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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 like density of states, Schl\"afli indices $\{p,q\}=\{10,3\}$ and $\{8,3\}$, as well as by flat bands, $\{8,8\}$. The Dirac density of states cuts off the logarithmic divergence of the staggered spin susceptibility and allows for a finite $U$ semi-metal to insulator transition. This transition has the same mean-field exponents as for the Euclidean counterpart. In the presence of flat bands we observe the onset of magnetic ordering at any finite $U$. The magnetic state at intermediate coupling can be described as a global antiferromagnet. It breaks the $C_p$ rotational and time reversal symmetries but remains invariant under combined $C_p \mathcal{T}$ transformations. The state is characterized by macroscopic ferromagnetic moments, that globally cancel. We observe that fluctuations on the boundary of the system are greatly enhanced: while spin wave calculations predict the breakdown of antiferromagnetism on the boundary but not in the bulk, QMC simulations show a marked reduction of the staggered moment on the edge of the system., Comment: 19 pages, 17 figures

Projection of the Coulomb potential onto flat bands paves the way to design various interactions in the particle-hole and particle-particle channels. Here we pose the question if we can use this mapping to overcome the negative sign problem for the simplest possible frustrated spin system consisting a trimer of spins-1/2 coupled with an antiferromagnetic exchange interaction. While the answer is negative, we show that we can map the sign problem for frustrated spin systems onto a problem where we need to simulate particle-hole symmetric systems but with long-ranged Coulomb interactions prevailing over short-ranged ones. While the latter systems are currently not accessible to auxiliary field determinant quantum Monte Carlo, this mapping motivates algorithmic development that may overcome this issue., Comment: 14 pages, 13 figures

We study a Su-Schrieffer-Heeger electron-phonon model on a square lattice by means of auxiliary-field quantum Monte Carlo simulations. The addition of a symmetry-allowed interaction permits analytical integration over the phonons at the expense of discrete Hubbard-Stratonovich fields with imaginary-time correlations. Using single-spin-flip and global updates, we investigate the phase diagram at the O(4)-symmetric point as a function of hopping $t$ and phonon frequency $\omega_0$. For $t=0$, where electron hopping is boson assisted, the model maps onto an unconstrained $\mathbb{Z}_2$ gauge theory. A key quantity is the emergent effective flux per plaquette, which equals $\pi$ in the assisted-hopping regime and vanishes for large $t$. Phases in the former regime can be understood in terms of instabilities of emergent Dirac fermions. Our results support a direct and continuous transition between a $(\pi,0)$ valence bond solid (VBS) and an antiferromagnetic (AFM) phase with increasing $\omega_0$. For large $t$ and small $\omega_0$, we find finite-temperature signatures, a disordered pseudogap phase, of a previously reported $(\pi,\pi)$ VBS ground state related to a nesting instability. With increasing $\omega_0$, AFM order again emerges., Comment: 17 pages, 19 figures

The two-dimensional Su-Schrieffer-Heeger model of electrons coupled to quantum phonons is investigated using Langevin dynamics within the framework of auxiliary-field quantum Monte Carlo. Based on an explicit determination of the density of zeros of the fermion determinant, it is argued that the method is efficient in the challenging adiabatic limit. Large-scale simulations at the O(4)-symmetric point establish that the ground state of the 2D SSH model undergoes a transition from a $(\pi,\pi)$ valence bond solid to an antiferromagnet with increasing phonon frequency, yet still in the adiabatic regime. The single-particle spectrum illustrates the renormalization of the electronic band and suggests the existence of a gapped polaronic band, whereas the particle-hole channels show gapless modes associated with long-range bond and magnetic order, respectively. The simulations are supplemented with a mean-field analysis and a self-consistent Born proximation., Comment: 14 pages, 16 Figures