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Quantum Field Theory Anomalies in Condensed Matter Physics

by R. Arouca, Andrea Cappelli, T. H. Hansson

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Submission summary

Authors (as registered SciPost users): Rodrigo Arouca · Andrea Cappelli · Hans Hansson
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-08-10
Date submitted: 2022-08-03 11:12
Submitted by: Arouca, Rodrigo
Submitted to: SciPost Physics Lecture Notes
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We give a pedagogical introduction to quantum anomalies, how they are calculated using various methods, and why they are important in condensed matter theory. We discuss axial, chiral, and gravitational anomalies as well as global anomalies. We illustrate the theory with examples such as quantum Hall liquids, Fermi liquids, Weyl semi-metals, topological insulators and topological superconductors. The required background is basic knowledge of quantum field theory, including fermions and gauge fields, and some familiarity with path integral and functional methods. Some knowledge of topological phases of matter is helpful, but not necessary.

Author comments upon resubmission

Let us thank the two referees for the throughout analysis of our lectures and their comments and suggestions. Please find our replies and corresponding changes in the file "refs_reply.pdf" available in the referee reply section.

List of changes

Here, we provide a point-by-point summary of the changes, detailed in the file "refs_reply.pdf" available in the referee reply section.

1-We added a remark to better relate ``geometric'' and ``topological'' in page 6 :
``Indeed, certain {\it global} quantities, like conductances, are topological and can only take discrete values.''

2-The following footnote is added before (2.9):

``This finite gauge transformation is $U(x)=\exp(-i \int^x dx' A_x(x')''

3-Footnote 3 is extended. It reads now
``Antiperiodic (Neveu-Schwartz) boundary conditions are standard for fermions, since the minus sign follows by bringing the fermion around the circle. Periodic (Ramond) conditions can also be chosen, they introduce additional features to be discussed in Sect. 9.3.''

4-Typos in equations (2.29), (3.35), (6.14), (6.16), and (6.17), and in the definition of axial gauge field below (6.13) were fixed.

5-We have slightly rewritten Section 6.4, emphasizing that we work in a frame where the covariant current is zero before the fields are applied. We have also added a reference to Bloch's result and the paper by Yamamoto.

6-We added the following phrase at the end of the paragraph after (7.13):
``The dots in (7.13) are other pseudoscalar quantities that are higher-order polynomials of the curvature and its covariant derivatives.''

7- We have added a reference to Kimura Prog.Theor.Phys. 42 (1969) 1191-1205 in the text after the result (7.15), before quoting the path-integral result by Fujikawa.

8-We added the following footnote at the end of the paragraph after (9.4):
``Note that the theta term (9.4) is purely imaginary both in the Euclidean formulation of Section 5.3 and in the Minkowskian version used for physical application, being subjected to TR invariance. Furthermore, this symmetry can also be directly formulated in Euclidean space, as described in Sect. 10.2''

9-We have added a footnote after (2.16) referring to section 3.3 for the complete discussion of the subtraction procedure (normal ordering):
``A detailed analysis of the subtraction procedure, unambiguously determining the finite terms, will be presented in section 3.3.''

10-We have added a footnote on page 28 (Sect. 3.5.1):
``See section 6.2 and appendix D.4, in particular.''

11-``We will return to the physical interpretation of $A_{5}$ in Sect. 6 and App. D discusses aspects of the mixed anomalies where both vector and axial (background) fields are present.'' in Sect. 2.2 changed to:
``We shall see in Sec. 6 that $A_5$ plays an important role in the physics of Weyl semimetals and the chiral magnetic effect, while in App. D we discuss aspects of the mixed anomalies where both vector and axial (background) fields are present.''

12-We removed $\rho$ in Eq. (6.10).

13-We introduced a vector notation for the Berry curvature in (6.12).

14-References to
[A] Adolfo G. Grushin, J\"orn W.F. Venderbos, Ashvin Vishwanath, and Roni Ilan
Phys. Rev. X 6, 041046 (doi:10.1103/PhysRevX.6.041046)

[B] Pavan Hosur, Shinsei Ryu, and Ashvin Vishwanath
Phys. Rev. B 81, 045120 (doi:10.1103/PhysRevB.81.045120)

were added.

Published as SciPost Phys. Lect. Notes 62 (2022)

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