Ground states of quasi-two-dimensional correlated systems via energy expansion

Abstract

We introduce a generic method for computing groundstates that is applicable to a wide range of spatially anisotropic 2D many-body quantum systems. By representing the 2D system using a low-energy 1D basis set, we obtain an effective 1D Hamiltonian that only has quasi-local interactions, at the price of a large local Hilbert space. We apply our new method to three specific 2D systems of weakly coupled chains: hardcore bosons, a spin-$1/2$ Heisenberg Hamiltonian, and spinful fermions with repulsive interactions. In particular, we showcase a non-trivial application of the energy expansion framework, to the anisotropic triangular Heisenberg lattice, a highly challenging model related to 2D spin liquids. Treating lattices of unprecedented size, we provide evidence for the existence of a quasi-1D gapless spin liquid state in this system. We also demonstrate the energy expansion-framework to perform well where external validation is possible. For the fermionic benchmark in particular, we showcase the energy expansion-framework's ability to provide results of comparable quality at a small fraction of the resources required for previous computational efforts.

Figures (17)

• Figure 1: Weakly interacting chains where particles can hop with transition probability $t =1$ () or $t_\perp < t$ (). All models we investigate also exhibit nearest neighbour repulsion with strength $V$ along the chains, and some also contain such repulsion in the the perpendicular direction, of strength $V_\perp$.
• Figure 2: Matrix elements of the creation operator in a system of size $L = 16$ and nearest neighbour repulsion $V = 2$ going from $N = 8$ to $N = 9$ particles in the center of the chain.
• Figure 3: \ref{['fig:rect-energies-over-V0', 'fig:rect-energies-over-tp']} show the energies of the system scaling with intrachain density repulsion and interchain coupling, respectively. In the former case, $t_\perp$ was fixed to be $0.1$ while in the latter $V = 2$. \ref{['fig:rect-bos-corr']} shows the correlator \ref{['eq:rect-hcb-density-density-corr']} along the strong coupling direction for different intra- chain couplings $t_\perp$ of a $16\times 12$ strip with repulsion $V = 2$. The relative error compared to exact data is shown on the right- hand $y$- axis ().
• Figure 4: Triangular lattice made up of weakly coupled chains where spins interact with coupling strength $J = 1$ () or $J_\perp$ (). The blue lines () show the choice of our elementary vectors. The unfinished lines in $x$- direction indicate while the horizontal direction has .
• Figure 5: \ref{['fig:spin-hcb-ee-correlator-J-sql', 'fig:spin-hcb-ee-correlator-J-triangular']} show the algebraic decay of the $S^zS^z$- correlator for $L = 16,20,24$ along the strong coupling direction and with an interchain coupling of $J_\perp / J = 0.1$ for the square ($\square$) and the triangular ($\triangle$) system, respectively (c.f. discussion below \ref{['eq:spin-hcb-perp']} and \ref{['eq:spin-hcb-perp_triangular']}). The inset of \ref{['fig:spin-hcb-ee-correlator-J-triangular']} shows $|C(x)|$ on a double logarithmic scale. \ref{['fig:spin-hcb-ee-correlator-Jperp']} shows $|C_\perp(y)|$ at $J_\perp = 0.1$ (c.f. discussion in main text). \ref{['fig:spin-hcb-ee-correlation-length']} shows the perpendicular correlation length $\xi_\perp(L)$ for multiple values of $J_\perp = 0.1, 0.15, 0.2$.