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Effective Field Theories for Topological States of Matter

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Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 239))

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Abstract

Since the discovery of the quantum Hall effect in the 1980s it has been clear that there exists states of matter characterized by subtle quantum mechanical effects that renders certain properties surprisingly stable against dirt and noise. The theoretical understanding of these topological quantum phases have continued to develop during the last few decades and it has really surged after the discovery of the time-reversal invariant topological insulators. There are many examples of topological phases that have been important for the theoretical understanding of topological states of matter as well as being of great physical relevance. In this chapter we will focus on some examples that we find particularly enlightening and relevant, but we will not make a complete classification. Some of the most important tools for the understanding of topological quantum matter are based on effective field theory methods. We shall employ two different types of effective field theories. The first, which is valid at intermediate length and time-scales, will not capture the physics at microscopic scales. Such theories are the analogs, for topological phases, of the Ginzburg–Landau theories used to describe the usual symmetry breaking non-topological phases. The second type of theories describe the physics on scales where non-topological gapped states would be very boring, namely at distances and times much larger than the correlation length and the time set by the inverse gap. On these scales everything is independent of any distance and the theories will be topological field theories, which do not describe any dynamics in the bulk, but do carry information about topological properties of the excitations, and also about excitations at the boundaries of the system. Finally, we will also study effective response actions. In a strict sense these are not effective theories, since they do not have any dynamical content, but encode the response of the system to external perturbations, typically an electromagnetic field. As we shall see, however, the effective response action for topological states can be used to extract parts of the dynamic theory through a method called functional bosonization.

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Notes

  1. 1.

    The topological interaction between anyons is a generalization of the Berry phase of \(-1\) acquired by the exchange of two identical fermions. This minus sign is directly related to Fermi statistics and in the same way the topological interaction is related to a specific “exclusion statistics”. This is the reason for the term “statistics” in anyonic statistics.

  2. 2.

    Most textbooks in condensed matter theory will cover the Ginzburg–Landau–Maxwell theory. For a modern text see e.g., [12]. Reference [13], by S. Weinberg, one of the founders of effective field theory, gives a good presentation from the field theoretic point of view. There are also several excellent recent textbooks, as for instance [14, 15], on the general subject of these notes.

  3. 3.

    This is not true in higher dimensions, and it is also not true for the longitudinal 2d conductance. Even for a pure sample does not equal the conductivity. For a rectangular Hall bar as in Fig. 1.1 the longitudinal conductance is \((W/L)\sigma \) where W and L are the widths and length of the bar respectively.

  4. 4.

    In some references you will come across the notion of “twisted boundary conditions” this is equivalent to inserting flux through the holes of the torus.

  5. 5.

    Note that the conventions for this field differ. We use the notation from [11], while in the work by Wen [17] the field here denoted by b is denoted by a.

  6. 6.

    Reference [24] treated the case where the ratio between the areas of the unit cells in the flux lattice and the Bravais lattis of the potential is a rational number q/p. In case one has to consider a larger unit cell, and each filled band will in general have a larger Hall conductance.

  7. 7.

    In a translationally invariant system, the shape of this zone is arbitrary, but the area is fixed to support n units of magnetic flux. In our case the shape has to be taken as to be commensurate with the Bravais lattice of the potential.

  8. 8.

    As explained in [36] a gauge invariant regularization of the ultraviolet divergence (e.g., using the Pauli-Villars method) gives rise to the anomaly term, but does not fix the sign.

  9. 9.

    See Sect. 16.3.3 in [14] for more details.

  10. 10.

    This has a direct consequence for the spectrum of Josephson junctions. In the first (real) case, the junction between two topological states with winding number \(\pm 1\), will host two Majorana zero-modes, which amounts to a single Dirac zero-mode, while in the second (complex) case such a junction will have no zero-mode, see e.g., [49].

  11. 11.

    The symmetry properties of the Lagrangian (1.111) are worth a comment. Under the parity transformation \((x,y)\rightarrow (-x,y)\) the two potentials transform as \((a_{0},a_{x},a_{y})\rightarrow (a_{0},-a_{x},a_{y})\) and \((b_{0},b_{x},b_{y})\rightarrow (-b_{0},b_{x},-b_{y})\), while under time reversal the transformations are, \((a_{0},a_{x},a_{y})\rightarrow (a_{0},-a_{x},-a_{y})\) and \((b_{0},b_{x},b_{y})\rightarrow (-b_{0},b_{x},b_{y})\), respectively. The unusual transformation properties of the potential \(b_{\mu }\) follow from those of the vortex current. It is easy to check that the BF action is invariant under both P and T.

  12. 12.

    This is a toy model not only because we use a relativistic form for the kinetic energy, but also because we use \(2+1D\) Maxwell theory, which amounts to a logarithmic Coulomb interaction. The generalization to the more realistic case is straightforward, and the result is qualitatively the same. The derivation, however, becomes less transparent. For the interested reader [53] is a good reference to see how to include fermions and also see how the chiral d-wave case works.

  13. 13.

    Note that in spite of the relativistic form we normalize the kinetic term such that \(|\phi |^{2}\) has the dimension of density.

  14. 14.

    The naive picture of the “composite bosons” flux-charge composites, is however slightly misleading since that would implies that you get a phase \(2\times k2\pi \) when taking one particle a full turn around another; there are equal contributions from the charge circling the flux and the flux circling the charge. This is not what happens, the correct phase is \(k2\pi \) corresponding to the exchange phase \(k\pi \) [60].

  15. 15.

    You might object since \(\mathscr {A}_{\mu }(\mathbf {b})\) is an operator within the full Hilbert space \(\mathscr {H}\), so one should be able to define its trace. That is, in principle, a correct assumption but the trace of \(\mathscr {A}_{\mu }(\mathbf {b})\) does not depend only on the structure of the fibers but of the full Hilbert space \(\mathscr {H}\) and it does not necessarily equal \( \mathscr {A}_{\alpha \,\mu }^{\ \alpha }(\mathbf {b})\), in a particular basis. The reason is the fact that the basis \(\{ \left| {\mathbf {b};\alpha } \right\rangle \} \) varies with \(\mathbf {b}\); \(\mathscr {A}_{\alpha \,\mu }^{\ \beta }(\mathbf {b})\) is the representation of the operator \(\mathscr {A}_{\mu }(\mathbf {b})\) where the bras are written in the basis \(\{ \left| {\mathbf {b}+b^{\mu };\alpha } \right\rangle \} \) and the kets in the basis \(\{ \left| {\mathbf {b};\alpha } \right\rangle \), see (1.172).

  16. 16.

    The fact that the first Chern number is an integer, proven here, hold for a general closed manifold as e.g., the torus. The proof is more involved since one has to divide B into more regions, but the arguments are analogous to the one for the sphere.

  17. 17.

    In case you do not know how to handle actions that are first order in time derivatives, you can learn in e.g., [65].

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Acknowledgements

We thank Maria Hermanns for making useful comments on a preliminary version of these notes, and we also thank Christian Spånslätt for reading and commenting the manuscript. We are very thankful to Marcus Berg and Sergej Moroz, who helped us find many typos and errors.

The nice hand-drawn pictures are provided by Sören Holst and the derivation of the parity anomaly given in Sect. 1.8.4 was shown to THH by the late professor Ken Johnson of MIT.

TKK acknowledge the Wenner-Gren foundations and stiftelsen Olle Engkvist Byggmästare for financial support during 2019 and 2018 respectively.

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Authors and Affiliations

  1. Department of Physics, Stockholm University, 10691, Stockholm, Sweden

    Thors Hans Hansson

  2. Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden

    Thors Hans Hansson

  3. Department of Physics, KTH Royal Institute of Technology, 10691, Stockholm, Sweden

    Thomas Klein Kvorning

  4. Department of Physics, University of California, Berkeley, CA, 94720, USA

    Thomas Klein Kvorning

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  1. International Institute of Physics-UFRN, Natal-Rn, Rio Grande do Norte, Brazil

    Alvaro Ferraz

  2. Theory Division, Saha Institute of Nuclear Physics, Kolkata, India

    Kumar S. Gupta

  3. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada

    Gordon Walter Semenoff

  4. International Institute of Physics-UFRN, Natal-Rn, Rio Grande do Norte, Brazil

    Pasquale Sodano

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Hansson, T.H., Klein Kvorning, T. (2020). Effective Field Theories for Topological States of Matter. In: Ferraz, A., Gupta, K., Semenoff, G., Sodano, P. (eds) Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory. Springer Proceedings in Physics, vol 239. Springer, Cham. https://doi.org/10.1007/978-3-030-35473-2_1

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