Exact Diagonalization of Heisenberg $\mathrm{SU}(N)$ Models

Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of $\mathrm{SU}(N)$, an orthonormal basis labeled by the set of standard Young tableaux in which the matrix of the Heisenberg $\mathrm{SU}(N)$ model (the quantum permutation of $N$-color objects) takes an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of the singlet space on $n$ sites increases very fast with $N$, this formulation allows us to extend exact diagonalizations of finite clusters to much larger values of $N$ than accessible so far. Using this method, we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid for SU(10).

Figure 1

(a) Example of a Young tableau: $\alpha =[3,2,2]$; (b) Integers $d_{i,N}$ that enter the numerator of the dimension of $\alpha$; (c) Hook lengths $l_i$; (d) Examples of standard tableaux ranked according to the last letter sequence. (e) Normal product state $|{\mathrm{\Phi }}_1^{[3,2,2]}⟩=|AAABBCC⟩$.

Figure 2

Real-space correlations $⟨P_{0j}⟩-1/N$ for various $\mathrm{SU}(N)$ models and cluster sizes on the square lattice with periodic boundary conditions: SU(5) (tilted 25 and 20 site cluster), SU(8) (16 sites), and SU(10) (20 sites). The black dot is the reference site 0. Positive (negative) correlations are depicted as blue (red) disks with an area proportional to the absolute value of the correlation. The correlations for SU(5) on the ($5\times 5$) 25-site cluster are shown and discussed in the Supplemental Material [28]. Table: dimension $f^{[n/N,\dots ,n/N]}$ of the singlet subspace in which the permutation Hamiltonian has been diagonalized, approximate dimension $(n-1)!/(n/N){!}^N$ of the Hilbert space used in standard ED [29], ground states energies per site ${\mathcal{E}}_{\mathrm{GS}}$.

Figure 3

Energy spectra (in units of $J$) for SU(5) (20 sites), SU(8) (16 sites), and SU(10) (20 sites), plotted as a function of the quadratic Casimir operator $C_2$ (See Refs. [28, 38]). Different irreps with the same $C_2$ [e.g., $C_2=2$ for SU(5)] are represented with different colors. Below the SU(5) tower of states, sketch of the long-ranged color ordered pattern consistent with the real-space correlations of Fig. 2 and with flavor-wave theory.

Figure 4

Dimer-dimer correlations $⟨P_{ij}{P}_{\mathrm{ref}}⟩-⟨P_{\mathrm{ref}}{⟩}^2$ for SU(8) (16 sites) and SU(10) (20 sites). The reference bond is shown in black while positive (negative) correlations are shown as solid blue (dashed red) lines, with a thickness proportional to the dimer-dimer correlation.