Renormalization algorithm for the calculation of spectra of interacting quantum systems

We present an algorithm for the calculation of eigenstates with definite linear momentum in quantum lattices. Our method is related to the density matrix renormalization group, and makes use of the distribution of multipartite entanglement to build variational wave functions with translational symmetry. The algorithm is applied to the study of bilinear-biquadratic $S=1$ chains, in particular to the region of phase space between the dimerized and ferromagnetic phases.

Figure 1

Schematic picture of the projected entangled multipartite states presented in the text.

Figure 2

Representation of the operators defined in Eqs. (14), which are used in the actualization of the effective Hamiltonian and norm operators. Empty and filled squares correspond to $E_1^{n,d}$ and $E_{{\sigma }^{\mu }}^{n,d}$, respectively.

Figure 3

Representation of Eqs. (15). Parentheses represent sites where matrices $A_{\left[n\right]}^s$ appear in (9). (•) corresponds to terms of the form $\chi \otimes {\left(A_{\left[n\right]}^s\right)}^{*}$ or $A_{\left[n\right]}^s\otimes \chi$, whereas $\left(\mu \right)$ corresponds to contractions with the operator ${\sigma }^{\mu }$, that is, terms of the form: ${\sum }_s\chi \otimes {\left(A_{\left[n\right]}^t\right)}^{*}{\sigma }_{t,s}^{\mu }$ or ${\sum }_s\left(A_{\left[n\right]}^t\right){\sigma }_{t,s}^{\mu }\otimes \chi$.

Figure 4

Representation of the procedure for initialization and actualization of operators in the calculation of $e^{n+1,n+d-1}$. Empty and gray squares are left and right operators, respectively, as defined in the text.

Figure 5

Actualization of the operator $h^{n+1,n+d-1}$ by using left and right operators.

Figure 6

Lowest states of a bilinear-biquadratic $S=1$ chain, $N=40$ sites, $D=10$. (a) $\theta =-\pi ∕2$, $E_0=-2.7976N$, (b) $\theta =-0.74\pi$, $E_0=-1.4673N$. Empty circles: lowest energy states. Filled circles: first branch of excitations. Estimated absolute error $\Delta E_k\approx 5\times {10}^{-3}$. Inset (a): error in the absolute energies $E_0^{\left[0\right]}$, $E_{\pi }^{\left[0\right]}$ as a function of $D$, estimated by comparison with $D=14$ calculations, for $\theta =-0.5\pi$. The convergence of the algorithm has also been checked with other values of $k$, yielding similar results. Inset (b): relative error as a function of $N$ in the first excited state energy with $k=0$, $\theta =-\pi ∕2$.

Figure 7

(a) Extrapolated dimer order parameter. (b) Exponent of the squared quantum nematic order parameter. In both cases $8<N<36$, and $D=12$ ($D=14$ for $N>28$).

Figure 8

(a) Finite size scaling of the gap between the lowest $k=0$, and $k=\pi$ states. The lines are the extrapolation to $N\to \infty$. (b) Scaled gap between the lowest energy $k=0$ states. In both plots $D=12$ ($D=14$, for $N=32$), and the error in $E_k$ is $={10}^{-3}$.