Numerical contractor renormalization method for quantum spin models

We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagomé lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagomé lattice we find that the previously proposed twofold degenerate $S=1∕2$ basis can account for a large number of phenomena of the spin $1∕2$ kagomé system. For spin $3∕2$, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\times 8$ lattice for the plaquette lattice or a 48-site kagomé lattice, which are beyond the possibilities of a standard exact diagonalization approach.

Figure 1

(a) 2-leg ladder. Basic block is a $2\times 2$ plaquette. (b) 3-leg torus with rung coupling $J_{\perp }$ and inter-rung coupling $J_{\Vert }$.

Figure 2

Ground-state energy per site and spin gap of a $2\times L$ Heisenberg ladder using CORE method with various range $r$ using PBC. For comparison, we plot the best known extrapolations (Ref. 10) with arrows.

Figure 3

(Color online) CORE coefficients [see Eq. (6)] for two coupled triangles as a function of the inter-rung coupling $\alpha =J_{\Vert }∕J_{\perp }$. The parameters were computed using range-2 CORE. The coefficients in panel (iii) have been divided by their values in the perturbative limit. They therefore all start at 1.

Figure 4

GS energy per site and spin gap for a $3\times L$ Heisenberg torus with $J_{\Vert }∕J_{\perp }=0.25$. Results are obtained using the CORE method at various range $r$.

Figure 5

Same as Fig. 4 for the isotropic case $J_{\Vert }=J_{\perp }=1$.

Figure 6

(Color online) Spinon dispersion relation (see text) as a function of longitudinal momentum (in units of $\pi$). We only plot the lowest branch corresponding to $k_y=\pm 2\pi ∕3$. The odd lengths run from 5 to 13. The lines are guide to the eyes for an extrapolation on both sides of $\pi ∕2$.

Figure 7

(a) The plaquette lattice. Full lines denote the plaquette bonds $J$, dashed lines denote the inter-plaquette coupling $J^{\prime }$. (b) The trimerized kagomé lattice. Full lines denote the up-triangle $J$ bonds, dashed lines denote the down-triangle coupling $J^{\prime }$. The standard kagomé lattice is recovered for $J^{\prime }∕J=1$.

Figure 8

(Color online) Density matrix weights of the two most important states on a strong ($J$-bonds) plaquette as a function of $J^{\prime }∕J$. These results were obtained by ED with the original Hamiltonian on a $4\times 4$ cluster.

Figure 9

(Color online) Low energy spectrum of two coupled plaquettes. The states targeted by the CORE algorithm are indicated by arrows together with their $\mathrm{SU}\left(2\right)$ degeneracy.

Figure 10

(Color online) Triplet gap for effective system sizes between 20 and 52 sites, as a function of the interplaquette coupling $J^{\prime }∕J$. For $J^{\prime }∕J⩾0.5$ a simple extrapolation in $1∕N$ is also displayed. These results compare very well with ED results on the original model (Ref. 19).

Figure 11

(Color online) Tower of states obtained with a range-2 CORE Hamiltonian on an effective $N=36$ square lattice (9-site CORE cluster) in different reduced momentum sectors. The tower of states is clearly separated from the decimated magnons and the rest of the spectrum.

Figure 12

Staggered moment per site as a function of the rescaled applied staggered field for the plaquette lattice and different values of $J^{\prime }∕J$. Circles denote the approximate crossing point of curves for different system sizes. We take the existence of this crossing as a phenomenological indication for the presence of Néel LRO. In this way the phase transition is detected between $0.5<J_c^{\prime }∕J<0.6$, consistent with previous estimates. The arrows indicate curves for increasing system sizes: 20, 32, 36, 40 and also 52, 64 for the isotropic case.

Figure 13

(Color online) Density matrix weights of the different total spin states in a triangle of a 12 site kagomé cluster with $S=1∕2$ and $S=3∕2$ spins. These results are obtained for the homogeneous case $\alpha =1$.

Figure 14

(Color online) Spectrum of two coupled triangles in the kagomé geometry with $S=1∕2$ spins. The entire lowest band containing 16 states is successfully targeted by the CORE algorithm.

Figure 15

(Color online) Spectrum of two coupled triangles in the kagomé geometry with $S=3∕2$ spins. The 16 states targeted by the CORE algorithm are indicated by the arrows and their degeneracies.

Figure 16

(Color online) Coefficients of the CORE range-two Hamiltonian for two coupled $S=1∕2$ triangles. The coefficients in panel (iii) have been divided by their values in the perturbative limit.

Figure 17

(Color online) Coefficients of the CORE range-two Hamiltonian for two coupled $S=3∕2$ triangles. The coefficients in panel (iii) have been divided by their values in the perturbative limit.

Figure 18

(Color online) Tower of states obtained with a range-two CORE Hamiltonian on an effective $N=27$ kagomé lattice (9-site CORE cluster). There is a large number of low-lying states in each $S$ sector. The symbols correspond to different momenta.

Figure 19

(Color online) Spin gap of the kagomé $S=1∕2$ model on various samples, obtained with the CORE method (range-two and three). Exact diagonalization result are also shown for comparison where available.

Figure 20

(Color online) Logarithm of the number of states within the magnetic gap. Results obtained with the CORE range-two Hamiltonian. For comparison exact data obtained in Refs. 21, 22 are shown. The dashed lines are linear fits to the exact diagonalization data.

Figure 21

Definition of chirality $\epsilon$ (see text for details).

Figure 22

The two-triangle problem. $\alpha$ is the coupling ratio $J^{\prime }∕J$.

Figure 23

Three ways of coupling the three spins $S$ on a triangle into a total spin $1∕2$ state. Each construction is related to the two others by the $3j$ symbols (see text).