We present a general derivation of semi-fermionic representation for spin operators in terms of a bilinear combination of fermions in real and imaginary time formalisms. The constraint on fermionic occupation numbers is fulfilled by means of imaginary Lagrange multipliers resulting in special shape of quasiparticle distribution functions. We show how Schwinger–Keldysh technique for spin operators is constructed with the help of semi-fermions. We demonstrate how the idea of semi-fermionic representation might be extended to the groups possessing dynamic symmetries. We illustrate the application of semi-fermionic representations for various problems of strongly correlated and mesoscopic physics.
The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.
Abstract We develop a unified framework for open quantum systems composed of many mutually interacting quantum spins, or any isomorphic systems like qubits and qudits, surrounded by one or more independent bosonic baths. Our framework, based on Schwinger–Keldysh field theory (SKFT), can handle arbitrary spin value S , dimensionality of space, and geometry, while being applicable to a large parameter space for system and bath. It can probe regimes in which non-Markovian dynamics and nonperturbative effects pose formidable challenges for other state-of-the-art theoretical methods. This is achieved by working with the two-particle irreducible (2PI) effective action, which resums classes of Feynman diagrams of SKFT to an infinite order. Furthermore, such diagrams are generated via an expansion in 1 / N , where N is the number of Schwinger bosons we employ to map spin operators onto canonically commuting ones, rather than via conventional expansion in system-bath coupling constant. We carefully benchmark our SKFT+2PI-computed results vs. numerically (quasi)exact ones from tensor network calculations applied to the archetypical spin-boson model where both methodologies are applicable. Additionally, we demonstrate the capability of SKFT+2PI to handle a much more complex spin-chain-boson model with multiple baths interacting with each spin where no benchmark from other methods is available at present. The favorable numerical cost of solving integro-differential equations produced by the SKFT+2PI framework with an increasing number of spins and time steps makes it a promising route for simulating driven-dissipative systems in quantum computing, quantum magnonics, and quantum spintronics.
The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semiclassical treatments, this theory allows us to describe a continuous transition between magnetically ordered and spin liquid states, as well as the continuous evolution of the corresponding excitation spectrum. The square lattice Heisenberg antiferromagnet is one of the first models that was approached with the Schwinger boson theory. Here we revisit this problem to reveal several subtle points that were omitted in previous treatments and that are crucial to further develop this formalism. These points include the freedom for the choice of the saddle point (Hubbard-Stratonovich decoupling and choice of the condensate) and the
1/N expansion in the presence of a condensate. A key observation is that the spinon condensate leads to Feynman diagrams that include contributions of different order in
1/N, which must be accounted for to get a qualitatively correct excitation spectrum. We demonstrate that a proper treatment of these contributions leads to an exact cancellation of the single-spinon poles of the dynamical spin structure factor, as expected for a magnetically ordered state. The only surviving poles are the ones arising from the magnons (two-spinon bound states), which are the true collective modes of an ordered magnet.
The fermion representation for
S=21 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=21 fermion representation is absent for
S>21. 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
px+ipy-wave spin liquid and the gapless
f-wave spin liquid. We compare our mean-field result with the experiment on
NiGa2S4, 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>21 spin liquids with fermionic spinon excitations.
A non-abelian spin liquid in triangular lattice spin-1 systems was recently formulated in the form of continuum field theory [T. Grover, and T. Senthil, Phys. Rev. Lett. 107, 077203 (2011); Cenke Xu, A.W.W. Ludwig, arXiv:1012.5671]. It has spin-1/2 bosonic spinons coupled to emergent quaternion gauge fields, and can be obtained by quantum disordering a non-collinear spin nematic order hypothesized to describe NiGa_2S_4 [H. Tsunetsugu, and M. Arikawa, J. Phys. Soc. Jpn. 75, 083701 (2006)], However a microscopic lattice description, e.g. the lattice spinon (mean-field) Hamiltonian and the spin wavefunction, has been missing, and it has been noted that the standard Schwinger boson or bosonic triplon representations of spin-1 cannot describe this spin liquid. In this paper a two-orbital Schwinger boson representation for spin-1 systems is developed and used to construct a mean-field description of this quaternion spin liquid. Projecting the mean-field state produces a prototype wavefunction, which is a superposition of close-packed AKLT loop configurations with nontrivial amplitudes. This new formalism and related wavefunctions may be generalized to higher spin systems and can possibly produce spin liquid states with even richer emergent gauge structures.