Density of States of Quantum Spin Systems from Isotropic Entanglement

We propose a method that we call isotropic entanglement (IE), which predicts the eigenvalue distribution of quantum many body (spin) systems with generic interactions. We interpolate between two known approximations by matching fourth moments. Though such problems can be QMA-complete, our examples show that isotropic entanglement provides an accurate picture of the spectra well beyond what one expects from the first four moments alone. We further show that the interpolation is universal, i.e., independent of the choice of local terms.

Figure 1

The exact diagonalization in dots and IE compared to the two approximations. The title parameters are explained in the section on numerical results.

Figure 2

An example where $\beta =1$: the quantum problem for all $d$ lies in between the iso ($p=0$) and the classical ($p=1$).

Figure 3

$N=11$ with full rank Wishart matrices as local terms.

Figure 4

Approximating the quantum spectrum by $H^{\mathrm{IE}}=\sum _{l=1}^2{Q}_l^T{H}_{l,\cdots ,l+4}{Q}_l$, where the interaction is taken to be 5-local Wishart matrices.

Figure 5

Local terms have a random binomial distribution.

Figure 6

Summary of isotropic entanglement. We have denoted the known classical and isotropic convolutions by $*$ and ${⊞}_{\mathrm{iso}}$, respectively. IE approximates the unknown quantum convolutions ${⊞}_q$.

Figure 7

IE vs Pearson and Gram-Charlier fits for local terms with binomial distribution.

Figure 8

IE vs Pearson and Gram-Charlier fits for local terms with Wishart distribution.