Ab initio optimization principle for the ground states of translationally invariant strongly correlated quantum lattice models

In this work, a simple and fundamental numeric scheme dubbed as ab initio optimization principle (AOP) is proposed for the ground states of translational invariant strongly correlated quantum lattice models. The idea is to transform a nondeterministic-polynomial-hard ground-state simulation with infinite degrees of freedom into a single optimization problem of a local function with finite number of physical and ancillary degrees of freedom. This work contributes mainly in the following aspects: (1) AOP provides a simple and efficient scheme to simulate the ground state by solving a local optimization problem. Its solution contains two kinds of boundary states, one of which play the role of the entanglement bath that mimics the interactions between a supercell and the infinite environment, and the other gives the ground state in a tensor network (TN) form. (2) In the sense of TN, a novel decomposition named as tensor ring decomposition (TRD) is proposed to implement AOP. Instead of following the contraction-truncation scheme used by many existing TN-based algorithms, TRD solves the contraction of a uniform TN in an opposite way by encoding the contraction in a set of self-consistent equations that automatically reconstruct the whole TN, making the simulation simple and unified; (3) AOP inherits and develops the ideas of different well-established methods, including the density matrix renormalization group (DMRG), infinite time-evolving block decimation (iTEBD), network contractor dynamics, density matrix embedding theory, etc., providing a unified perspective that is previously missing in this fields. (4) AOP as well as TRD give novel implications to existing TN-based algorithms: A modified iTEBD is suggested and the two-dimensional (2D) AOP is argued to be an intrinsic 2D extension of DMRG that is based on infinite projected entangled pair state. This paper is focused on one-dimensional quantum models to present AOP. The benchmark is given on a transverse Ising chain and 2D classical Ising model, showing the remarkable efficiency and accuracy of the AOP.

Figure 1

Three steps to implement ab initio optimization principle approach.

Figure 2

(a) The operator ${\widehat{F}}^{\partial }(s_i,s_{i+1})$ is written as a summation of ${\widehat{F}}^L(s_i,a)$ and ${\widehat{F}}^R(s_{i+1},a)$ with eigenvalue decomposition [Eq. (3)]. (b) Given by Eq. (4), $\widehat{\mathcal{F}}(S,a{a}^{\prime })$ with $S=(s_1,\cdots ,s_N)$ is obtained by acting ${\widehat{F}}^R(s_1,a)$ and ${\widehat{F}}^L(s_N,a^{\prime })$ on the first and last sites of $(\widehat{I}-\varepsilon {\widehat{H}}^B/2)$, respectively, with ${\widehat{H}}^B$ the bulk Hamiltonian of a supercell. (c) The relation between the total Hamiltonian $\widehat{H}$ and the operator $\widehat{\mathcal{F}}(S,a{a}^{\prime })$ given by Eq. (5).

Figure 3

Graphic representations of (a) the optimization function $\mathcal{F}$ in Eq. (7) that is to be maximized, (b) the operator $\widehat{\mathcal{M}}$ in Eq. (8), and (c) $\widehat{\mathcal{H}}$ in Eq. (9) that consists of (d) the bulk and boundary parts.

Figure 4

From the self-consistent equations [Eqs. (10, 11, 12)], the optimization function $\mathcal{F}$ in Eq. (7) can be written as $\mathcal{F}=\langle \mathrm{\Phi }\left|\widehat{\varrho }\right|\mathrm{\Phi }\rangle$, where one has $\widehat{\varrho }=\widehat{I}-\varepsilon \widehat{H}$ (green shadow) and $|\mathrm{\Phi }\rangle$ is the ground state in an MPS form (red shadow) given by Eq. (13). Then, from the eigenequation $\widehat{\varrho }|\mathrm{\Phi }\rangle =C|\mathrm{\Phi }\rangle$, an infinite TN can be constructed.

Figure 5

Graphic representations of (a) the tensor ring decomposition (TRD) given by Eq. (14) and (b) the rank-1 decomposition [24, 38] by Eq. (16). The rank-1 decomposition is special TRD with the ring rank $\chi =1$.

Figure 6

(a) The bond energy $E_{i,i+1}=\langle {\widehat{H}}_{i,i+1}\rangle$ versus position $i$ at $h=0.5, \chi =6$, and $\varepsilon ={10}^{-4}$ for different sizes of the supercell $N$. $E_{\mathrm{exact}}$ is the exact solution of the infinite chain by fermionization [48]. $E_{0,1}$ (and $E_{N,N+1}$) is the bond energy between two adjacent supercells. In the middle of the chain, $E_{i,i+1}$ converges to $E_{\mathrm{exact}}$ as $N$ increases, where the error is about $O\left({10}^{-7}\right)$. (b) For $N=10$ and $h=0.5, E_{1,2}$ (on the boundary of the supercell) and $E_{5,6}$ (in the middle) versus $\chi$ at $\varepsilon ={10}^{-3}$ and ${10}^{-4}$. One can see that the boundary effect decay with $\chi$, where $E_{5,6}$ converges accurately to $E_{\mathrm{exact}}$. The systematic error of $E_{1,2}$ is caused by the Trotter-Suzuki error $\sim O\left({\varepsilon }^2\right)$. By fitting at $\varepsilon ={10}^{-4}$, an exponential convergence $E_{5,6}=E_{\mathrm{exact}}-e^{-1.179\chi -8.2704}$ is found. For comparison, $E_{1,2}$ and $E_{5,6}$ obtained by exact diagonalization (ED) at $N=12$ with open and periodic boundary conditions are also shown.

Figure 7

The relative error of the average energy per site given by Eq. (21). By fitting, it is found that $\mathrm{\Delta }E$ satisfies a logarithmic relation with $N$ [Eq. (22)].

Figure 8

The entanglement entropy $S$ versus (a) the ring rank $\chi$ with different supercell size $N$ and (b) versus $N$ with different $\chi$. One can see that $S$ scales linearly with both ${log}_2\chi$ and ${log}_{2}N$, consistent with the conformal field theory and the former numeric results [49, 50, 51, 52, 53].

Figure 9

(a) The TN of the partition function of 2D Ising model on square lattice is formed by one inequivalent local tensor $F$ given in Eq. (24). The yellow shadow shows a tensor cluster with $L_x=3$ and $L_y=2$. (b) For different ($L_x,L_y$), the log-log plot of the error of free energy (per site) versus $\chi$ is shown at the critical temperature $T_c=2/ln(1+\sqrt{2})$, indicating an algebraic scaling between the error and $\chi$.

Figure 10

(a) Sketch of the optimization function $\mathcal{F}$ for a 2D quantum system, where the operator $\widehat{\mathcal{F}}$ is located in the center. The $x$ and $y$ denote the two spatial dimensions of the 2D model. The black bonds in the $z$ direction represent the physical indexes of the operator $\widehat{\mathcal{F}}$, and the ground state is a PEPS formed by the blue tensor $A$. The four red tensors $X^I, X^{II}, Y^I$, and $Y^{II}$ are the boundary tensors with ancillary indexes that play the role of “entanglement bath.” (b) The operator $\widehat{\mathcal{F}}$ of a 2D quantum system on a square lattice is constructed by choosing a square as the supercell. (c) The sketch of tensor ring decomposition (TRD) of a 3D TN, where the optimization function $\mathcal{F}$ is maximized.

Figure 11

The convergence of the relative error of the bond energy in the middle of the supercell $\mathrm{\Delta }{E}_{3,4}$ along with iteration time. The parameters are $h=0.5, N=6, \chi =8$, and $\varepsilon ={10}^{-4}$. One can see in (a) that with the standard AOP, $\mathrm{\Delta }{E}_{3,4}$ converges to $O\left({10}^{-6}\right)$ with only about 30 iterations. For comparison, in the contraction-truncation scheme (see the details in the main text in Appendix pp1), it takes $O\left({10}^4\right)$ iterations to converge approximately to $O\left({10}^{-4}\right)$. The iterations are stopped when the change of $\mathrm{\Delta }{E}_{3,4}$ after one iteration is smaller than ${10}^{-10}$. To reach such a convergence, the CPU time that AOP and contraction-truncation scheme takes are 69 and 3557, respectively.