Fermion path integrals and topological phases

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here a description is given in detail of what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples considered are time-reversal invariant topological insulators and superconductors in two or three space dimensions. Along the way, the precise meaning of the statement that in the bulk of a 3D topological insulator, the electromagnetic $\theta$ angle is equal to $\pi$, is clarified.

Figure 1

Spectral flow for a Dirac operator. The vertical axis parametrizes an eigenvalue $\lambda$ and the horizontal axis parametrizes a parameter $s$ on which the eigenvalues depend. In the case shown, the spectrum is the same at $s=1$ as at $s=0$, but there is a net upward flow of one eigenvalue through $\lambda =0$ between $s=0$ and $s=1$. This leads to a sign change of the fermion path integral $Z_{\psi }$.

Figure 2

A mirror-symmetric construction of a three-manifold $Y$, by gluing together identical pieces $Y_1$ and $Y_2$ along their common boundary $M$, with reversed orientation for $Y_2$.

Figure 3

A topological insulator supported on the right half of $\mathbb{R}\times S^2$ (solid black region).

Figure 4

A topological invariant $\mathfrak{I}$ defined for a manifold $X$ is said to be a cobordism invariant if it vanishes whenever $X$ is the boundary of a manifold $Z$ of one dimension more. If $X$ has some structure [such as an orientation, a spin structure, or a U(1) gauge field] that is required in defining $\mathfrak{I}$, then this structure is required to extend over $Z$.

Figure 5

The simplest illustration of the statement that cobordism invariance leads to relations that are natural in topological field theory. The connected sum $M$ of two $D$ manifolds $M_1$ and $M_2$ is made by cutting small holes out of $M_1$ and $M_2$ and gluing them together along their boundaries. If the space of physical states on a sphere $S^{D-1}$ is one dimensional (as expected in a unitary topological field theory), one can deduce a universal relation between partition functions: $Z_M=g{Z}_{M_1}{Z}_{M_2}$, where $g$ is a constant characteristic of the theory. Cobordism invariance implies such a relation with $g=1$, because $M$ is cobordant to the disjoint union of $M_1$ and $M_2$. This cobordism is sketched in this figure (for the case $D=1$ with all manifolds being circles). Topological field theories derived from cobordism invariants always satisfy the condition that the space of physical states on any ($D-1$) manifold has a dimension of 1, which is why they lead to a relation of the given kind.

Figure 6

The complete but not compact manifold $X^{\prime }$ is built by gluing onto the manifold $X$, which has a boundary $Y$, and a semi-infinite tube $Y\times {\mathbb{R}}_{+}$.

Figure 7

By gluing together two oriented four-manifolds $X$ and $X^{\prime }$ that have the same boundary $Y$ (after reversing the orientation of $X^{\prime }$ so that the orientations are compatible), we build a closed oriented four-manifold $\widehat{X}$. $\widehat{X}$ is a spin manifold if $X$ and $X^{\prime }$ have spin structures that agree along $Y$.

Figure 8

A spacetime $X$ with boundary $Y$ that is built by gluing together mirror image copies of some manifold $X_0$. The copies are denoted $X_0$ and ${X_0}^{\prime }$. $Y$ is likewise the union of mirror image pieces. $X_0$ has two types of boundaries: $M$, along which it is glued to ${X_0}^{\prime }$, and also its intersection with $Y=\partial X$. $X_0$ has corners along $M\cap Y$.

Figure 9

Spectral flow for real fermions. The spectrum is symmetric under $\lambda \leftrightarrow -\lambda$. The fermion path integral changes sign whenever an eigenvalue pair flows through 0, so we count $\mathrm{mod}2$ how many times this happens in interpolating from an initial metric $g_0$ to its transform $g_0^{\varphi }$ under some diffeomorphism $\varphi$. In the example shown, this occurs once, so $\mathrm{Pf}(\mathsf{D})$ is odd under $\varphi$.