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arXiv:2212.07837

Bilinear Majorana representations for spin operators with spin magnitudes $S>1/2$

Yannik Schaden, Johannes Reuther

Published Dec 15, 2022• 0 citations
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Abstract

We present a classification of bilinear Majorana representations for spin-S operators, based on the real irreducible matrix representations of SU(2). We identify two types of such representations: While the first type can be straightforwardly mapped onto standard complex fermionic representations of spin-S operators, the second type realizes spin amplitudes S=s(s+1)/4 with s∈N and can be considered particularly efficient in representing spins via fermions. We show that for s=1 and s=2 this second type reproduces known spin-1/2 and spin-3/2 Majorana representations and we prove that these are the only bilinear Majorana representations that do not introduce any unphysical spin sectors. While for s>2, additional unphysical spin spaces are unavoidable they are less numerous than for more standard complex fermionic representations and carry comparatively small spin amplitudes. We apply our Majorana representations to exactly solvable small spin clusters and confirm that their low-energy properties remain unaffected by unphysical spin sectors, making our representations useful for auxiliary-particle based methods.

DOI
10.17169/refubium-39443

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Bilinear Majorana representations for spin operators with spin magnitudes S>1/2S>1/2S>1/2
We present a classification of bilinear Majorana representations for spin-S operators, based on the real irreducible matrix representations of SU(2). We identify two types of such representations: While the first type can be straightforwardly mapped onto standard complex fermionic representations of spin-S operators, the second type realizes spin amplitudes S=s(s+1)/4 with s∈N and can be considered particularly efficient in representing spins via fermions. We show that for s=1 and s=2 this second type reproduces known spin-1/2 and spin-3/2 Majorana representations and we prove that these are the only bilinear Majorana representations that do not introduce any unphysical spin sectors. While for s>2, additional unphysical spin spaces are unavoidable they are less numerous than for more standard complex fermionic representations and carry comparatively small spin amplitudes. We apply our Majorana representations to exactly solvable small spin clusters and confirm that their low-energy properties remain unaffected by unphysical spin sectors, making our representations useful for auxiliary-particle based methods.
We present a classification of bilinear Majorana representations for spin-SSS operators, based on the real irreducible matrix representations of SU(2). We identify two types of such representations: While the first type can be straightforwardly mapped onto standard complex fermionic representations of spin-SSS operators, the second type realizes spin amplitudes S=s(s+1)/4S=s(s+1)/4S=s(s+1)/4 with $s\ensuremath{\in}\mathbb{N}$ and can be considered particularly efficient in representing spins via fermions. We show that for s=1s=1s=1 and s=2s=2s=2 this second type reproduces known spin-1/21/21/2 and spin-3/23/23/2 Majorana representations and we prove that these are the only bilinear Majorana representations that do not introduce any unphysical spin sectors. While for s>2s>2s>2, additional unphysical spin spaces are unavoidable they are less numerous than for more standard complex fermionic representations and carry comparatively small spin amplitudes. We apply our Majorana representations to exactly solvable small spin clusters and confirm that their low-energy properties remain unaffected by unphysical spin sectors, making our representations useful for auxiliary-particle based methods.
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.
Based on the Dirac spinor representation of the SO(4) group, we discuss the relationship between three types of representation of spin in terms of Majorana fermions, namely the Kitaev representation, the SO(3) representation, and the SO(4) chiral representation. Comparing the three types, we show that the Hilbert space of the SO(3) representation is different from the other two by requiring a pairing of sites, but it has the advantage over the other two in that no unphysical states are involved. As an example of its application, we present an alternative solution of the Kitaev honeycomb model. Our solution involves no unphysical states which enables a systematic calculation of physical observables. Finally, we discuss an extension of the model to a more general exactly soluble Z2{Z}_{2}Z2​ gauge theory interacting with complex fermions.
We revisit the fermionic parton approach to S=1/2S=1/2S=1/2 quantum spin liquids with SU(2)\mathrm{SU}(2)SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with S=1S=1S=1 and S=0S=0S=0 Majorana fermion excitations coupled to a deconfined Z2{Z}_{2}Z2​ gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2)\mathrm{SU}(2)SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2{Z}_{2}Z2​ vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1S=1S=1 Majorana spin liquids may occur in models and in real systems.
The Majorana representation of spin 1/21/21/2 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)O(N)O(N) quantum Heisenberg antiferromagnet for general N.N.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,N=3,N=3, however, the representation fails and gives an incorrect gap in the O(3)O(3)O(3) Heisenberg chain. We trace this failure at N=3N=3N=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 NNN but irrelevant at $N=\ensuremath{\infty}.$ Specifically at N=3,N=3,N=3, there is an unphysical ``longitudinal'' mode. A correct treatment of these difficulties must precede applications of the Majorana representation of spin-1/21/21/2 operators to systems of physical interest.
Abstract Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker–Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker–Planck and stochastic equation form, including dissipation through particle losses.