Chiral spin liquids with crystalline Z2 gauge order in a three-dimensional Kitaev model

Abstract

Chiral spin liquids (CSLs) are time-reversal symmetry breaking ground states of frustrated quantum magnets that show no long-range magnetic ordering, but instead exhibit topological order and fractional excitations. Their realization in simple and tractable microscopic models has, however, remained an open challenge for almost two decades until it was realized that Kitaev models on lattices with odd-length loops are natural hosts for such states, even in the absence of a time-reversal symmetry breaking magnetic field. Here we report on the formation of CSLs in a three-dimensional Kitaev model, which differ from their widely studied twodimensional counterparts, namely, they exhibit a crystalline ordering of the Z2 gauge fluxes and thereby break some of the underlying lattice symmetries. We study the formation of these unconventional CSLs via extensive quantum Monte Carlo simulations and demonstrate that they are separated from the featureless paramagnet at high temperatures by a single first-order transition at which both time-reversal and lattice symmetries are simultaneously broken. Using variational approaches for the ground state, we explore the effect of varying the Kitaev couplings and find at least five distinct CSL phases, all of which possess crystalline ordering of the Z2 gauge fluxes. For some of these phases, the complementary itinerant Majorana fermions exhibit gapless band structures with topological features such as Weyl nodes or nodal lines in the bulk and Fermi arc or drumhead surface states.

Thermodynamic classification of three-dimensional Kitaev spin liquids

“…7 and 8 in classifying the aforementioned Majorana metals. We show that the finite-temperature behavior is very systematic, except for two cases where additional frustration and/or spontaneous symmetry breaking effects become important [17][18][19] . In particular, we find a strong correlation between the transition temperature and the local flux/vison gap, whereas there is no correlation to the minimal loop length 20 .…”

“…In this context, we discuss the different behaviors of the relevant thermodynamic observables and relate them to elementary geometric properties of the underlying lattices (e.g., its fundamental symmetries), and to the different Majorana (semi-)metal ground states that these systems exhibit. We also review results from former works of our collaboration 18,19 on the more exotic systems (8,3)c (section IV D), which exhibits "gauge frustration", and the non-bipartite lattice (9,3)a (section IV E), which intertwines gauge ordering and timereversal symmetry breaking (due to the odd length of its elementary plaquettes). Finally, we sum up the conclusions following from our studies and give an outlook on further research directions in the field of 3D Kitaev systems in section VI.…”

Theory of collective topologically-protected Majorana fermion excitations of networks of localized Majorana modes

“…A Hamiltonian of the form Eq. 2 also describes the low-energy physics of Kitaev-type models of Majorana spin liquids and mean field theories for Majorana spin liquids [30][31][32][33][34][35][36][37][38] Motivated by this, Fig. 2 displays another example of R-type region, this time of relevance to the effect of vacancies in the Kitaev-like model on the startriangle lattice or wine glass lattice [35].…”

“…where a rr = −a r r are real-valued amplitudes that couple modes at sites r and r , and the factor of 4 is merely a matter of convention. As already noted, the quadratic Hamiltonian H Majorana is also of interest in the context of various generalizations of Kitaev's honeycomb model to other three-coordinated lattices [30][31][32][33][34][35][36][37][38].…”

“…This is clearly of significance in the context of various proposals for physical platforms for Majorana computation, since these collective topologically-protected Majorana excitations of the network as a whole can serve as qubits in a Majorana quantum computer even when the original Majorana qubits of the network are destroyed by strong mixing amplitudes. Our approach can also be used to understand vacancyinduced Curie tails in Kitaev-type models of Majorana spin liquids and mean field theories for such a spin liquid state [30][31][32][33][34][35][36][37][38] The remainder of this artice is organized as follows: In Sec. II, we introduce the Majorana network Hamiltonian of interest to us, summarize the classical results of Tutte, Lovász, and Anderson on the dimension of the null space of real skew-symmetric (equivalently, pure imaginary hermitean) matrices, and review how they determine the number of topologically-protected zero-energy Majorana excitations of the network as a whole.…”