin liquid state. As an example illustrating the basic properties of this state I consider a model of Kondo lattice. It is shown that this model has a singlet ground state; elementary excitations have a spectral gap and are S=1 real (Majorana) fermions.A novel approach to S=1/2 antiferromagnets with strong quantum fluctuations based on the representation of spin-1/2 operators as bilinear forms of real (Majorana) fermions is suggested. This representation does not contain unphysical states and thus does not require the imposition of constraints on the fermionic Hilbert space. This property allows one to construct a simple and effective mean field theory of the spin liquid state. As an example illustrating the basic properties of this state I consider a model of Kondo lattice. It is shown that this model has a singlet ground state; elementary excitations have a spectral gap and are S=1 real (Majorana) fermions.

arXiv:cond-mat/9206007Physical Review LettersOct 5, 199288 citations

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B. Sriram Shastry, Diptiman Sen

We study the one-dimensional Heisenberg antiferromagnet with s= using a Majorana representation of the s= spins. A simple Hartree-Fock approximation of the resulting model gives a bilinear fermionic description of the model. This description is rotationally invariant and gives power-law correlations in the ``ground state'' in a natural fashion. The excitations are a two-parameter family of particles, which are spin-1 objects. These are contrasted to the ``spinon'' spectrum, and the technical aspects of the representation are discussed, including the problem of redundant states.We study the one-dimensional Heisenberg antiferromagnet with s= using a Majorana representation of the s= spins. A simple Hartree-Fock approximation of the resulting model gives a bilinear fermionic description of the model. This description is rotationally invariant and gives power-law correlations in the ``ground state'' in a natural fashion. The excitations are a two-parameter family of particles, which are spin-1 objects. These are contrasted to the ``spinon'' spectrum, and the technical aspects of the representation are discussed, including the problem of redundant states.

arXiv:cond-mat/9603179Physical review. B, Condensed matterFeb 1, 199735 citations

Fermionic theory for quantum antiferromagnets with spin S>12

Zheng-Xin Liu, Yi Zhou, Tai Kai Ng

The fermion representation for $S=\frac{1}{2}$ spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of $S=\frac{1}{2}$ fermion representation is absent for $S>\frac{1}{2}$ . As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D) $S=1$ and $S=3/2$ antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a $S=1$ open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude $S$ is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated $S=1$ AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped ${p}_{x}+i{p}_{y}$ -wave spin liquid and the gapless $f$ -wave spin liquid. We compare our mean-field result with the experiment on ${\text{NiGa}}_{2}{\text{S}}_{4}$ , which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study $S>\frac{1}{2}$ spin liquids with fermionic spinon excitations.The fermion representation for $S=\frac{1}{2}$ spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of $S=\frac{1}{2}$ fermion representation is absent for $S>\frac{1}{2}$. As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D) $S=1$ and $S=3/2$ antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a $S=1$ open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude $S$ is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated $S=1$ AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped ${p}_{x}+i{p}_{y}$-wave spin liquid and the gapless $f$-wave spin liquid. We compare our mean-field result with the experiment on ${\text{NiGa}}_{2}{\text{S}}_{4}$, which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study $S>\frac{1}{2}$ spin liquids with fermionic spinon excitations.

arXiv:1005.5626Physical Review BOct 13, 201037 citations

Physics of Resonating Valence Bond Spin Liquids

Julia Wildeboer

This thesis will investigate various aspects of the physics of resonating valence bond spin liquids. After giving an introduction to the world that lies beyond Landau's priciple of symmetry breaking, e.g. giving an overview of exotic magnetic phases and how they can be described and: possibly) found, we will study a spin-rotationally invariant model system with a known parent Hamiltonian, and argue its ground state to lie within a highly sought after exotic phase, namely the Z $_{2}$ quantum spin liquid phase. A newly developed numerical procedure --Pfaffian Monte Carlo-- will be introduced to amass evidence that our model Hamiltonian indeed exhibits a Z $_{2}$ quantum spin liquid phase. Subsequently, we will prove a useful mathematical property of the resonating valence bond states: these states are shown to be linearly independent. Various lattices are investigated concerning this property, and its applications and usefullness are discussed. Eventually, we present a simplified model system describing the interplay of the well known Heisenberg interaction and the Dzyaloshinskii-Moriya: DM) interaction term acting on a sawtooth chain. The effect of the interplay between the two interaction couplings on the phase diagram is investigated. To do so, we employ modern techniques such as the density matrix renormalization group: DMRG) scheme. We find that for weak DM interaction the system exhibits valence bond order. However, a strong enough DM coupling destroys this order.This thesis will investigate various aspects of the physics of resonating valence bond spin liquids. After giving an introduction to the world that lies beyond Landau's priciple of symmetry breaking, e.g. giving an overview of exotic magnetic phases and how they can be described and: possibly) found, we will study a spin-rotationally invariant model system with a known parent Hamiltonian, and argue its ground state to lie within a highly sought after exotic phase, namely the Z$_{2}$ quantum spin liquid phase. A newly developed numerical procedure --Pfaffian Monte Carlo-- will be introduced to amass evidence that our model Hamiltonian indeed exhibits a Z$_{2}$ quantum spin liquid phase. Subsequently, we will prove a useful mathematical property of the resonating valence bond states: these states are shown to be linearly independent. Various lattices are investigated concerning this property, and its applications and usefullness are discussed. Eventually, we present a simplified model system describing the interplay of the well known Heisenberg interaction and the Dzyaloshinskii-Moriya: DM) interaction term acting on a sawtooth chain. The effect of the interplay between the two interaction couplings on the phase diagram is investigated. To do so, we employ modern techniques such as the density matrix renormalization group: DMRG) scheme. We find that for weak DM interaction the system exhibits valence bond order. However, a strong enough DM coupling destroys this order.

Jan 1, 20130 citations

The Majorana representation of spin

1/2

could be useful in the theory of strongly correlated electrons because it respects isotropy in spin space and has no constraints. To judge its reliability, we apply it to the

O(N)

quantum Heisenberg antiferromagnet for general

N.

N\ensuremath{\rightarrow}\ensuremath{\infty}

we find a resonant valence bond-like spin-liquid ground state in precise agreement with previous results from an SU(\ensuremath{\infty}) extrapolation. At

N=3,

however, the representation fails and gives an incorrect gap in the

O(3)

Heisenberg chain. We trace this failure at

N=3

to (i) a redundancy of states and (ii) the presence of a local gauge invariance both of which are intrinsic properties of the Majorana representation of spins at any

N

but irrelevant at

N=\ensuremath{\infty}.

Specifically at

N=3,

there is an unphysical ``longitudinal'' mode. A correct treatment of these difficulties must precede applications of the Majorana representation of spin-

1/2

operators to systems of physical interest.

The bond operator representation of quantum

S=\frac{1}{2}

spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.

The fermion representation for

S=\frac{1}{2}

spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of

S=\frac{1}{2}

fermion representation is absent for

S>\frac{1}{2}

. As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D)

S=1

and

S=3/2

antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a

S=1

open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude

S

is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated

S=1

AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped

{p}_{x}+i{p}_{y}

-wave spin liquid and the gapless

f

-wave spin liquid. We compare our mean-field result with the experiment on

{\text{NiGa}}_{2}{\text{S}}_{4}

, which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study

S>\frac{1}{2}

spin liquids with fermionic spinon excitations.

_{2}

quantum spin liquid phase. A newly developed numerical procedure --Pfaffian Monte Carlo-- will be introduced to amass evidence that our model Hamiltonian indeed exhibits a Z

_{2}

quantum spin liquid phase. Subsequently, we will prove a useful mathematical property of the resonating valence bond states: these states are shown to be linearly independent. Various lattices are investigated concerning this property, and its applications and usefullness are discussed. Eventually, we present a simplified model system describing the interplay of the well known Heisenberg interaction and the Dzyaloshinskii-Moriya: DM) interaction term acting on a sawtooth chain. The effect of the interplay between the two interaction couplings on the phase diagram is investigated. To do so, we employ modern techniques such as the density matrix renormalization group: DMRG) scheme. We find that for weak DM interaction the system exhibits valence bond order. However, a strong enough DM coupling destroys this order.