Time-Dependent Variational Principle

4.2.4 Conclusion

With all improvements mentioned, time evolution with the Krylov method becomes rel-atively fast and well-controlled. Furthermore, the Krylov method can be easily applied to all kind of Hamiltonians without any changes of the method and generalises very eas-ily to other tensor network states such as binary tree tensors. The only parts needing adaptation for other network topologies are the calculation of expectation values as well as the operator application with simultaneous orthogonalisation against previous Krylov vectors.

In Fig. 4.3 the runtime changes of the different improvements can be seen for a simple imaginary-time evolution. In this case all improvements lead to a reduction of the runtime by a factor 3.5. Depending on the model and the form of the time evolution the runtime reduction can be even bigger.

A^∗₁ A^∗₂ A^∗₃ A^∗₄ A1 A2 A3 A4

B₆^∗ B₇^∗ B₈^∗ B₉^∗ B₁₀^∗ B6 B7 B8 B9 B10

|σ1i |σ2i |σ3i |σ4i |σ5i |σ6i |σ7i |σ8i |σ9i |σ10i

|σ1i |σ2i |σ3i |σ4i |σ5i |σ6i |σ7i |σ8i |σ9i |σ10i

Figure 4.4: Graphical representation of the projectorPˆ₄^L⊗1ˆ5⊗Pˆ₆^R, which is representative for the terms of the first sum in Eq.(4.3.5). TheAand B tensors are the left- and right-normalised tensors of the state|ψi. The terms of the second sum are constructed in the same manner, by replacing the straight vertical line in the fifth bond by the corresponding A matrices.

with|al−1iA and |bliB as defined in Eq. (2.1.17) and Eq. (2.1.17) respectively, the projec-tors take the following form

Pˆ_l^L = ^X

al−1

|al−1iAhal−1|A, (4.3.3)

Pˆ_l^R =^X

bl

|bli^Bhbl|^B. (4.3.4)

The first sum consists of terms that project all tensors to the left and right of site l on the state|ψi while the site l stays unchanged, see Fig. 4.4 for a graphical representation.

Thus, they filter for all states that have only changes on a single tensor. The second term removes all those states that are identical with the state|ψi itself.

The key idea is now to insert the projector in the time dependent Schrödinger equation

∂

∂t|ψi=−iPˆT|ψiHˆ|ψi

=−i

L

X

l=1

Pˆ_l−1^L,|ψi⊗1ˆl⊗Pˆ_l+1^R,|ψiHˆ|ψi+i

L−1

X

l=1

Pˆ_l^L,|ψi⊗Pˆ_l+1^R,|ψiHˆ|ψi. (4.3.5) This equation is still not solvable, but we can approximate it by solving each term indi-vidually. By applying a symmetrised second error Suzuki-Trotter decomposition similarly to TEBD we separate the time evolution of|ψiinto individual and sequentially applicable time evolutions. If we define

ˆh1,l = ˆP_l−1^L ⊗^l1ˆl⊗Pˆ_l+1^R H,ˆ (4.3.6)

ˆh2,l = ˆP_l^L⊗Pˆ_l+1^R H,ˆ (4.3.7)

we can write the time evolution operator with the symmetrised second order Suzuki-Trotter decomposition as

U(tˆ +δt) =e^−ih1,1δt2e^ih2,1δt2e^−ih1,2δt2 . . . e^−ih1,2δt2e^−h2,1δt2e^−ih1,1δt2 +O(δt³). (4.3.8)

This means we have to solve Lequations describing a forward time evolution of the form

∂

∂t|ψi=−iPˆ_l−1^L ⊗1ˆl⊗Pˆ_l+1^R Hˆ|ψi, (4.3.9) and L−1 terms describing a backward time evolution written as

∂

∂t|ψi= +iPˆ_l^L⊗Pˆ_l+1^R Hˆ|ψi. (4.3.10)

We can obtain effective single-site equations by multiplying each individual equation of the form of Eq. (4.3.9) with |ψi from the left but omitting the tensor on sitel. Another set of equations can be obtained by multiplying each individual equation of the form of Eq. (4.3.10) from the left with the state |ψi having an open bond between site l−1 and l. Thus, the equations can be written as

∂

∂tMl =−iHˆ_l^effMl, (4.3.11)

∂

∂tCl =iHˆ˜_l^effCl, (4.3.12)

with Ml being just the tensor of |ψi on site l while Cl is the singular value matrix between site l and l+ 1 obtained after bringing all tensors to the left of site l + 1 into the left-canonical form and all tensors to the right of sitel into right canonical form with successively applied SVDs. We refer from defining the effective matrices ˆ˜H_l^eff and ˆH_l^eff of the two time evolution equations in form of equations, but rather give a graphical definition in Fig. 4.5 and Fig. 4.6. At this stage we just have to use one of the usual methods for exponentiating a matrix to solve all 2L−1 equations successively and apply the resulting operators in the correct order to |ψi according to Eq. (4.3.8).

TDVP is advantageous for multiple reasons: First, it is unitary, i.e. preserves both the norm and the energy of the time-evolved state by construction. Second, it exists in a site and two-site version with the site version scaling similar to the single-site DMRG. This is the case because TDVP in the limit of taking t→ −i∞ represents a ground state search. In other words, TDVP and DMRG without subspace expansion just differ in the sense that in TDVP the local eigensolver is replaced by a local exponentiation.

Third, it calculates the time-evolved state directly instead of building up a subspace, which potentially can involve heavy calculations including basis states with high bond dimensions. Finally, it requires only a single MPO representation of the Hamiltonian ˆH.

We do not have to analytically separate the Hamiltonian as in TEBD.

One of the disadvantages of TDVP is that even if one time step with time δt has an error of O(δt³), we observe relatively large errors due to the projection in the tangent space itself if the Hamiltonian has long range interaction (i.e. not nearest-neighbour).

This is not surprising since even for a Hamiltonian with nearest-neighbour interaction the single-site TDVP induces a projection error while two-site TDVP does not^[46]. Therefore, it seems reasonable that a higher order TDVP like three-site or even more would be able

A1 A2 A3 A4

A^∗₁ A^∗₂ A^∗₃ A^∗₄ A1 A2 A3 A4

A^∗₁ A^∗₂ A^∗₃ A^∗₄

M5 B6 B7 B8 B9 B10

B₆^∗ B₇^∗ B₈^∗ B₉^∗ B₁₀^∗ B6 B7 B8 B9 B10

B₆^∗ B₇^∗ B₈^∗ B₉^∗ B₁₀^∗

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10

Figure 4.5: Graphical representation of the right-hand side of the effective single-site time for-ward evolving Schrödinger equation Eq. (4.3.11). The first row from the bottom is the original MPS in the mixed-canonical representation, the second row is the Hamiltonian, the third and fourth row is representing the the projector Pˆ₄^L⊗ˆ15 ⊗Pˆ₆^R and the last row is the state |ψi with site 5 missing, which we multiplied from the left to the Schrödinger equation. Because of the orthogonalisation properties of the A and B matrices, the upper two rows simplify to unit matrices.

to overcome this drawback at the cost of longer calculation times. Unfortunately, this error increases with smaller time steps indicating a competition between the time-step error and the truncation error.

After testing the presented time evolutions extensively, two-site, second-order TDVP is the method of choice for all our calculations if not stated otherwise.