Spin liquid phases of alkaline-earth-metal atoms at finite temperature

We study spin liquid phases of spin-5/2 alkaline-earth-metal atoms on a honeycomb lattice at finite temperatures. Our analysis is based on a Gutzwiller projection variational approach recast to a path-integral formalism. In the framework of a saddle-point approximation we determine spin liquid phases with lowest free energy and study their temperature dependence. We identify a critical temperature, where all the spin liquid phases melt and the system goes to the paramagnetic phase. We also study the stability of the saddle-point solutions and show that a time-reversal symmetry breaking state, a so-called chiral spin liquid phase, is realized even at finite temperatures. We also determine the spin structure factor, which, in principle, is an experimentally measurable quantity and is the basic tool to map the spectrum of elementary excitations of the system.

Figure 1

Illustration of the unit cell considered here. The unit cell is surrounded by the dashed line and contains six sites. The numbers close to the nodes (blue in the color version) label the sublattice index (the sites inside the unit cell), while the $(m,n)$ pair indexes the unit cell itself. There are six Hubbard-Stratonovich fields inside each unit cell labeled by ${\chi }_1...{\chi }_6$ and three other independent ones between the neighboring unit cells ${\chi }_7...{\chi }_9$. We also show the elementary lattice vectors connecting the neighboring unit cells.

Figure 2

The saddle-point solutions for the three lowest free energy spin liquid states. (a) The chiral spin liquid state. (b) The quasiplaquette state. (c) The plaquette state.

Figure 3

The free energy per plaquette $F_{\mathrm{mf}}(T,V)/V$ of the three lowest lying saddle-point solutions.

Figure 4

The lowest static eigenvalues of the curvature of the effective action in the Brillouin zone at zero temperature for the chiral spin liquid phase (a), and for the quasiplaquette phase (b).

Figure 5

The static structure factor $\mathcal{S}(\mathbf{q},0)$ in the Brillouin zone at zero temperature for the chiral spin liquid phase (a), and for the quasiplaquette phase (b).

Figure 6

The spectral density ${\rho }_{\mathrm{tot}}\left(\omega \right)$ at zero temperature of the chiral spin liquid phase (a), the quasiplaquette phase (b), and the plaquette phase (c).

Figure 7

Feynman graphs representing the expansion Eq. (A1). We have plotted the first order (a), the second order (b), and the nth order (c) diagrams. The straight lines represent the free fermion propagator, while the wiggly line is for the $\Sigma$ vertex. The sum of the incoming momentum has to be zero.