Finite-system DMRG algorithm

After obtaining the approximate block representations for a lattice of desired size L built up by the infinite-system DMRG algorithm, application of the finite-system DMRG algorithm

E

m₄^E σ₁ σ₂ σ₃

σ_l−1

m_l−2S σ_l m_l+1^E ml+2^E

σ_l+1 σ_l

l−1^S

m

Environment Block System Bolck

H_S^l−1 H_E^l+2

ml+2^E

σ_l+1 σ_l

l−1^S

m

System Bolck Environment Block

Environment Block System Bolck

H H_E

H_E

l−2 l+1

4 S

Environment Block System Bolck

H_S^l−1 H^l+2

Figure 1.4: Depiction of the finite-system DMRG algorithm.

provides high accuracy results. Each block basis is optimized by sweeping two exactly repre-sented site blocks through a lattice of fixed size L. Because the size of the lattice is fixed, the system block grows, while the environment block shrinks (see Fig. 1.3). The reduced trans-formation only takes place on the system block, while the environment block is transformed by using previously stored environment blocks. In each step of a particular sweep, the basis of the system block is optimized.

For the consistency with the discussion of the infinite-system DMRG algorithm, Sec. 1.2.2, we assume that the lattice has reflection symmetry. One then only needs to do sweeps through a half of the lattice. The finite-system DMRG algorithm of a reflection symmetric lattice con-sists of the following steps:

1. After the desired lattice size L is reached in the second step of the infinite-system DMRG algorithm, one has obtained the superblock basis{|m^S_l−1σ_lσ_l+1m^E_l+2i}.

2. Switch the roles of system block and the environment, see Fig. 1.4 and steps 2-4 in the infinite-system DMRG algorithm. Construct the new reduced basis{|m^E_l+1i}of the system block.

3. Use a stored block of appropriate size as an environment block to form the new su-perblock in the basis{|m^S_l−2σ_l−1σ_lm^E_l+1i}, see Fig. 1.4.

4. Repeat steps 2-3 above until the environment block shrinks to only one site. The su-perblock is now in the basis{|σ₁σ₂σ₃m^E₄i}. Note that every system block is stored at each step above. Here a right-to-left half-sweep is finished.

5. Switch the roles of system block and the environment. Repeat steps 2-3 above, until the starting point superblock with the basis{|m^S_l−1σ_lσ_l+1m^E_l+2i}is reached. Here a left-to-right half-sweep is finished. These two half-sweeps make up one sweep in finite-system DMRG algorithm.

6. Repeat steps 2-5, until convergence of the ground state energy is obtained.

7. Carry out the measurements.

Notice that the assumption of reflection symmetry is not necessary; it is possible to general-ize the finite-system DMRG algorithm to lattices with no reflection symmetry by sweeping through the whole lattice. In addition, in some cases, simultaneously targeting a few of energy eigenstates may improve the convergence of the ground state.

To improve the efficiency of the calculations, one usually fixes the quantum number of the superblock, drastically shrinking the basis. For example, the z-component of total spin Szis fixed in the calculation of the ground state of the Heisenberg model; the z-component of total spin Sz and the total electron number are fixed in the calculation of the ground state of the Hubbard model. Besides using U(1) symmetry of the Hamiltonian, other symmetries are also can be applied in the DMRG calculations, e.g., S U(2) symmetry [38].

The measurements are usually performed after several finite-system DMRG sweeps have been carried out. By representing the operator A in the superblock basis, the expectation value of the operator A,hψ|A|ψi, can be directly calculated, where the current wave function has the representation|ψi ≡ψ_m^S

l−1σ_lσ_l+1m^E_l+2. We take the Heisenberg model as an example. For a local operator S^z_j, the expectation value is given by

hψ|S^z_j|ψi= X

m^S_l−1σ_lσ_l+1m^E_l+2,σ^′_l

ψ^∗_mS

l−1σlσl+1m^E_l+2[S^z_j]σlσ^′_lψ_m^S

l−1σ^′_lσl+1m^E_l+2 (1.33) and similarly for other local operators. This formula gives a exact evaluation ofhψ|A|ψiwith the approximate wave functionψ_m^S

l−1σlσl+1m^E_l+2. The only error comes from the reduced basis.

For the expectation value of two operators on two different sites, such as the spin-spin correlation functionhψ|S^z_jS^z_k|ψi, how to keep track of these two operators depends on whether j and k are located on the different block or not. If j and k are on different blocks,hψ|S^z_jS_k^z|ψi can be evaluated using

hψ|S^z_jS^z_k|ψi= X

m^S_l−1σlσl+1m^E_l+2,m^′_l^S

−1m^′_l+2^E

ψ^∗_mS

l−1σ_lσ_l+1m^E_l+2[S^z_j]_m^S

l−1m^′S_l−1[S_k^z]_m^E

l+2m^′E_l+2ψ_m′S

l−1σ_lσ_l+1m^′_l+2^E , (1.34) where one keeps track of [S^z_j]_m^S

l−1m^′_l^S₋₁ and [S^z_k]_m^E

l+2m^′_l+2^E independently. If j and k are on the same block,hψ|S^z_jS^z_k|ψishould be evaluated using

hψ|S^z_jS^z_k|ψi= X

m^S_l−1σ_lσ_l+1m^E_l+2,m^′S_l

−1

ψ^∗_mS

l−1σlσl+1m^E_l+2[S^z_jS^z_k]_m^S

l−1m^′_l−^S₁ψ_m′S

l−1σlσl+1m^E_l+2, (1.35) where one does not keep track of [S^z_j]_m^S

l−1m^′′_l−1^S and [S^z_j]_m′′S

l−1m^′_l−1^S separately. The reason is that P

m^′′_l−^S₁[S^z_j]_m^S

l−1m^′′_l−^S₁[S^z_j]_m′′S

l−1m^′_l−^S₁ ≈[S^z_jS^z_j]_m^S

l−1m^′_l−^S₁, whereP

m^′′_l−^S₁|m^′′_l−^S₁ihm^′′_l−^S₁| ≈1 for the truncated basis {|m^′′S_l−1i}.

The other important operator that must be evaluated is the Hamiltonian operator. By multiplying Eq. (1.32) byψ^∗_{i j} from the left, one obtains the expression for the expected value of the energy,

hψ|H|ψi= X

i ji^′j^′

ψ^∗_{i j}[H]i j,i′j^′ψ_i′j^′

= X

i ji^′j^′

(ψ^∗_{i j}[H_S^l]_ii′ψ_i′j+ψ^∗_{i j}ψ_{i j}′[H^l+1_E ]_{j j}′ +ψ^∗_{i j}[ ˆσ_l]_ii′ψ_i′j^′[ ˆσ_l+1]_{j j}′).

(1.36)

One can also increase the efficiency by applying the so-called wave function transfor-mation in the finite-system DMRG algorithm [39]. A approximate wave function generated from the previous sweep step using the wave function transformation can reduce the number of Davidson steps or Lanczos steps substantially. For the sake of simplicity, we will not in-clude any truncations in the basis transformation procedure, which meansP

m|mihm|=1. We begin with the wave function of the superblock

|ψi= X

m^S_l−1σ_lσ_l+1m^E_l+2

ψ_m^S

l−1σlσl+1m^E_l+2|m^S_l−1i|σli|σl+1i|m^E_l+2i. (1.37)

In step 2 of the finite-system DMRG algorithm, Eq. (1.37) is transformed to

|ψi= X

m^S_l−1σ_lm^E_l+1

ψ_m^S

l−1σlm^E_l+1|m^S_l−1i|σ_li|m^E_l+1i, (1.38) where

ψ_m^S

l−1σlm^E_l+1 = X

σ_l+1m^E_l+2

ψ_m^S

l−1σlσl+1m^E_l+2hm^E_l+1|m^E_l+2σ_l+1i, (1.39) by inserting P

m^E_l+1|m^E_l+1ihm_l+1^E | = 1 into Eq. (1.37). Step 3 of the finite-system DMRG algo-rithm further transforms Eq. (1.38) to

|ψi= X

m^S_l−2σ_l−1σ_lm^E_l+1

ψ_m^S

l−2σl−1σlm^E_l+2|m^S_l−2i|σ_l−1i|σ_li|m_l+1^E i, (1.40) where

ψ_m^S

l−2σl−1σlm^E_l+2 =X

m^S_l−1

ψ_m^S

l−1σlm^E_l+1hm^S_l−2σ_l−1|m^S_l−1i. (1.41) These two steps shift the positions of the single sites one place from right to left.

The wave function transformation can be generized to the MPS algorithm, which was introduced by by ¨Ostlund and Rommer [40]. One can define

hm^S_l−2σ_l−1|m^S_l−1i=A^m

S l−2,m^S

l−1

l−1 [σ_l−1], (1.42)

hm_l+3^E σ_l+2|m^E_l+2i=A^m

E l+2,m^E_l+3

l+2 [σ_l+2], (1.43)

where the matrix A_l[σ_l] was used to treat the fixed point (i.e., a site independent A[σ]) to which the DMRG eventually converges. From Eq. (1.42), one then obtains

|m^S_l−1i= X

m^S_l

−2σl−1

A^m

S l−2,m^S

l−1

l−1 [σ_l−1]|m^S_l−2i|σ_l−1i, (1.44) where

X

σ_l

A^S_l^†[σl]A^S_l [σl]= 1, (1.45)

|m_l+2^E i= X

m^E_l+3σ_l+2

A^m

E l+2,m^E_l+3

l+2 [σ_l+2]|σ_l+2i|m^E_l+3i, (1.46)

and

X

σ_l

A_l^E[σ_l]A^E†_l [σ_l]= 1. (1.47) One can carry out a recursion step to obtain|m^S_l−2iexpressed in terms of|m^S_l−3iusing Eq. (1.44).

One stops the recursion at site N when the left block basis|m^S_Ni=|σ₁. . . σ_Niin the dimension D. One then obtains

|m^S_l−1i= X

m^S_N,...m^S_l

−2σ_N+1...σ_l−1

A^m

S N,m^S_N+1

N+1 [σ_N+1]. . .A^m

S l−2,m^S_l

−1

l−1 [σ_l−1]|m^S_Ni|σ_N+1. . . σ_l−1i

= X

m^S_N+1,...m^S_l

−2σ1...σl−1

A^m

S N,m^S_N+1

N+1 [σ_N+1]. . .A^m

S l−2,m^S_l

−1

l−1 [σ_l−1]|σ₁. . . σ_l−1i. (1.48)

Similarly, one can expand|m_l+1^E ias

|m_l+2^E i= X

m_l+3^E ...m_R^Eσ_l...σ_M−1

A^m

E l+2,m^E_l+3

l+2 [σl+2]. . .A^m

E M−1,m^E_M

M−1 [σM−1]|σl. . . σM−1i|m^E_Mi

= X

m_l+3^E ...m^E_M

−1σl+2...σL

A^m

E l+2,m_l+3^E

l+2 [σ_l+2]. . .A^m_M^EM₋⁻₁^1,mEM[σ_M−1]|σ_l+2. . . σ_Li, (1.49) where the recursion is stpped at site M, and the small right block M has the basis{|m^E_Mi} of the dimension of D, where|m^E_Mi=|σ_M. . . σ_Li.

Substituting Eqs. (1.48) and (1.49) into Eq. (1.37), and rewritingψ_m^S

l−1σlσl+1m^E_l+2 in the form ψ^mSl−1ml+2E [σ_lσ_l+1], one obtains a D×D wave function,

|ψi= X

σ₁...σ_N

A_N+1[σ_N+1]. . .A_l−1[σ_l−1]ψ[σ_lσ_l+1]A_l+2[σ_l+2]. . .A_M−1[σ_M−1]m^S_Nm^E_M

|σ₁. . . σ_Li. (1.50) The procedure in the finite-system DMRG sweeps can be recognized again in Eq. (1.50). The wave function coefficients ψ[σ_lσ_l+1] are updated at every step in a right to left sweep. Fur-thermore, a new optimized matrix A_l+2[σ_l+2] is obtained by using the updated wave function to form the reduced density matrixρ^l+2_E . Note that, the matrices A_l[σ_l] and A_l+1[σ_l+1] are not needed to represent the current DMRG wave function. However, they will be generated in the next steps. From this view of point, the DMRG can be viewed as an optimization procedure for a variational matrix-product wave function.

The finite-system DMRG algorithm is also important for the Suzuki-Trotter adaptive time-dependent DMRG (see sec.1.3.2). In the Suzuki-Trotter adaptive time-time-dependent DMRG, one

directly applies the time-evolution operator to the two site blocks l and l+1 in a finite-system sweeping procedure to advance the wave function one time step.

Im Dokument Explicit Exchange Interaction and Decoherence Dynamics in One-Dimensional Quantum Systems (Seite 30-36)