DMRG truncation

Let us again assume that we are interested in calculating the ground state of a quantum lattice system, e.g., a simple spin chain as in Eq. (3.1). Let us now divide the system into two subsystems which we call “block A” and “block B”. For each of these blocks we have a complete orthonormal basis, {|ii} for blockA and {|ji}

for blockB (see Fig. 3.2). The basis dimension of blockA (B) isD_A (D_B).

Figure 3.2.: Division of the system into two blocks. The red circles mark the “sites”, i.e., the spins.

A normalized state |ψiof the complete system, also called “superblock” (because it consists of two “blocks”), can be expanded in the following way:

|ψi=

DA

X

i=1 DB

X

j=1

ψij|ii|ji, X

i,j

|ψ_ij|²= 1. (3.4)

The state |ψi can, e.g., be the ground state of the system, but the following state-ments are true for an arbitrary (normalized) superblock state. Before we proceed with the derivation of the “optimal” truncation scheme, we shortly recapitulate the concept of reduced density matrices [87].

Reduced density matrices: If we have a system divided into two blocks (sub-systems)AandB, and the system is in a general mixed stateρ, the reduced density matrices ρA and ρB are defined as [87]

ρ_A≡Tr_Bρ and ρ_B≡Tr_Aρ , (3.5)

where TrB{·}denotes a partial trace over all basis states|jiof blockB, i.e., TrB{·} ≡ P

jhj| · |ji, and likewise for Tr_A{·}. This procedure yields an operator that acts only

3.2. Basis truncation

on the Hilbert space of subsystem A or B. ρ_A and ρ_B both have the standard properties of a density matrix [88]. The expectation value of an operator O that also only acts on one of the subsystems, e.g., subsystemA, can then be calculated as [87]

Tr{ρ O}= Tr_A{ρ_AO}. (3.6)

If the system is in a pure state |ψi = P

i,jψij|i, ji with |i, ji ≡ |ii|ji, so that ρ = |ψihψ| = P

i,i⁰,j,j⁰ψ_ijψ_i⁰_j⁰|i, jihi⁰, j⁰| (we assume here and in the following real coefficients), the reduced density matrices are calculated as follows:

ρ_A = Tr_B{ |ψihψ| }=X

i,i⁰,j

ψ_ijψ_i⁰_j|iihi⁰|,

ρB = TrA{ |ψihψ| }= X

i,j,j⁰

ψijψij⁰|jihj⁰|. (3.7)

As will be shown later, both density matrices have the same nonzero eigenvalues.

The problem of finding the optimal approximation

|ψi˜ =

m

X

a=1 DB

X

j=1

ψ˜_aj|ai|ji (3.8)

of|ψiwith onlym < D_Abasis states for blockAhas a solution [86] as will be shown in the following. Mathematically, we are solving the following problem (we closely follow the derivation in Ref. [6]): We look for a transformation Uai to orthonormal states

|ai=X

i

U_ai|ii ≡X

i

hi|ai|ii, ha|a⁰i=δ_aa⁰ (3.9)

and coefficients ˜ψaj, so that the square of the distance,

||ψi − |ψi|˜ ², (3.10)

is minimal. Inserting (3.4), (3.8), and (3.9) yields (again assuming real coefficients)

||ψi − |ψi|˜ ² = 1 +X

a,j

ψ˜_aj² −2X

a,i,j

ψ˜ajUaiψij. (3.11)

Minimizing this expression with respect to ˜ψajleads to the condition ˜ψaj =P

iUaiψij, so that

||ψi − |ψi|˜ ² = 1−X

a,j

ψ˜_aj² . (3.12)

Finding the minimum of this expression is equivalent to finding the maximum of X

a,j

ψ˜_aj² = X

a,i,i⁰,j

U_aiU_ai⁰ψ_ijψ_i⁰_j ≡ X

a,i,i⁰

U_aiU_ai⁰(ρ_A)_ii⁰, (3.13) where (ρ_A)_ii⁰ are the matrix elements of the reduced density matrix (see Eqs. 3.7) of block A in the basis {|ii}. This means that the eigenvalues wa of ρA have the properties 0≤w_a≤1 andPDA

a=1w_a= 1. The equation can now be rewritten using U_ai =hi|ai:

X

a,j

ψ˜_aj² =

m

X

a=1

ha|ρ_A|ai. (3.14)

This expression is maximal if we choose the|aito be the eigenvectors ofρ_Awith the largest eigenvalues [6, 86]. Summarizing, if we want to keep only m basis states for blockA,||ψi − |ψi|˜ ² is minimal and hence the wave function approximation optimal if we approximate

|ψi=

DA

X

i=1 DB

X

j=1

ψij|ii|ji (3.15)

by

|ψi˜ =

m

X

a=1 DB

X

j=1

ψ˜_aj|ai|ji, (3.16)

with ˜ψaj = P

ihi|aiψ_ij and |ai being the m eigenvectors of ρA with the largest eigenvalues w_a. The error of this approximation is

||ψi − |ψi|˜ ² = 1−

m

X

a=1

w_a≡∆w . (3.17)

The quantity ∆w is called “truncated weight” [6] and is a measure for the error due to the truncation. For many systems, the error of the energy per site that was calculated within the truncated basis using DMRG is approximately proportional to the truncated weight [6, 89].

Using the so-called Schmidt decomposition [87,90,91] leads to a more direct deriva-tion of the optimal truncaderiva-tion procedure. If we again start from Eq. (3.4), the sin-gular value decomposition of the matrix ψ_ij leads to the Schmidt decomposition of

|ψi [6, 87, 90]:

|ψi=

DSchmidt

X

α=1

√w_α|α_Ai|α_Bi, (3.18)

3.2. Basis truncation

with |α_Ai (|α_Bi) being the eigenvectors of the reduced density matrix of block A (B) and wabeing the eigenvalues, i.e.,

ρ_A=

D_Schmidt

X

α=1

w_α|α_Aihα_A|, ρ_B=

D_Schmidt

X

α=1

w_α|α_Bihα_B|, (3.19) with DSchmidt ≤ min(DA, DB). Both reduced density matrices have the same D_Schmidt nonzero eigenvalues (the remaining DA−D_Schmidt eigenvalues for block A and D_B−D_Schmidt eigenvalues for block B are zero). Since 0 ≤√

w_α ≤ 1, it is immediately clear, which block basis states contribute most to the state|ψi, namely the eigenvectors of the reduced density matrices with the largest eigenvalues. The Schmidt decomposition is furthermore enlightening because it reveals the degree of entanglement between the two blocks for the state|ψi. The von Neumann entropy of the reduced density matrices can be used as a measure for the entanglement [92]:

SvN=−X

α

wαlnwα. (3.20)

If the superblock is in a product state, only one of the eigenvalues of the reduced density matrices is nonzero, i.e., equal to one for this case, and the entanglement is zero. If all reduced density matrix eigenvalues have the same value, the entanglement is maximal [87]. Keeping them density matrix eigenvectors with the largest eigen-values thus also means maintaining the maximum amount of entanglement between the blocks [6]. We can furthermore define the truncated entropy,

∆S =−

DSchmidt

X

α=m+1

wαlnwα, (3.21)

which can also be used as a measure for the accuracy of a calculation [93–95].

A reduction to m normalized density matrix eigenvectors leads to a state |ψi˜ which is not normalized, since P

a,jψ˜_aj² = 1−∆w. However, since ∆w 1, the estimation of the error in Eq. (3.17) still approximately holds for a normalized state

|φi ≡ 1

√1−∆w|ψi˜ =

m

X

α=1

√wα

√1−∆w|α_Ai|α_Bi (3.22)

as a more detailed calculation shows:

||ψi − |φi|² =

m

X

α=1

√ w_α−

√wα

√1−∆w 2

+

DSchmidt

X

α=m+1

w_α 1−∆w

= 2−∆w−2√

1−∆w+ ∆w

1−∆w. (3.23)

If we now use √

1−∆w≈1−∆w/2 and ∆w/(1−∆w)≈∆w, we arrive at

||ψi − |φi|² =||ψi − 1

√1−∆w|ψi|˜ ² ≈∆w , (3.24)

which holds for small ∆w. ∆w is indeed usually very small in calculations, often much smaller than 10⁻⁴.

The statement about the error is in general, however, only true if the block bases are complete before the truncation. In practice, one works with truncated bases for both blocks. This induces the so-called “environmental error” [6, 89]. The envi-ronmental error can, however, be minimized by employing the finite-size algorithm with its “sweeps” (see Sec. 3.3). Before we proceed to the description of the DMRG algorithm, we introduce the notion of “target states”.

Target states: The states that are to be approximated are called target states.

This can, e.g., be the ground state or the lowest energy eigenstate in some symmetry subspace. It is important to note that a target state does not need to be an energy eigenstate. It is possible to target more than one state at the same time. In this case, there are two possible strategies for how to build the reduced density matrices (with |ψ_kidenoting the target states):

• One might build and diagonalize the reduced density matrices separately for the individual target states and then choose several of the eigenstates of the individual density matrices as the new basis states [96].

• One can build a weighted density matrix ρ=X

k

p_k|ψ_kihψ_k|, X

k

p_k= 1, (3.25)

which is used for the calculation of the reduced density matrices (3.7). Then, themeigenstates of the reduced density matrices with the largest eigenvalues are chosen [86]. The states|ψ_ki need not be orthogonal [88].

We have always used the second approach in our implementation. The weights were chosen equal, if |ψ_ki are energy eigenstates. For the dynamical DMRG (DDMRG) method (see Sec. 3.7), the weighting is slightly different.

Im Dokument Calculation of static and dynamical properties of giant magnetic molecules using DMRG (Seite 32-36)