Reconstructing entanglement Hamiltonian via entanglement eigenstates

Entanglement Hamiltonian holds crucial clues in understanding quantum entanglement and its underlying phenomenon in strongly correlated systems. To date, however, a generic recipe to map out the operator form of entanglement Hamiltonian remains outstanding. Here, we present a systematic framework which explicitly reconstructs entanglement Hamiltonian based on the information of one entangled mode of the reduced density matrix. We demonstrate its successful application to quantum spin lattice models. The obtained entanglement Hamiltonians accurately recover the expectations from analytical theories and faithfully capture all features of the reduced density matrices, which are evidenced by the agreement between the original and reconstructed full entanglement spectra and the high density matrix fidelity.

Figure 1

(a) One-dimensional spin-$1/2$ chain and (b) spin ladder made of two coupled periodic spin-$1/2$ chains. The dashed line shows the entanglement bipartition into two subsystems $A$ (red) and $B$ (blue), each of which encloses $L$ sites.

Figure 2

(a) Spatially varying coupling strengths $J_{n,n+1}$ of the parent Hamiltonian of bipartition of a spin-$1/2$ Heisenberg chain. The black dashed lines show the CFT predicted envelope function $f_{\mathrm{env}}\left(\tilde{n}\right)=\tilde{n}(1-\tilde{n})$ and $\tilde{n}=\frac{n+(n+1)}{2L}$. (b) Direct comparison $-log{p}_n$ of the ES, $q_n$ by Eq. (C10) and the eigenvalues ${\varepsilon }_n$ of parent Hamiltonian $H_P$. The black line represents the best linear fit, with the slope $\sim 1.00032$ and intercept $\sim {10}^{-5}$. Inset: one-to-one comparison of $-log{p}_n$ of the ES and eigenvalues ${\varepsilon }_n$ grouped by quantum number $S_A^z$ in subsystem $A$. Different symbols show the results computed on $2\times L$ systems: blue triangular $\left(L=10\right)$, red circles $\left(L=12\right)$, green diamonds $\left(L=14\right)$, and navy squares $\left(L=16\right)$.

Figure 3

(a) EH parameters $J_n/J_1$ versus $\theta$ (up to fourth nearest neighbor) for a given size $L=12$. Inset: density matrix fidelity $F({\rho }_A,\varrho )$ between ${\rho }_A$ and $\varrho$ versus $\theta$. (b) Comparison EH parameters $J_2/J_1$ and the perturbation theory Eq. (9) (red dashed line). Different symbols stand for $L=8$ (green), $L=10$ (blue), and $L=12$ (black). $\theta \to \pi /2$ relates to the strong interchain coupling limit. (c) Wave function fidelity $|\langle {\varphi }_E^0|{\phi }_A^0\rangle |$ as a function of $\theta$, where $|{\phi }_A^0\rangle$ and $|{\varphi }_E^0\rangle$ is the ground state of ${\widehat{H}}_A$ and $H_E$, respectively.

Figure 4

(a) Spatially varying coupling strengths $J_{n,n+1}$ of the parent Hamiltonian of bipartition of a one-dimensional spin-$1/2$ chain model at $XY$ point $\left(\mathrm{\Delta }=0\right)$. The black dashed lines show the CFT predicted envelope function $f_{\mathrm{env}}\left(\tilde{n}\right)=\tilde{n}(1-\tilde{n})$ and $\tilde{n}=\frac{n+(n+1)}{2L}$. (b) Direct comparison $-log{p}_n$ of the ES, $q_n$ by Eq. (C10) and the eigenvalues ${\varepsilon }_n$ of parent Hamiltonian $H_P$. The black line represents the best linear fit, with the slope $\sim 1.000090$ and intercept $\sim {10}^{-9}$. Inset: one-to-one comparison of $-log{p}_n$ of the ES and eigenvalues ${\varepsilon }_n$ grouped by quantum number $S_A^z$ in subsystem $A$. (c) Inverse participation ratio of eigenstates of parent Hamiltonian $H_P$. Different symbols show the results computed on $2\times L$ systems: blue triangular $\left(L=10\right)$, red circles $\left(L=12\right)$, green diamonds $\left(L=14\right)$, and navy squares $\left(L=16\right)$.

Figure 5

(a) Entanglement spectra $\left({\xi }_i-{\xi }_0\right)$, obtained from reduced density matrix, are grouped by total momentum $K$ along the chain direction. (b) Energy spectra $\left(E_i-E_0\right)$ of reconstructed entanglement Hamiltonian $H_E$. In (a),(b), the lowest spectra branch is fitted as $v|sinK|$ by red dashed line. (c) Direct comparison of entanglement spectra $\left({\xi }_i-{\xi }_0\right)$ and energy spectra $\left(E_i-E_0\right)$. All low-lying spectra are computed on $2\times L$ ladders shown in black circles $\left(L=10\right)$, green squares $\left(L=12\right)$, and blue diamonds $\left(L=14\right)$. Here we set $\theta =\pi /3$ and $\mathrm{\Delta }=1.0$.

Figure 6

Coupling strength of entanglement Hamiltonian of spin-$1/2XXZ$ model: $\widehat{H}={\sum }_{n=1}^{2L}{\widehat{h}}_{n,n+1}={\sum }_{n=1}^{2L}{S}_n^x{S}_{n+1}^x+S_n^y{S}_{n+1}^y+\mathrm{\Delta }{S}_n^z{S}_{n+1}^z$. Different panels show various $\mathrm{\Delta }>1$. The subsystem size is chosen to be $L_A=12$. The squared dots and crossed dots respectively represent the longitudinal and transverse coupling strength $J_{n,n+1}^{zz},J_{n,n+1}^{xy}$. The dashed lines show various envelope function for entanglement Hamiltonian $f_{\mathrm{CFT}}\left(x\right)=B(L_A-x)x/L_A(B$ is a renormalized factor).