Good book on field theory

It is Fields and Rings by Kaplansky!

it seems to be a small pamphlet but is concisely and very well composed. Only about 70 pages on Field Theory but almost all major results are there. What’s more, the exercises there are so good that I hav seen many people recommending them, including our Prof. Jacob who puts quite a few in his problem sets. The book is also what Prof. Jacob covered in his first eight week lectures.

I think the Ring Theory part, which is unusually put after Field Theory in the book, is also excellent lecture note. Anyway,  Kaplansky is a big guy in Algebra!


There may be other good books on Field Theory, such as Dummit’s and Rotman’s.


Talk by Dominic


Topological cystalline insulator (Liang Fu, 2011)

Interacting SPTs/SETs

Internal symmetry (charge conservation, Z^T_2, etc.)

Bosonic SPTs: Group cohomology classification H^{n+1}(G,U(1)) [Chen et. al. 2011]

Bosonic SETs G-crossed braided tensor categories [Barkeshli,,2014]

0) How do you gauge internal symmetry?

  1. How do you”gauge”spatial symmetry”?
  2. Classification of phases (U(1)) of phases)

If we have a U(1) symmetry \vec{A},

S_{SPT} \rightarrow S_{SPT}[\vec{A}]\xrightarrow{integrate out SPT} \frac{n}{2\pi} \int \varepsilon_{\mu\nu\lambda} A_\mu \partial_\nu A_\lambda

Hall conductance, integer


S_{SPT}\xrightarrow{gauge}S_{SPT}[A]\xrightarrow{integrate out SPT} Topological term, gauge field

[Dijgraaf Witten,1991] H^{d+1}(G,U(1))

What is a gauge field: it is a connection on a principal G-bundle. A physical picture of this: branch cuts, across which a particle state |\psi\rangle becomes g|\psi\rangle, where g is element of the group. (In case of \mathbb{R}^n, the group is GL). Branch cuts always have codim=1.

Examples (pass)

“Topological crystalline liquid”: topological part of it (TQFT) does not care about lattice. (liquid in the sense that it doesn’t care about the lattice)

E.g. G=1, X=\mathbb{R}^2, M = \mathbb{R}^2, f(z) = z^2,


Gauge field: M\rightarrow BG, Classifying space: BG=EG/G

“Crystalline gauge field” M\rightarrow X//G=(X\times EG)/G

H^{d+1}(BG,U(1))=H^{d+1}(G,U(1)), H^{d+1}(X//G,U(1)): equivariant cohomology (crystalline SPTs)



Some Latex Skills

I learnt something when

Define function ( use of [2] as “2 arguments”, #1 as “the first argument”),

use [T1]{fontenc} for inputing letters with accents (i.e. french words)

fontawesome for internet icons

fontspec to specify font for each use: use \newfontfamily{][]{}[] to define font for each context, where font size, font type (bold, italic, ..), letter spacing, etc. We can use \scalebox (from graphicx) to horizontally stretch letters

use tikz to plot: draw; node for plots with text; (to plot a rectangular missing one side, must plot three sides separately)

The importantce of fonts; fonts convention for resume

Berry Phase and Berry Curvature

The most horrible moment is not when you find something you have not learned and calculated; it is when you find something you have learned and calculated, but have no idea of this…

(In fact I have learned the intrinsic anomalous hall effect, and calculated the unquantized Hall conductivity in Cenke’s homework 4. But at that time I had no concept of AHE… I have also learned berry phases in undergrad, but at during work on Cenke’s homework I can make no connection to the old one I learned, nor could I relate berry phase to topology…)

Of course, Berry phase effects on electronic properties (Rev. Mod. Phys. 82, 1959 (2010)) is the reference for berry phase. This is an undergrad-level review paper, and whose content should be known to any undergrad who wants to do theoretical physics, by the end of their undergrad. I should have read it!

Condition under which berry phase cannot be neglected: “in systems with broken either time-reversal or inversion symmetries, their proper description requires the use of the full velocity formula 3.6.”

What is berry phase: note that a Hamiltonian is always a map from some parameter space to some target space; and the eigenstates are also map from this parameter space to a target space. The parameter space can be k space, or magnetic field space and also on. (The important thing is: a parameter space is a manifold, so we can consider topology on it!)

If we examine the map of an eigenstate on a closed path in the parameter space, we will find that this eigenstate, when coming back to its original place, accumulates a phase \gamma = i\oint d\mathbf{R}\cdot \langle \psi(\mathbf{R})|\nabla_{\mathbf{R}}|\psi(\mathbf{R})\rangle . (Note in this description, we have used adiabatic assumtion.) Of course, this phase is unique to quantum mechanics – it reflects that U(1) projective nature of physical states in the description of physical states (i.e. the Hilbert space). This \gamma is the berry phase.

As noted above, the phase \gamma is obtained only when we have a closed path in the parameter space. So how do we create a closed path in it? The review paper tells us this. In the case of the parameter space being a BZ, we can create a closed path by either applying a magnetic field (so that electrons go under cyclic motion) or apply high enough electric field (so that electrons drift on the dispersion, and due to the periodic nature of BZ they go back to their original spots). For a spin coupled to a magnetic field, we can let the magnetic field rotate in time (so that the magnetic field draws a circle in the parameter space of itself, which is simply S^2).

But the more intrinsic concept than berry phase is berry curvature — using Stoke’s theorem we can always transform the line integral of berry phase (it is really A_\mu(\mathbf{R})=i \langle \psi(\mathbf{R})|\nabla_{\mathbf{R}_\mu}|\psi(\mathbf{R})\rangle that we are integrating over the closed path!) to a surface integral of  F_{\mu\nu} = \partial_{[\mu}A_{\nu]} (in a more mathematically correct sense, a two-form integral on the parameter manifold). This object $F_{\mu\nu}$ is the berry curvature of the parameter space. The fundamental idea is that this berry curvature is intrinsic to the parameter space — it is the geometry of the parameter space.  (Note: strictly speaking, we cannot say a BZ is a parameter space — what we should say is that a dispersion band on BZ is a parameter space. This is because a BZ can have many bands on it, and each band has its own geometry and can define its own berry curvature!)

Now that we have defined berry curvature, what can we do with it? Note that this parameter space of \mathbf{R}, together with the berry curvature F_{\mu\nu} on it, is a space with a gauge field. So any way we study a gauge theory — we will use it to study the parameter space and berry curvature!

For derivation of TKNN invariant (quantization of \sigma_{xy} in QHE) and AHE (where \sigma_{xy} is not quantized), and other examples, see the review paper.

Introductory Reading for QSLs

From Leon:

Classical SL: =spin ice, e.g. in Dy2Ti2O7, large spin, frustration, but eventually order at ~0.5K

QSL: VBS(valence bond solid) is not QSL (b/c breaks translation, and more importantly, not long-range entangled) and have been found in many materials; RVB(resonating valence bond) is QSL(Anderson predict it to be physics of High Tc SC in 1987)

Understanding of existence of different QSLs: low energy excitation due to long-range VB configuration, also possible in short-range RVB by simply rearranging configurations. (And hence this is the motivation for classification of QSL, e.g. PSG.)

Defining property of QSL: long range entanglement; fractional excitation(spinon)(they can be shown to be equivalent); no symmetry breaking. Fractional excitations also occur in spin ice, but not true coherent quasiparticles.

long range entanglement = fractional (spinon) excitation = deconfined phase = boundary tensionless (spinons can space very long while only causing finite amount of energy). Spin ice becomes confined before 0 K, but QSL remains deconfined at 0 K.

 Spinons are well established in one-dimensional (1D) systems, in which they occur as domain walls(Fig. 4a). A spinon can thus be created similarly to a monopole in spin ice, by flipping a semi-infinite string of spins. A key difference, however, is that in one dimension the only boundary of such a string is its end point, so the string is guaranteed to cost only a finite energy from this boundary. By contrast, in two or three dimensions, the boundary of a string extends along its full length. A string would naturally be expected to have a tension (that is, there is an energy cost proportional to its length). String tension represents confinement of the exotic particle, as occurs for quarks in quantum chromodynamics. This is avoided in spin ice by the special form of the nearest-neighbour Hamiltonian. However, when spin ice is in equilibrium, corrections to this form would be expected to lead to monopole confinement at low temperatures. In a true 2D or 3D QSL, the string associated with a spinon remains robustly tensionless even at T = 0 K, owing to strong quantum fluctuations (Fig. 4c). This can be understood from the quantum superposition principle: rearranging the spins along the string simply reshuffles the various spin or valence-bond configurations that are already superposed in the ground state. Detailed studies of QSL states have shown that higher-dimensional spinons can have varied character. They may obey Fermi–Dirac36, Bose–Einstein37 or even anyonic statistics (see page 187). They may be gapped (that is, require a non-zero energy to excite) or gapless, or they may even be so strongly interacting that there are no sharp excitations of any kind38..


Experimental technique: nuclear magnetic resonance (NMR) and muon spin resonance experiments prove magnetic moments; specific heat (compare with theory); thermal transport and elastic/inelastic neutron scattering (excitation properties).

Materials are distinguished by their degree of Mott insulator (U/t).