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

Abstract

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 mimic the interactions between a supercell and the infinite environment, and the other give 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 clue 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, and etc., providing a unified perspective that is previously missing in this fields; 4) AOP as well as TRD gives novel implications to existing TN-based algorithms: a modified iTEBD is suggested and the 2D AOP is argued to be an intrinsic 2D extension of DMRG that is based on infinite projected entangled pair state.