We develop a mean-field theory of the 1D spin-1 antiferromagnetic Heisenberg model, using a two-orbital fermionic representation for the spin-1 operators. Two ansatz with different orbital pairing symmetries are discussed and the projected mean-field ground states with their corresponding parent Hamiltonians have also been obtained. Then we study the U(1) spin liquid state with large spinon Fermi surfaces. The physical conductivity obtained through Ioffe-Larkin composition rule can support surface plasmon modes propagating along the interface between the spin liquid and a linear medium, which can be excited using a Kretschmann-Raether configuration. Next we generalize the linear current response to including nonlinear effects under the perturbation of a time dependent but spatially uniform electric field in the framework of an effective U(1) gauge theory derived through a Fermi-liquid-like approach. The nonlinear susceptibility can support waves with triple frequency inside the spin liquids through the process of third harmonic generation. As a byproduct during the Fermi-liquid approach, we generalize the standard Landau Fermi liquid theory to describe the intrinsic anomalous Hall conductivity, based on some general symmetric considerations. Fitting result of the anomalous Hall conductivity is well consistent with the experimental data, and behaves better in lower frequency range than the simple two-band model.
In this paper we study the nonlinear current response of
U(1) spin liquids with large spinon Fermi surfaces under the perturbation of a time-dependent ac electric field
E(t) within the framework of an effective
U(1) gauge theory. In particular, the third-order nonlinear current response to ac electric fields is derived. We show that as in the case of linear current response, an in-gap power-law $(\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\eta}})$ response is found for the nonlinear current at low frequency. The nonlinear susceptibility may also induce through process of third harmonic generation propagating EM wave with frequency $3\ensuremath{\omega}$ inside the spin liquids.
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.
Quantum spin liquids are novel phases of matter whose physical properties cannot be captured within the conventional framework of phase transitions. These quantum states are characterized by excitations, called spinons, carrying a fractional value of the integer spin. A description based on a mean-field approximation leads to effective gauge theory. Due to this gauge freedom symmetry group operations act via projective symmetry group (PSG) representations, which can be used to classify quantum states beyond the conventional method. This introducing discussion is found in Chapters 2-6. A preceding PSG classification on the square lattice, which took spin rotational symmetry breaking terms into account, showed that the resulting mean-field states could exhibit topologically non-trivial spinon bands. In Chapter 7, only those el- ements of this Classification are studied, which lead to short-ranged mean-field models. The mutual effect of spinons and visons is scrutinized by adding static configurations of a background gauge field. Spinons and visons form composite ob- jects, which in the case of topological spinon bands, are described by Majorana zero modes. An exact mapping corroborates the relation to topological supercon- ductors and vortices. The effect of multiple visons is investigated, and a possible experimental signature is exposed. Chapter 8 explains how spinon mean-field theory can be used to derive a phe- nomenological effective model for the compound Ca 10 Cr 7 O 28 . A material that was recently proposed as a quantum spin liquid. This low-energy theory consists of spinon hopping processes. The underlying bilayer kagome structure can be simpli- fied to an effective honeycomb lattice. The dynamical spin structure factor is calcu- lated and compared to neutron scattering data. It is discussed how f -wave pairing of spinons can explain small deviations from a linear temperature dependency observed in heat capacity measurements at the low temperatures. This pairing mechanism breaks the U (1) invariance group of the pure hopping model and renders a coherent picture of Ca 10 Cr 7 O 28 being a Z 2 quantum spin liquid. Chapter 9 carries out a PSG classification for the simple-, body centered-, and face centered cubic lattices. Despite a large number of possible PSG representations, only very few describe short-ranged mean-field states. These mean-field models are constructed up to third neighbor couplings and calculated in a self-consistent manner. It is found that the energy is lower for non-trivial sign patterns induced by non-trivial representations of the symmetry generators. Higher neighbor extensions, including pairing terms, yield lower mean-field ground state energies compared to extensions with only hopping terms. For the fcc lattice, a first neighbor state could be identified, which exhibits symmetry protected zero-energy modes.
We study 1-dimension (1-d) and 2-dimensional (2-d) quantum anti-ferromagnets. Following AKLT's theory of valent bond solid (VBS) states, we find a family of topologically nontrivial 1-d VBS states for spin systems and provide a method to construct their parent Hamiltonians. Then we generalize the fermionic representation of S = 1/2 to arbitrary spin, and find an important difference between integer spins and half-odd-integer spins. Depending on different particle number constrains we develop different versions of mean field theory for spin models. We find that the mean field with restored particle-hole symmetry can reproduce Haldane's conjecture for 1-d Heisenberg models. Applying the mean field theory to 2-d anti-ferromagnetic models, we find that S = 1 spin liquids may exist in triangle lattice. Comparing with the experimental results of NiGa2S4, we argue that the antiferro-nematic f-wave spin liquid is a possible ground state. On the other hand, to check whether the material ZnCu3(OH)6Cl2 is really an U(1) Dirac spin liquid, we derive the Raman scattering operators of different symmetry channels for Hubbard model on Kagome lattice. Then we numerically calculate the Raman cross-section according to the mean field spin liquid state. The low energy continuum part of our results qualitatively match the experimental data.
We study the transverse dynamical spin susceptibility of the two-dimensional U(1) spinon Fermi-surface spin liquid in a small applied Zeeman field. We show that both short-range interactions, present in a generic Fermi liquid, as well as gauge fluctuations, characteristic of the U(1) spin liquid, qualitatively change the result based on the frequently assumed noninteracting spinon approximation. The short-range part of the interaction lead to a collective ``spinon spin wave'' mode, which splits off from the two-spinon continuum at a small momentum and disperses downward. Gauge fluctuations renormalize the susceptibility, providing nonzero power-law weight in the region outside the spinon continuum and giving the spin wave a finite lifetime, which scales as momentum squared. We also study the effect of Dzyaloshinskii-Moriya anisotropy on the zero momentum susceptibility, which is measured in electron spin resonance (ESR), and obtain a resonance linewidth linear in temperature and varying as
B2/3 with magnetic field
B at low temperatures. Our results form the basis for a theory of inelastic neutron scattering, ESR, and resonant inelastic x-ray scattering studies of this quantum spin-liquid state.
Recently, thermal Hall effect has been observed in the paramagnetic state of volborthite, which consists of distorted kagome layers with
S=1/2 local moments. Despite the appearance of magnetic order below
1K, the response to external magnetic field and unusual properties of the paramagnetic state above
1K suggest possible realization of exotic quantum phases. Motivated by these discoveries, we investigate possible spin liquid phases with fermionic spinon excitations in a nonsymmorphic version of the kagome lattice, which belongs to the two-dimensional crystallographic group
p2gg. This nonsymmorphic structure is consistent with the spin model obtained in the density functional theory calculation. Using projective symmetry group analysis and fermionic parton mean field theory, we identify twelve distinct
Z2 spin liquid states, four of which are found to have correspondence in the eight Schwinger boson spin liquid states we classified earlier. We focus on the four fermionic states with bosonic counterpart and find that the spectrum of their corresponding root
U(1) states features spinon Fermi surface. The existence of spinon Fermi surface in candidate spin liquid states may offer a possible explanation of the finite thermal Hall conductivity observed in volborthite.