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Matrix Product States: Foundations

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Quantum Many-Body Physics of Ultracold Molecules in Optical Lattices

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

The bulk of the numerical results for many-body systems contained in this thesis are obtained by variational algorithms on a class of states known as matrix product states (MPSs).

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Notes

  1. 1.

    By which we mean a spectral gap in the thermodynamic limit, E 1 − E 0 > 0 where E 1 is the energy of the first excited state and E 0 is the energy of the ground state. Finite-sized systems whose infinite counterparts are gapless typically have a gap which vanishes as an inverse polynomial in the system volume.

  2. 2.

    For a counterexample in a non-translationally invariant 1D chain, see [17]. Here, a variant of the Dasgupta–Ma–Fisher renormalization group procedure for random spin chains is used to explicitly construct a spin chain satisfying a volume law. Additionally, as this system is not translationally invariant, the volume law depends crucially on how the system is divided. In fact, there exist divisions for which the entropy of entanglement is identically zero.

  3. 3.

    Rigorously, the central charge is determined by the coefficient of the anomalous term in the commutator of the energy-momentum tensor at two different positions [31]. Hence, c describes the behavior of a conformally invariant system when a macroscopic length scale is introduced.

  4. 4.

    That is to say, the χ that is required to accurately reflect the entanglement structure grows exponentially with the von Neumann entropy.

  5. 5.

    Strictly speaking, this is true only for fixed error. If we require that the error be bounded by an inverse polynomial in the system size, finding a ground state with the given representation is still very difficult [34].

  6. 6.

    By isometric, we mean that this matrix has orthonormal rows. If it were square, it would be unitary, but we are transforming from a χ 2 dimensional space to a χ dimensional space, and so only the rows are orthonormal.

  7. 7.

    That is, at the first iteration we map from the product of the vacuum and a block \(B^{\left (0\right )}\) to a new set of states which is of course identical to the block \(B^{\left (0\right )}\). A similar comment applies to the last iteration.

  8. 8.

    Note that, strictly speaking, DMRG does not maximize entanglement due to a renormalization of the density-matrix spectra induced by truncation [37]. However, for most physical systems this intuition does not cause any pitfalls.

  9. 9.

    This should not be confused with the rank of a matrix, which is the number of nonzero singular values. We shall avoid confusion in this text by referring to the number of nonzero singular values as the matrix rank or rank of a matrix and referring to the definition given here as the tensor rank, the rank of a tensor, or a rank-r tensor.

  10. 10.

    Double precision General Matrix Multiply.

  11. 11.

    Basic Linear Algebra Subprograms, a large collection of numerical routines which were designed to take advantage of the cache structure of modern computers. Using BLAS routines versus naive loops for contractions leads to speedups often of a factor of 4 or more, even when using aggressively optimizing compilers.

  12. 12.

    That is, the algorithm scales as \(\mathcal{O}\left (\chi ^{5}\right )\) but χ PBC = χ OBC for a fixed error in the improved ansatz.

  13. 13.

    Note that this does not affect the canonical form of these two tensors.

  14. 14.

    See Eq. (6.69) for the condition that a left-canonical MPS on an infinite lattice is normalized.

  15. 15.

    Strictly speaking, the transfer operator using the MPS tensors at site j take χ j ×χ j matrices to χ j−1 ×χ j−1 matrices with the given order of operations, but the great numerical use is in infinite systems where χ is uniform across all bonds.

  16. 16.

    For critical systems, the diverging of the correlation length requires us to consider systems with an infinite number of sites. For an MPS algorithm which operates in this limit, see Chap. 8.

  17. 17.

    These irreps are only one-dimensional in the Abelian case. In the non-Abelian case it is still true that the space decomposes into degeneracy spaces and irreps; however, the irreps can have dimensions larger than one.

  18. 18.

    In the physics literature, the Clebsch–Gordan coefficients typically refer to this unitary transformation for the case of SU(2). Here we use it in the more general mathematical sense as the unitary matrix connecting the tensor product of the representation spaces of two irreps of a group to a direct sum of irreducible representation spaces [68].

  19. 19.

    Here we say only that the eigenvectors can be chosen in this way to account for possible energetic degeneracies of two states with differing total particle number. In the presence of such a degeneracy, any linear combination of the degenerate states is also an eigenstate.

  20. 20.

    This construction can also be compared with circuit diagrams demonstrating conservation of charge as specified by Kirchoff’s laws.

  21. 21.

    This is the most significant difference between the Abelian and non-Abelian cases. In the latter, the allowed irreps are enumerated by the Clebsch–Gordan series, and the elements of the unitary matrix relating \(t_{q_{\left (\alpha i\right )}}\) to \(t_{q_{\alpha }}\) and \(t_{q_{i}}\) are nontrivial.

  22. 22.

    This is in fact the definition of a perfect hash function, which is harder to find than a hash function in general but can be explicitly found for our purposes.

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  1. Colorado School of Mines, Golden, CO, USA

    Michael L. Wall

  2. JILA, NIST, and University of Colorado, Boulder, CO, USA

    Michael L. Wall

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Wall, M.L. (2015). Matrix Product States: Foundations. In: Quantum Many-Body Physics of Ultracold Molecules in Optical Lattices. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-14252-4_6

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