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Introduction to Quantum Many-Body Physics

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Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems

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Abstract

In the introduction to his 1929 paper on many-electron systems (Dirac, Proc R Soc Lond Ser A, 123:714, 1929, [1]), Dirac envisioned that the two problems facing quantum mechanics were “in connection with the exact fitting in of the theory with relativity ideas” on the one hand, and the fact that “the exact application of these laws leads to equations much too complicated to be soluble” on the other.

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Notes

  1. 1.

    We assume the basic postulates and formalism of quantum mechanics [4, 5] to be in place, as well as all mathematical concepts and techniques. We refer to Ref. [6] for a modern overview.

  2. 2.

    We will always ignore the mathematical difficulties of defining this thermodynamic limit.

  3. 3.

    The concepts in this section can be found in more detail in standard textbooks on quantum many-body systems [9, 10, 13,14,15].

  4. 4.

    The number of lattice sites is denoted as \(|\mathcal {L}|\), which, in the thermodynamic limit, is sent to infinity.

  5. 5.

    We denote \(\vec {n}_i\) as the lattice vector of site i.

  6. 6.

    Note that the static structure factor is regained from the dynamic correlations as

    $$\begin{aligned} s(\vec {q}) = \int \mathrm {d}\omega \; S(\vec {q},\omega ). \end{aligned}$$
  7. 7.

    We refer to Chap. 3 for a worked-out version of the particle interpretation of elementary excitations.

  8. 8.

    In the following chapters we will also mention a few experimental realizations of the specific models that we will study.

  9. 9.

    We refer to Ref. [23] for all details on magnetic materials.

  10. 10.

    The results mentioned in this section are laid out in exquisite detail in Ref. [52].

  11. 11.

    See also Ref. [73].

  12. 12.

    This definition of the two-particle S matrix will be discussed in full detail in Chap. 3.

  13. 13.

    This section should be more or less self-contained, but the reader is referred to review papers [189,190,191] for more details.

  14. 14.

    The generic case excludes the cases where the MPS can be written as (i) the superposition of multiple translation-invariant MPSs with smaller bond dimension or (ii) the superposition of p p-periodic states each of which can be written as an MPS. The condition under which an MPS is generic is related to the injectivity of the MPS, which means that by concatenating enough A tensors the map from the virtual space to the physical one becomes injective, and to the fact that there is only eigenvalue of \(\mathcal {E}\) on the unit circle [221].

  15. 15.

    We refer to the reviews [189, 191, 202] for a detailed account.

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    Laurens Vanderstraeten

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Vanderstraeten, L. (2017). Introduction to Quantum Many-Body Physics. In: Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-64191-1_2

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