Curvature-induced frustration in the $XY$ model on hyperbolic surfaces

We study low-temperature properties of the $XY$ spin model on a negatively curved surface. Geometric curvature of the surface gives rise to frustration in local spin configuration, which results in the formation of high-energy spin clusters scattered over the system. Asymptotic behavior of the spin-glass susceptibility suggests a zero-temperature glass transition, which is attributed to multiple optimal configurations of spin clusters due to nonzero surface curvature of the system. It implies that a constant ferromagnetic spin interaction on a regular lattice can exhibit glasslike behavior without possessing any disorder if the lattice is put on top of a negatively curved space such as a hyperbolic surface.

Figure 1

(a) A heptagonal lattice with $l=3$ in the Poincaré disk. The circumference of the disk represents the points at infinity. (b) Parallelism between two spins in the Poincaré disk. Parallel transport of an arrow from $i\text{th}$ to $j\text{th}$ site along the geodesic curve (dotted) rotates the arrow by ${\psi }_{ij}$.

Figure 2

(Color online) Snapshots of spin configurations with $l=7$, where a darker line represents a higher-energy bond. (a) A high-temperature regime with $T=1.00$ and (b) a closer look at the surface. (c) A low-temperature regime with $T\approx 0.06$ and (d) its magnified image, where clustering structures are clearly visible. The bond energy and temperature $T$ are in units of $J$ and $J/k_B$, respectively.

Figure 3

(Color online) (a) Autocorrelations of spin configurations, given in Eq. (2), together with fitting results with a stretched-exponential form. The results are obtained for $T=0.250$, 0.167, 0.125, 0.100, 0.083, and 0.067, from bottom to top. (b) Relaxation time $t_0$ as a function of inverse temperature from the stretched-exponential fit. Inset: $b$ as a function of $1/T$ from the same fit. It approaches unity at high $T$ leading to a simple exponential form.

Figure 4

(Color online) (a) Spin-glass susceptibility ${\chi }_{SG}$ as a function of $T$. (b) ${\chi }_{SG}$ plotted against $N^{-1}$ at $T=0.100$, 0.083, 0.071, and 0.063, from bottom to top. It does not exhibit a power law with respect to $N$ for any finite $T$.