Geometrically frustrated coarsening dynamics in spinor Bose-Fermi mixtures

Coarsening dynamics theory describes equilibration of a broad class of systems. By studying the relaxation of a periodic array of microcondensates immersed in a Fermi gas, which mediates long-range spin interactions to simulate frustrated classical magnets, we show that coarsening dynamics can be suppressed by geometrical frustration. The system is found to eventually approach a metastable state which is robust against random field noise and characterized by finite correlation lengths together with the emergence of topologically stable ${\mathrm{Z}}_2$ vortices. We find universal scaling laws with no thermal-equilibrium analog that relate the correlation lengths and the number of vortices to the degree of frustration in the system.

Figure 1

Spinor Bose-Fermi mixture consisting of an array of microcondensates immersed in a degenerate Fermi gas. The condensates, whose magnetizations are represented by blue solid arrows, are located at the lattice sites of an optical lattice (grid), while spin-1/2 fermions (brown spheres with arrows) move freely (zigzag trajectories) in a harmonic trapping potential [magenta (light gray)].

Figure 2

Triangular (a) and kagome (b) lattices. Each plaquette $i$ contains the magnetizations ${\mathbf{S}}_{i,1},{\mathbf{S}}_{i,2},{\mathbf{S}}_{i,3}$ at three vertices and the spin chirality ${\mathbf{C}}_i$ defined by Eq. (11). In the ground state of an antiferromagnet, the magnetizations form an angle of ${120}^{\circ }$ with one another due to frustration. The magnetizations in blue shaded plaquettes are identical in the ground state, and the winding number of ${\mathrm{Z}}_2$ vortices is calculated along the directed loops (big red triangles with arrows) connecting these plaquettes (see the main text for details). The vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{b}}_1,{\mathbf{b}}_2$ are coordinate axes. (c) The spin and chirality vectors form a structure similar to a tetrahedron with color-labeled vertices.

Figure 3

Time dependences of nearest-neighbor spin and chirality correlations of the system on triangular and kagome lattices. Here, the time is measured in units of $\hslash /J_1$, where $J_1>0$ is the nearest-neighbor antiferromagnetic spin interaction. For the kagome lattice, the next-nearest-neighbor spin interaction $J_2=-0.1J_1$ is added to lift the degeneracy in the ground-state manifold.

Figure 4

Fourier transform $g_{\mathrm{s}}\left(\mathbf{k}\right)$ of the spin correlation function of the system in (a) triangular and (b) kagome lattices at time $J_{1}t/\hslash =100$. Here $k_1$ and $k_2$ are the components of momentum $\mathbf{k}$ along the directions given by vectors ${\tilde{\mathbf{a}}}_1$ and ${\tilde{\mathbf{a}}}_2$, which are reciprocal to the basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ of the Bravais lattice, respectively ($|{\mathbf{a}}_1|=|{\mathbf{a}}_2|=a$). For the kagome lattice, the next-nearest-neighbor spin interaction $J_2=-0.1J_1$ is added to lift the degeneracy in the ground-state manifold.

Figure 5

Time-dependent spin-correlation and chirality-correlation lengths, ${\xi }_{\mathrm{s}}$ and ${\xi }_{\mathrm{c}}$, for the triangular and kagome lattices. Here, time is measured in units of $\hslash /J_1$, where $J_1>0$ is the nearest-neighbor antiferromagnetic interaction, and the correlation lengths are measured in units of the system's size $L$. For the kagome lattice, the next-nearest-neighbor interaction $J_2=-0.1J_1$ is added to lift the degeneracy in the ground-state manifold. The stepwise feature in the evolutions of ${\xi }_{\mathrm{s}}$ and ${\xi }_{\mathrm{c}}$ is a consequence of the fact that the correlation lengths are only determined by integer multiples of the lattice constant (see the text for details). The early stage of the evolution ($J_{1}t/\hslash \le 70$) is not shown here as the initial ferromagnetic spin order remains dominant and thus the antiferromagnetic spin correlation length is ill-defined.

Figure 6

Time evolutions of the spin-correlation length (black, solid) and chirality-correlation length (red, dashed) of the system on the triangular lattice with a random magnetic field of strength $B_0=\hslash J_1/g_{\mathrm{L}}{\mu }_{\mathrm{B}}$, where ${\mu }_{\mathrm{B}}$ is the Bohr magneton. The random field varies both temporally and spatially with a fixed magnitude $\left|\mathbf{B}(\mathbf{r},t)\right|=B_0$ but pointing in a random direction.

Figure 7

Time evolutions of the spin-correlation length (black, solid) and chirality-correlation length (red, dashed) of the system in the triangular lattice with a small tunneling rate of atoms between lattice sites. The tunneling is introduced by adding a hopping term $t{\sum }_m{\sum }_{\langle i,j\rangle }(a_{m,j}^{†}{a}_{m,i}+a_{m,i}^{†}{a}_{m,j})$ with $t=0.1J_1$ to the Hamiltonian of the system. A repulsive onsite spin-independent interaction of bosons $J_{00}{\sum }_{m{m}^{\prime }}{\sum }_j{a}_{m,j}^{†}{a}_{m^{\prime },j}^{†}{a}_{m^{\prime },j}{a}_{m,j}$ with $J_{00}=10J_1$ is also added to avoid the collapse of the system.

Figure 8

Spatial distributions of (a) the winding number of ${\mathrm{Z}}_2$ vortices and (b)–(d) three components $S_x,S_y,S_z$ of the magnetization of the system in the triangular lattice at time $J_{1}t/\hslash =100$. The winding number and the magnetization are represented by the gray scale and the color gauge, respectively. Here $x_1$ and $x_2$ are the spatial coordinates of the Bravais lattice in the directions given by unit vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ in Fig. 2. The arrows indicate the position of a representative ${\mathrm{Z}}_2$ vortex.

Figure 9

Dependencies of the spin-correlation length ${\xi }_{\mathrm{s}}$ (black triangles), the chirality-correlation length ${\xi }_{\mathrm{c}}$ (red squares), and the number of ${\mathrm{Z}}_2$ vortices $N_{\mathrm{v}}$ (green circles) on the ratio of the next-nearest-neighbor interaction $J_2<0$ to the nearest-neighbor one $J_1>0$ for the system on the kagome lattice. They are evaluated at a fixed time $t=200\hslash /J_1$ in the dynamics of the system and displayed in the logarithmic scales. The correlation lengths are measured in units of the system size. The averages are taken over 10 initial states with random phases in the spinor components. Straight lines show the least-square fittings of the corresponding numerical data.

Figure 10

Critical exponents $\alpha ,\beta$, and $\gamma$ for three different conditions with the initial state being either ferromagnetic or paramagnetic (the polar state) and for different values of the energy damping rate $\mathrm{\Gamma }$.

Figure 11

Critical exponents $\alpha ,\beta$, and $\gamma$ for $J_2<0$ and $J_2>0$ with all the other parameters and the initial condition being identical.

Figure 12

Energy landscape of the lowest-energy band of an antiferromagnet in the kagome lattice with a negative next-nearest-neighbor interaction $J_2=-0.1J_1$.

Figure 13

Energy landscape of the first low-energy band of an antiferromagnet in the kagome lattice with a positive next-nearest-neighbor interaction $J_2=0.1J_1$.

Figure 14

Energy landscape of the second low-energy band of an antiferromagnet in the kagome lattice with a positive next-nearest-neighbor interaction $J_2=0.1J_1$.

Figure 15

Plot of $\alpha \left(x\right)$ given by Eq. (B4).

Figure 16

Plot of $\beta (x,y)$ given by Eq. (B5) as a function of $y$ for $x=0.5$ which is relevant to the values of the system's parameters in Sec. 2.