Entanglement-Guided Search for Parent Hamiltonians

We introduce a method for the search of parent Hamiltonians of input wave functions based on the structure of their reduced density matrix. The two key elements of our recipe are an ansatz on the relation between the reduced density matrix and parent Hamiltonian that is exact at the field theory level, and a minimization procedure on the space of relative entropies, which is particularly convenient due to its convexity. As examples, we show how our method correctly reconstructs the parent Hamiltonian correspondent to several nontrivial ground state wave functions, including conformal and symmetry-protected-topological phases, and quantum critical points of two-dimensional antiferromagnets described by strongly coupled field theories. Our results show the entanglement structure of ground state wave functions considerably simplifies the search for parent Hamiltonians.

Figure 1

Schematics of the parent Hamiltonian search. The starting point is the wave function of interest $|\mathrm{\Psi }⟩$ (with a half-partition reduced density matrix $\rho$), a set of local operators $\{O_{\alpha ,r}\}$, and an initial guess for their coefficients $w_{\alpha }^0$. The relative entropy $S$ between $\rho$ and an ansatz Bisognano-Wichmann reduced density matrix $\sigma (\{O_{\alpha ,r},w_{\alpha }^n\})$ is evaluated at each step of the minimization procedure. The procedure is then stopped once the desired accuracy ($\epsilon$) is reached: the final outcomes are the couplings $w_{\alpha }^{*}$ of the parent Hamiltonian, and the entanglement inverse temperature $\beta$. The inset shows a sample of our results for the bilayer Heisenberg model at the critical point (see text). Different lines correspond to minimization from different initial sets ${\overrightarrow{w}}_0$. Convergence within 0.1% of the correct value of $g$ (interlayer coupling; blue and green) and $\beta$ (red lines) is typically reached after 30 (gray area) and $\simeq 200$ steps, respectively.

Figure 2

Parent Hamiltonian search for the XXZ chain. (a) Ratios $J_{\alpha }=w_{\alpha }/\beta$ (blue lines) and $\beta$ (red lines) as functions of the minimization steps. We compare with the exact coupling and entanglement temperature. Here, $L=12$ and $\mathrm{\Delta }=1$. We consider the full basis of operators $\{w_{\alpha }\}$ (see text). (b) Error, Eq. (8), as function of the minimization steps; the value $\mathrm{\Delta }$ labels the different input states used, ground state of the respective $XXZ$ Hamiltonian. (c) System size scaling of the ratios of the average converged next-nearest neighbor (NNN) couplings $w_{\alpha }^{\mathrm{NNN}}$ and the nearest neighbor (NN) $w_{xx}=w_{xx}^{\mathrm{NN}}$ at $\mathrm{\Delta }=0.5$. Finite size effects are scaling to the exact value enlarging the system size. (d) Average convergence steps over initial realizations of the coupling as a function of system size. The convergence rate typically improves with system size.

Figure 3

Landscape of the relative entropy landscape between $\rho$ (obtained from the ground state at $\mathrm{\Delta }=1$) and ${\sigma }_{\mathrm{BW}}(\mathrm{\Delta },\beta )$ for $L=12$: the minimum is at (1.0,4.0). The insets shows a planar cut ($\mathrm{\Delta }=1$) for different system sizes.

Figure 4

Parent Hamiltonian search for the Haldane chain with $L=8$. (a) $J_{\alpha }=w_{\alpha }/\beta$ and $\beta$ as functions of the minimization steps; the initial state is at $\mathrm{\Delta }=1$. The inset shows a magnification up to 14 steps of $|1-w_{\alpha }/\beta |$: convergence to the correct solution at the ${10}^{-4}$ level is typically achieved after 6 steps. (b) Error as a function of minimization steps for different values of $\mathrm{\Delta }$.

Figure 5

(a)–(b) Error between the exact and the BW reduced density matrix, Eq. (8), as a function of temperature for the 2D Heisenberg model (a), and as a function of $g$ for the bilayer Heisenberg model at $\beta =3.3$ (b). (c) Error as a function of the minimization step for $L=8$; the error decreases quickly, without any plateau as in the 1D case. (d) Converged inter-to-intralayer coupling versus linear system size.