Matrix product states - Wave-function based approach for the RDMF 145

9. Wave-function based approach for the RDMF 145

9.6. Matrix product states

The use of matrix product states (MPS, [White, 1992, 1993; Östlund and Rommer, 1995;

Schollwöck, 2005; Schollwöck, 2011]⁶) as the ansatz for the many-particle wave functions in solid-state physics and quantum chemistry [Chan and Sharma, 2011; Marti and Reiher, 2011; Kurashige, 2014] has emerged as a new powerful method. We have discussed matrix product states (MPS) and matrix product operators (MPO) in detail in section 8.8. As a systematically improvable ansatz, MPS are suitable for the evaluation of the density-matrix functional.

9.6.1. Constrained optimization over MPS

We propose to solve the minimization problem of the density-matrix functional given by Eq. (9.27) and the constraints in Eq. (9.28) and Eq. (9.29) with MPS-parametrizations of the many-particle wave functions. For simplicity we only consider a single many-particle wave function at zero temperature, i.e. Levy’s functional F_Levy^W^ˆ [ρ⁽¹⁾] of Eq. (5.25). The problem to be solved is

F_Levy^W^ˆ [ρ⁽¹⁾] = min

{|Ψi}hΨ|Wˆ|Ψi (9.135)

with the constraints

1 = hΨ|Ψi, (9.136)

ρ⁽¹⁾_β,α =^X

hΨ|ˆc^†_αˆc_β|Ψi. (9.137) The corresponding Lagrange function is

L(|Ψi, λ, h_α,β) = hΨ|Wˆ|Ψi −λ(hΨ|Ψi −1)−^X

α,β

h_α,βhΨ|ˆc^†_αcˆ_β|Ψi −ρ⁽¹⁾_β,α (9.138) with the Lagrange multipliers λ for the norm constraint and h_α,β for the density-matrix constraint. For an MPS-representation of the many-particle wave function

|Ψi= ^X

j1,...,jL∈{0,↑,↓,↑↓}

M^(1),j¹M^(2),j²· · ·M^(L),j^L|j₁, ..., j_Li (9.139)

M^(1),j¹M^(2),j²· · ·M^(L),j^L = (9.140)

the entries of the matrices M^(i),jⁱ are the variational parameters and one would obtain a highly nonlinear constrained minimization problem. To avoid this non-linearity, we adopt the DMRG-like iterative minimization described in section 8.8.4. We propose to use the augmented-Lagrangian formalism described in section 9.3.3 to map the constrained min-imization problem to a series of unconstrained minmin-imization problems. The augmented Lagrangian that corresponds to the given constrained minimization problem is

L(|Ψi, λ, µλ, hα,β, µα,β) =L(|Ψi, λ, hα,β) + µ_λ

2 (hΨ|Ψi −1)² +^X

α,β

µ_α,β 2

hΨ|ˆc^†_αˆc_β|Ψi −ρ⁽¹⁾_β,α². (9.141)

6A more complete list of references can be found in the reviews [Schollwöck, 2005], [Verstraete et al., 2008] and [Schollwöck, 2011].

9.6. Matrix product states The unconstrained subproblems defined by Eq. (9.48) have here the form

min|Ψi L(|Ψi, λ, µ_λ, h_α,β, µ_α,β) (9.142) with fixed Lagrange multipliers and penalty parameters. We propose to solve an un-constrained problem with a DMRG-like iterative minimization. That means, that the minimization is performed stepwise by only minimizing over a subset of the matrices, i.e.

a single M^(l),j^l in the single-site approach or M^(l),j^l and M^(l+1),j^l+1 in the two-site ap-proach, while the other matrices are not changed. We consider the two-site approach and assume that the in the current step the matrices M^(l),j^l and M^(l+1),j^l+1 at site l and l+ 1 are to be optimized. We also assume that the MPS representation is in a mixed-canonical form

|Ψi= ^X

j1,...,jL

A^(1),j¹· · ·A^(l−1),j^l−1M^(l),j^lM^(l+1),j^l+1B^(l+2),j^l+2· · ·B^(L),j^L|j₁, ..., j_Li (9.143) with left-normalized matrices A^(i),jⁱ and right-normalized matricesB^(i),jⁱ. The MPOs for the interaction operator ˆW,

Wˆ = ^X

j1,...,jL,j₁⁰,...,j_L⁰

W^(1),j¹^,j¹⁰W^(2),j²^,j²⁰ · · ·W^(L−1),j^L−1^,j^L−1⁰ W^(L),j^L^,j⁰^L|j₁, ..., j_Lihj₁⁰, ..., j_L⁰|

W^(1),j¹^,j¹⁰W^(2),j²^,j⁰²· · ·W^(L−1),j^L−1^,j^L−1⁰ W^(L),j^L^,j⁰^L = (9.144) and the operators ˆc^†_αˆc_β

c^†_αcˆβ = ^X

j1,...,jL,j₁⁰,...,j_L⁰

C^(1),j¹^,j

0 1

α,β · · ·C^(L),j^L^,j

0 L

α,β |j1, ..., jLihj₁⁰, ..., j_L⁰ |

C^(1),j¹^,j

0 1

α,β · · ·C^(L),j^L^,j

0 L

α,β = (9.145)

are constructed. The minimization will be performed over elements ofP^(l),j^l^,j^l+1, which is the contraction

P^(l),j^l^,j^l+1 =M^(l),j^lM^(l+1),j^l+1, (9.146) to make use of the larger ansatz space. Then we implicitly form the local matrix-product representations of the interaction operator ˆW,

W^(l),j^l^,j^l+1^,j

0 l,j⁰_l+1

il−1,il+1,i⁰_l−1,i⁰_l+1 = , (9.147)

and the corresponding local matrix-product representations of the operators ˆc^†_αˆc_β

C^(l),j^l^,j^l+1^,j

0 l,j_l+1⁰

α,β;il−1,i_l+1,i⁰_l−1,i⁰_l+1 = . (9.148)

Please note, that at this point we don’t restricted the form of the interaction operator.

The derivative of the augmented Lagrangian in Eq. (9.141) with respect to P^(l),j^l^,j^l+1 takes the form

∂L(P^(l),j^l^,j^l+1, λ, µ_λ, h_α,β, µ_α,β)

∂(P^(l),j^l^,j^l+1)^∗ = ^X

j_l⁰,j⁰_l+1

W^(l),j^l^,j^l+1^,j^l⁰^,j⁰^l+1P^(l),j⁰^l^,j^l+1⁰

−^hλ−µ_λc_normP^(l),j^l^,j^l+1ⁱP^(l),j^l^,j^l+1

−^X

α,β

hh_α,β−µ_α,βc_α,βP^(l),j^l^,j^l+1ⁱ ^X

j_l⁰,j_l+1⁰

C^(l),j^l^,j^l+1^,j⁰^l^,j^l+1⁰ P^(l),j^l⁰^,j⁰^l+1

(9.149) with

c_norm(P^(l),j^l^,j^l+1) = ^X

j⁰_l,j_l+1⁰

P^(l),j⁰^l^,j^l+1⁰ P^(l),j^l⁰^,j^l+1⁰ ^∗−1 (9.150) c_α,β(P^(l),j^l^,j^l+1) = ^X

jl,jl+1,j⁰_l,j_l+1⁰

C^(l),j^l^,j^l+1^,j^l⁰^,j^l+1⁰ P^(l),j^l⁰^,j^l+1⁰ P^(l),j^l^,j^l+1^∗ −ρ⁽¹⁾_β,α. (9.151) In contrast to the minimization of the total energy over MPS discussed in section 8.8.4, the derivative in Eq. (9.149) is not linear inP^(l),j^l^,j^l+1 and the minimization condition can-not be written as an eigenvalue problem. However, we can solve the local minimization problem overP^(l),j⁰^l^,j^l+1⁰ with a conjugate-gradient approach or the limited memory BFGS-method [Broyden, 1970; Fletcher, 1970; Goldfarb, 1970; Shanno, 1970]. We have imple-mented this scheme with the MPS-library ITensor⁷ by extending the existing DMRG-like ground state search to the minimization of the augmented Lagrangian with a conjugate-gradient method. This scheme could also be used to enforce other constraints on the MPS as long as the constraints can be written as an expectation value of some operator.

9.6.2. Example results

We investigate the convergence of the iterative solution of the constrained minimization problem proposed in the previous section. We consider the case of a half-filled 8-site Hub-bard chain with an interaction strength ofU/t= 2. We have obtained the ground state of this system with the total-energy minimization implemented in ITensor. The ground state can be represented as an MPS with a maximal bond dimension of 120 and a maximal trun-cation error of 10⁻¹⁰. We have also obtained the one-particle reduced density matrix ρ⁽¹⁾ of this ground state and consider the evaluation of the density-matrix functional F^W^ˆ[ρ⁽¹⁾] for this one-particle reduced density matrix with the approach presented in section 9.6.1.

Figure 9.5 shows the interaction energy hΨ|Wˆ|Ψiafter each solution of an unconstrained subproblem, i.e. a two-site minimization problem. The convergence with the number of sweeps is moderately slower than the convergence of the total energy in a total-energy minimization for the same system, which is also shown in Figure 9.5 for comparison. A complete sweep through the 8-site system consists of 14 two-site problems. However, ev-ery minimization of an unconstrained subproblem is much more computationally involved than in a DMRG-ground-state search, because it is a non-linear problem and instead of

7ITensor C++ library (version 2.1.1), http://itensor.org/

9.6. Matrix product states

0 1 2 3

1 15 29

density-matrix functional evaluation total-energy minimization

de vi at io n fr om fin al va lu e/ t

step i

Figure 9.5.: Convergence of the interaction energy (red line) during the DMRG-like con-strained minimization algorithm proposed in section 9.6.1 for the one-particle reduced density matrix of the numerically exact ground state of a half-filled 8-site Hubbard chain with interaction strength U/t = 2. A step denotes the solution of the two-site unconstrained minimization problem at a bond. The black line shows the convergence of the total energy during a conventional DMRG-like total-energy minimization. One sweep through the 8-site system consists of 14 steps. Thus, the figure shows three sweeps.

one MPO for the Hamiltonian there are ∈ O(L²) MPOs for the interaction Hamiltonian and the operators ˆc^†_αcˆ_β. Figure 9.6 shows the maximal absolute constraint violation af-ter each solution of an unconstrained subproblem. The presented results show that the modification of the DMRG-like total-energy minimization to a DMRG-like augmented-Lagrangian based iterative solution of the constrained minimization problem over matrix product states preserves the fast convergence speed in terms of sweeps of the DMRG-like total energy minimization. Together with a reduction of the entanglement entropy by a transformation of the one-particle basis discussed in section 8.8, the parametrization of a many-particle wave function as a matrix product state could be a valuable approach for the evaluation of the density-matrix functional within hybrid theories that combine DFT and RDMFT. The transformation of the one-particle basis does not lead to an increase of the bond dimension of the MPOs here because the MPO of the interaction Hamiltonian is not affected by the unitary transformation and the elements of the one-particle reduced density matrix are evaluated in the transformed basis.

0.0001 0.001 0.01 0.1 1

1 15 29

m ax . ab s. co ns tr ai nt vi ol at io n

step i

Figure 9.6.: Convergence of the maximal absolute constraint violation during the DMRG-like constrained minimization algorithm proposed in section 9.6.1 for the one-particle reduced density matrix of the numerically exact ground state of a half-filled 8-site Hubbard chain with interaction strength U/t = 2. A step denotes the solution of the two-site unconstrained minimization problem at a bond. One sweep through the 8-site system consists of 14 steps. Thus, the figure shows three sweeps.

Im Dokument New methods for the ab-initio simulation of correlated systems (Seite 184-189)