Infinite time-evolving block decimation algorithm beyond unitary evolution

Physical Review B - Condensed Matter and Materials Physics

Volumn 78, Issue 15, 2008, Pages

  Orus R ^a   Vidal, G ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55349127399     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.78.155117     Document Type: Article

References (45)

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35. The canonical form is unique up to a choice of phases ei α. Two canonical forms for |Ψ, (Γ,λ) and (Γ′, λ′), are related by (Γ′) αβ i = ei α Γ αβ i e-i β and λ′ =λ.
36. There are states of a chain, such as the cat state limN→ c0 |0 N + c1 |1 N, | c0 | 2 + | c1 | 2 =1, for which the dominant eigenvalue is degenerate. The iMPS description needs to be supplemented with an extra tensor, sitting at infinite, that determines the boundary conditions (in this case the values of c0 and c1). Here we will not consider such cases.
37. Non-Hermitian eigenvalue problems also occur in the context of transfer matrix DMRG. For instance, see, N. Shibata,, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/36/37/201 36, R381 (2003).
38. A Cholesky decomposition can also be used to obtain two lower triangular matrices for X and Y (see e.g. http://en.wikipedia.org/wiki/ Cholesky_decomposition).
39. This bound is optimal as long as χ>κ. In the case κχ the efficiency can be improved by using alternative contractions.
40. Truncating a bond index so as to retain the χ largest Schmidt coefficients λα is optimal in that it maximizes the overlap between the initial and truncated states. In the present case we use this recipe to truncate all bond indices of the iMPS at once. This is no longer expected to be optimal, but it is simple and seen to produce very satisfactory results.
41. Another way to turn an iMPS { ΓA, λA, ΓB, λB } into the canonical form is by using the algorithm of Ref. to simulate a large sequence of trivial two-site gates (that is, gates that implement the identity operator) alternatively acting on even and odd bonds. It is seen that after each update the iMPS is closer to the canonical form. In practice, the orthonormalization strategy explained in this paper is more efficient and precise.
42. For a nonsymmetric Hamiltonian K2, Q is decomposed into two different matrices Q= Q1 Q2 (e.g., through a singular value decomposition). If K2 changes along different lattice directions (anisotropic model), then we will decompose two matrices Qx and Qy. In both situations one can proceed in a similar way as in the symmetric, isotropic case.
43. Our construction was inspired by a similar one in F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.220601 96, 220601 (2006);
44. where finite systems were analyzed by mapping the partition function into a PEPS. Here we skip the map into PEPS and significantly reduce simulation costs by decreasing the bond dimension of the resulting 2D tensor network from d2 to d.
45. Another good reason to use the canonical form of an iMPS is that it simplifies the comparison between two states. As a criterion for convergence of the sequence | Ψp in Eq. 19, we require that the Schmidt coefficients λ of the iMPS have converged with respect to p within some accuracy.