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Register a new userProjective symmetry group classification of chiral $mathbb{Z}_2$ spin liquids on the pyrochlore lattice: application to the spin-$1/2$ XXZ Heisenberg model
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We give a complete classification of fully symmetric as well as chiral $mathbb{Z}_2$ quantum spin liquids on the pyrochlore lattice using a projective symmetry group analysis of Schwinger boson mean-field states. We find 50 independent ansatze, including the 12 fully symmetric nearest-neighbor $mathbb{Z}_2$ spin liquids that have been classified by Liu et al. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.075125]. For each class we specify the most general symmetry-allowed mean-field Hamiltonian. Additionally, we test the properties of a subset of the spin liquid ansatze by solving the mean-field equations for the spin-$1/2$ XXZ model near the antiferromagnetic Heisenberg point. We find that the ansatz with the lowest energy at mean-field level is a chiral spin liquid that breaks the screw symmetry of the lattice modulo time reversal symmetry. This state has a different symmetry than the previously studied monopole flux state. Moreover, this chiral spin liquid state has a substantially lower energy than all other symmetric spin liquid states, suggesting that it could be a stable ground state beyond the mean-field approximation employed in this work.
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We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin $S=1/2$. We then introduce a systematic generalization of the approach for symmetric $mathbb{Z}_2$ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral $mathbb{Z}_2$ states, we study the subset of U(1) phases variationally in the antiferromagnetic $J_1$-$J_2$-$J_d$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.
Guided by the recent discovery of SU($2$)$_1$ and SU($3$)$_1$ chiral spin liquids on the square lattice, we propose a family of generic time-reversal symmetry breaking SU($N$)-symmetric models, of arbitrary $Nge 2$, in the fundamental representation, with short-range interactions extending at most to triangular units. The evidence for Abelian chiral spin liquid (CSL) phases in such models is obtained via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small clusters are carried out up to $N=10$, revealing (in some range of the Hamiltonian parameters) a bulk gap and ground-state degeneracy on periodic clusters as well as linear dispersing chiral modes on the edge of open systems, whose level counting is in full agreement with SU($N$)$_1$ Wess-Zumino-Witten conformal field theory predictions. Using an SU($N$)-symmetric version of iDMRG for $N=2,3$ and $4$ to compute entanglement spectra on (infinitely-long) cylinders in all topological sectors, we provide additional unambiguous signatures of the SU($N$)$_1$ character of the chiral liquids. An SU($4$)-symmetric chiral PEPS is shown to provide a good variational ansatz of the $N=4$ ground state, constructed in a manner similar to its $N=2$ and $N=3$ analogs. The entanglement spectra in all topological sectors of an infinitely long cylinder reveal specific features of the chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Results for the correlation lengths suggest some form of long-range correlations in SU($N$) chiral PEPS, which nevertheless do not preclude an accurate representation of the gapped SU($N$) CSL phases. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups.
Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as Volborthite and Herbertsmithite that have Kagome like structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at $m=frac{1}{3}$ is considered to be the most prominent. Here we study the problem of the antiferromagnetic spin-1/2 quantum XXZ Heisenberg model on a Kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated Kagome lattice we can now formalize the Jordan-Wigner transformation on the Kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of $frac{1}{3}$-filling ($m=frac{1}{3}$ plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of $XY$ anisotropy the ground state at the $1/3$ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction $1/2$ and that at the $5/9$ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction $2/3$.
Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor $J_1$ and second-neighbor $J_2$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. Neel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio $J_2/J_1$, we find strong evidence for a continuous transition to a nonmagnetic phase at $J_2/J_1 approx 0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence bond state gives an excellent approximation to the exact ground-state energy. For $0.23 lesssim J_2/J_1 lesssim 0.4$, a gapless $Z_2$ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the Neel order melts, this ordered state becomes clearly favored only for $J_2/J_1 gtrsim 0.3$. Finally, for $0.36 lesssim J_2/J_1 le 0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the Neel antiferromagnet from the plaquette valence-bond solid.
We consider the pyrochlore-lattice quantum Heisenberg ferromagnet and discuss the properties of this spin model at arbitrary temperatures. To this end, we use the Greens function technique within the random-phase (or Tyablikov) approximation as well as the linear spin-wave theory and quantum Monte Carlo simulations. We compare our results to the ones obtained recently by other methods to corroborate our findings. Finally, we contrast our results with the ones for the simple-cubic-lattice case: both lattices are identical at the mean-field level. We demonstrate that thermal fluctuations are more efficient in the pyrochlore case (finite-temperature frustration effects). Our results may be of use for interpreting experimental data for ferromagnetic pyrochlore materials.
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