Entanglement distance between quantum states and its implications for a density-matrix renormalization group study of degenerate ground states

We study the concept of entanglement distance between two quantum states, which quantifies the amount of information shared between their reduced density matrices (RDMs). Using analytical arguments combined with density-matrix renormalization group (DMRG) and exact diagonalization (ED) calculations, we show that for gapless systems the entanglement distance has power law dependence on the energy separation and subsystem size, with ${\alpha }_E$ and ${\alpha }_{\ell }$ exponents, respectively. Using conformal field theory (CFT) we find ${\alpha }_E=2$ and ${\alpha }_{\ell }=4$ for Abelian theories with $c=1$, as in the case of free fermions. For non-Abelian CFTs ${\alpha }_E=0$, and ${\alpha }_{\ell }$ is twice the conformal dimension of the thermal primary fields. For instance, for $Z_3$ parafermion CFT ${\alpha }_E=1$ and ${\alpha }_{\ell }=4/5$. For gapped 1+1 dimensional (1+1D) fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of ${\alpha }_E$ and ${\alpha }_{\ell }$. These two branches are related to momentum transfers near zero and $\pi$. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground states (GSs) of 2+1D topological states have maximum entanglement distance. In contrast, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we discuss the computational complexity of DMRG algorithms that are capable of finding all degenerate GSs.

Figure 1

The two measures of entanglement distance between the GS and excited states obtained by creating an electron hole excitation $\left(c_{k_F+\mathrm{\Delta }k}^{†}{c}_{k_F}\right)$ in gapless 1+1D free fermions with nearest neighbor hopping $t_1=1$ at half-filling, and $\mathcal{N}=1000$. The blue (orange) color represents excitations with $\left|\mathrm{\Delta }k\right|<\pi /2$ ($>\pi /2$) momentum transfer. Insets are log-log plots showing the power law behavior for small $\ell$ and $\mathrm{\Delta }E$. (a) ${\epsilon }_1$ for the two lowest excited states vs $\ell$. (b) ${\epsilon }_1$ for all electron-hole excited states ($\ell =50$). There are oscillations of period $\mathcal{N}/\ell$ around the saturation point. (b),(c) Similar quantities for ${\epsilon }_2^{\chi }$ with $\chi =2^6$. Both measures for entanglement distance grow monotonically with $\ell$. They also start growing for small $\mathrm{\Delta }E$ and then saturate and oscillate around the saturation point. The power law growth of entanglement distance for small $\ell$ and $\mathrm{\Delta }E$ suggests ${\alpha }_E=1.9$ (1.6), ${\alpha }_{\ell }=3.9$ (3.9), and ${\beta }_{\ell }\left(2^6\right)=0.86$ (0.98) for the blue (orange) branch, close to our theoretical predictions.

Figure 2

DMRG results with $\chi =20$ for the entanglement distance metrics between the GS and excited states of a critical $Z_3$ parafermion chain (equivalent to a three-state Potts model) of length $\mathcal{N}=100$. These results suggest that ${\alpha }_E$ is approximately vanishing and ${\alpha }_{\ell }=0.82$, indicating the scaling dimension of the thermal operator must be around 0.41; both are consistent with our theoretical predictions. Also, we find that ${\beta }_E\left(\chi \right)=\chi /20+1.3$ ($\ell =30$) with a high accuracy (see Appendix pp2 for more results) and ${\beta }_{\ell }\left(\chi \right)$ has a weaker $\chi$ dependence.

Figure 3

The same plot as Fig. 1 for gapped 1+1D free fermions with staggered chemical potential ${\mu }_n={(-1)}^n$. The excited states are bifurcated into two branches distinguished by the momentum transfer. The power law behavior of the entanglement distance suggests ${\alpha }_{\ell }=3.8$ (2.2) for the blue (orange) branch.

Figure 4

The same plot as Fig. 2, for a gapped $Z_3$ parafermion chain in the trivial phase. Both metrics of entanglement distance saturate when $\ell$ becomes comparable with the correlation length.

Figure 5

Comparing the results of the single-state DMRG with the first multistate DMRG algorithm discussed in the paper for the lowest energy eigenvalues. We consider a Laughlin state at $1/3$ filling, torus geometry, with $N_e=12$ electrons and $L_y=15$ circumference. We have considered Haldane's $V_1$ pseudopotential [53] for the interaction term. We have implemented center-of-mass momentum conservation mod $N_e$, thus the three topological sectors have the same total momentum (mod $N_e$). The single-state DMRG with $\chi =150$ finds only one of the three degenerate ground states (enlarging bond dimension does not help in finding more states). On the other hand, the first multistate DMRG algorithm finds all three degenerate ground states for both $\chi =150$, and $\chi =450$. Although the ground-state energy of $\chi =150$ is nonzero and larger than its true value, the excitation energies ($E_i-E_0$) are estimated well. Furthermore, we see that the first multistate DMRG with ${\chi }_{\mathrm{off}}=450$ gives a ground-state energy close to zero (and that of the single-state DMRG method targeting one ground state), consistent with our theoretical expectations [see Eq. (13)].

Figure 6

Comparison of the results of the single-state DMRG with the second multistate DMRG algorithm introduced in this paper for the lowest energy eigenvalues. In this case there is a $Z_3$ parafermion chain of length $\mathcal{N}=90$ with six domain walls leading to $\left(\text{GSD}\right)=27$. Again single-state DMRG cannot find all ground states even with ${\chi }_{\mathrm{tot}}=80$, while the second multistate DMRG can easily find all 27 states with ${\chi }_{\mathrm{tot}}=20$.

Figure 7

DMRG results for ${\epsilon }_2^{\chi }$ in a critical $Z_3$ parafermion chain of $N=100$ total length measured at $\ell =30$ for three different values of bond dimension, $\chi$. These results suggest that ${\beta }_E\left(\chi \right)=\chi /20+1.3$ ($\ell =30$) with a high accuracy.