The Kitaev model with an applied magnetic field in the [Formula: see text] direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at [Formula: see text] and a second transition at a higher field [Formula: see text] to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function [Formula: see text] and the Fermi surface [Formula: see text] of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.
Recent experimental evidence for a field-induced quantum spin liquid (QSL) in
α-RuCl
3 calls for an understanding for the ground state of honeycomb Kitaev model under a magnetic field. In this work we address the nature of an enigmatic gapless paramagnetic phase in the antiferromagnetic Kitave model, under an intermediate magnetic field perpendicular to the plane. Combining theoretical and numerical efforts, we identify this gapless phase as a
U(1) QSL with spinon Fermi surfaces. We also reveal the nature of continuous quantum phase transitions involving this
U(1) QSL, and obtain a phase diagram of the Kitaev model as a function of bond anisotropy and perpendicular magnetic field.
In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.
It is now well established that the Kitaev honeycomb model in a magnetic field along the [111] direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-Abelian QSL at low fields
H<Hc1 and a partially polarized phase at high fields
H>Hc2. Here, we analyze the low-field and high-field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization and density matrix renormalization group methods. We find that the energy to create a bound state of two-magnon ${\mathrm{\ensuremath{\Delta}}}_{p}$ becomes lower than the energy to create a single spin flip ${\mathrm{\ensuremath{\Delta}}}_{s}$ near
Hc2. In the entire Kitaev spin liquid, ${\mathrm{\ensuremath{\Delta}}}_{p}<{\mathrm{\ensuremath{\Delta}}}_{s}$ and both gaps vanish at
Hc2. We make testable predictions for magnon pairing that could be observable in Raman scattering measurements on Kitaev QSL candidate materials.
The possible existence of a quantum spin liquid (QSL) phase, an exotic state of matter with long-range quantum entanglement and fractionalized excitations, in
α-RuCl
3 has sparked widespread interests in exploring QSLs in various Kitaev models under magnetic fields. Recently, a
K-
J-
Γ-
Γ′ model has been proposed to accurately describe the compound by fitting the measured thermodynamic data, where
J is Heisenberg interaction, and
Γ,
Γ′ are off-diagonal exchanges on top of the dominant Kitaev coupling
K. Based on this effective model, an intermediate QSL phase in presence of an out-of-plane fields along the
[111] direction, between the low-field zigzag order and high-field polarized phase, has been predicted. By combining density matrix renormalization group (DMRG), exponential tensor renormalization group (XTRG), and variational Monte Carlo (VMC) calculations, we address the nature of this QSL phase in the honeycomb
K-
J-
Γ-
Γ′ model under the
[111]-direction field. Our DMRG calculations find the algebraic-like decay of spin correlation function, the finite spin scalar chiral order, and lattice nematic order. Together with the XTRG results of power-law specific heat at low temperature, our findings naturally suggest a gapless nematic chiral spin liquid. On the other hand, our VMC study finds a gapped nematic chiral spin liquid with the variational energy very close to that obtained in DMRG. As VMC finds a very small gap that is beyond the current resolution of both ground-state DMRG and finite-temperature XTRG calculations on finite-width cylinders, we resort the full clarification for the nature of the intermediate QSL to future studies. Lastly, we discuss the implications of our results to the recent experiment on
α-RuCl
3 and the QSL-like phase in a similar
K-
Γ-
Γ′ model.
The main question we address is how to probe the fractionalized excitations of a quantum spin liquid (QSL), for example, in the Kitaev honeycomb model. By analyzing the energy spectrum and entanglement entropy, for antiferromagnetic couplings and a field along either [111] or [001], we find a gapless QSL phase sandwiched between the non-Abelian Kitaev QSL and polarized phases. Increasing the field strength towards the polarized limit destroys this intermediate QSL phase, resulting in a considerable reduction in the number of frequency modes and the emergence of a beating pattern in the local dynamical correlations, possibly observable in pump-probe experiments.
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the
[001] magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with
KX=KY,
KX+KY+KZ=−3K, and
KZ<−K, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at
hc1=gμBHz1/K, to another gapless QSL with
C6 rotational symmetry at
hc2, and to a new
U(1) gapless QSL between
hc3 and
hc4, respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic
C6 gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields
hc1,
hc2,
hc3, and
hc4 come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.