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arXiv:cond-mat/0305064 · Physical Review Letters

Spin-Spin Correlators in the Majorana Representation

Alexander Shnirman, Yuriy Makhlin

Published Nov 10, 2003• 58 citations
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Abstract

In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit's state.

DOI
10.1103/physrevlett.91.207204

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In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit's state.
Using the Majorana fermion representation of spin-1/2 local moments, we show how the dynamic spin correlation and susceptibility are obtained directly from the one-particle Majorana propagator. We illustrate our method by applying it to the spin dynamics of a nonequilibrium quantum dot, computing the voltage-dependent spin relaxation rate and showing that, at weak coupling, the fluctuation-dissipation relation for the spin of a quantum dot is voltage dependent. We confirm the voltage-dependent Curie susceptibility recently found by Parcollet and Hooley [Phys. Rev. B 66, 085315 (2002)]].
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 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.
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.
In this paper we resolve a contradiction between the fact that the method based on the Majorana representation of spin-12\frac{1}{2}21​ is exact and its failure to reproduce the perturbative Bloch-Redfield relaxation rates. Namely, for the spin-boson model, direct application of this method in the leading order allows for a straightforward computation of the transverse spin correlations; however, for the longitudinal spin correlations it apparently fails in the long-time limit. Here we indicate the reason for this failure. Moreover, we suggest how to apply this method to allow, nevertheless, for simple and accurate computations of spin correlations. Specifically, we demonstrate that accurate results are obtained by avoiding the use of the longitudinal Majorana fermion and that correlations of the remaining transverse Majorana fermions can be easily evaluated using an effective Gaussian action.
We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of Spin(16,C)\text{Spin}(16,\mathbb{C})Spin(16,C) invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group E8{E}_{8}E8​ is playing in these considerations.