Analysis of quantum spin models on hyperbolic lattices and Bethe lattice

We study the Hubbard and Heisenberg models on hyperbolic lattices with open boundary conditions by means of mean-field approximations, spin-wave theory and quantum Monte Carlo (QMC) simulations. For the Hubbard model we use the auxiliary-field approach and for Heisenberg systems the stochastic series expansion algorithm. We concentrate on bipartite lattices where the QMC simulations are free of the negative sign problem. The considered lattices are characterized by a Dirac li...

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the $\{8,3\}$ lattice and find a gapped $\ma...

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the euclidean case. We explain this anomaly by a careful re-derivation of the Kramers--Wann...

To explore the non-Euclidean generalization of higher-order topological phenomena, we construct a higher-order topological insulator model in hyperbolic lattices by breaking the time-reversal symmetry (TRS) of quantum spin Hall insulators. We investigate three kinds of hyperbolic lattices, i.e., hyperbolic $\{4,5\}$, $\{8,3\}$ and $\{12,3\}$ lattices, respectively. The non-Euclidean higher-order topological behavior is characterized by zero-energy effective corner states appe...

We have extended the canonical tree tensor network (TTN) method, which was initially introduced to simulate the zero-temperature properties of quantum lattice models on the Bethe lattice, to finite temperature simulations. By representing the thermal density matrix with a canonicalized tree tensor product operator, we optimize the TTN and accurately evaluate the thermodynamic quantities, including the internal energy, specific heat, and the spontaneous magnetization, etc, at ...

We show that the simple update approach proposed by Jiang et. al. [H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is an efficient and accurate method for determining the infinite tree tensor network states on the Bethe lattice. Ground state properties of the quantum transverse Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The transverse Ising model is found to undergo a second-order quantum phase transition with a diverg...

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 s...

Motivated by the AdS/CFT correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary s...

Hyperbolic lattices are a revolutionary platform for tabletop simulations of holography and quantum physics in curved space and facilitate efficient quantum error correcting codes. Their underlying geometry is non-Euclidean, and the absence of Bloch's theorem precludes the straightforward application of the often indispensable energy band theory to study model Hamiltonians on hyperbolic lattices. Motivated by recent insights into hyperbolic band theory, we initiate a crystall...

The Bethe lattice Ising model -- a classical model of statistical mechanics for the phase transition -- provides a novel and intuitive understanding of the prototypical relationship between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary...