Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation—indicated by a larger leading ground-state coefficient—is obtained when combining the symmetric group ${\mathcal{S}}_n$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group $\mathrm{U}\left(n\right)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

Figure 1

Weight of leading CSF for the compact ordered 6-site chain with OBC. The labels on the lattice sites refer to the order in which the spins are coupled in the GUGA CSF formalism. Thus, for example, the second site from left with label “3” is coupled in the 3rd position in the CSF.

Figure 2

Cumulative doublet coupling of “meta-spin-$\frac{1}{2}$” in the most compact order.

Figure 3

(a) Lowest single CSF energies and (b) corresponding weight $|c_0|$ for the 6-site ring as a function of permutations. The permutations/orderings are ordered by decreasing maximum weight $|c_0|$ in the ground-state wave function (only the first 100 highest weighted permutations are shown for clarity).

Figure 4

Exchange matrix elements $X_{ij}$ for the 6-site chain reference CSFs: (a) $|ududud\rangle$ (natural), (b) $|uuuddd\rangle$ (bipartite), and (c) $|uududd\rangle$ (compact), with the respective order in parenthesis. The nonzero $J_{ij}$ values, due to the corresponding orderings, are indicated by the black rectangles/frames. They act as a “mask” in the product $X_{ij}{J}_{ij}$, see Eq. (29), and for a given CSF, the “optimal” lattice ordering yields the lowest possible diagonal matrix element, Eq. (29). (Color online: blue/red colors are a guide to the eye indicating the sign and magnitude of the respective values.)

Figure 5

(a) $\langle S_1^z·S_x^z\rangle$, (b) $\langle S_5^z·S_x^z\rangle ,$ and (c) $\langle S_{10}^z·S_x^z\rangle$ exact and single-CSF spin-spin correlation functions for the 10-site chain with OBC.

Figure 6

(a) Single CSF spin-spin correlation function for the compact ordering vs chain size and an exponential fit to the even (green-dashed line) and odd spin-spin correlation (red-dashed line) for the 36-site single CSF in the compact order, with $a_{\mathrm{even}}=0.153,b_{\mathrm{even}}=0.203,a_{\mathrm{odd}}=-0.25$, and $b_{\mathrm{odd}}=0.203$. (b) OBC Single CSF spin-spin correlation function for the compact ordering compared with exact results obtained with the $\mathcal{H}\mathrm{\Phi }$ software package [166] for 30 sites. (c) Power fit to the even and odd spin-spin correlation for the 30-site exact spin-spin correlation function displayed on a double logarithmic scale, with $a_{\mathrm{even}}=0.133,b_{\mathrm{even}}=-0.969,a_{\mathrm{odd}}=-0.218$, and $b_{\mathrm{odd}}=-1.049$.

Figure 7

Analytic TDL spin-spin correlation function [167] with power-law fits (solid lines) for even and odd lattice sites, with $a_{\mathrm{even}}=0.107,b_{\mathrm{even}}=-0.820,a_{\mathrm{odd}}=-0.148$, and $b_{\mathrm{odd}}=-0.975$, compact single CSF spin-spin correlation function with exponential fit (dashed lines) and $n=30$ exact results (ED) with OBC [166].

Figure 8

(a) OBC Néel state and single CSF energies (diagonal Hamiltonian matrix element) of the natural and compact orderings vs the number of sites compared with GUGA-FCIQMC results—using a CSF basis and the compact ordering—and exact Bethe Ansatz [172] and ED results, obtained with $\mathcal{H}\mathrm{\Phi }$ [166]. (Energies/diagonal matrix elements are divided by the corresponding number of sites $L$ for comparative reasons.) (b) OBC weights of the Néel state and the leading CSFs of the natural and compact orderings vs the inverse lattice size obtained with GUGA-FCIQMC. These weights are an approximation of the exact $|c_0^{ex}|$ of the ground-state wave function obtained via, e.g., ED.

Figure 9

Difference of GUGA-FCIQMC energy per site results compared to ED results [166] of the 20- (a) and 30-site (b) 1D Heisenberg model with OBC for different orderings (natural and compact) and a SD-based results (SDs) vs the number of walkers $N_w$.

Figure 10

Difference of GUGA-FCIQMC energy per site results compared to $M=500$ DMRG reference results [107, 173, 174, 175] of the 10- (a), 20- (b), and 30-site (c) 1D Hubbard model with OBC for different orderings (natural and compact) and SD-based results (SDs) vs the number of walkers $N_w$.

Figure 11

Energy difference per site to the $M=500$ DMRG reference results of the 10- (a), 20- (b), and 30-site (c) hydrogen chain in a minimal basis for different orderings (natural and compact) and SD-based results (SDs) vs the number of walkers $N_w$.

Figure 12

30-site hydrogen chain DMRG energy difference to $M=500$ Fiedler (natural order) reference result for different orbital ordering with and without spin-adaptation vs matrix dimension $M$.

Figure 13

Sketch of the mixed stochastic minimization procedure.