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

http://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.3266

http://users.physik.fu-berlin.de/~pelster/Koenigstein/becca1.pdf

From Leon:

http://www.nature.com/nature/journal/v464/n7286/pdf/nature08917.pdf

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).

Cauchy theorem: a good place to review elementary group theory stuff

Cauchy’s theorem in group theory is an excellent place to review fundamental group stuff because (one of) its proof(s) involves the concepts of normal group, centralizer, orbit-stabilizer theorem and conjugacy class equation, etc. On the other hand, it is a crucial step to many important $p$-group results, including Sylow’s theorems.

Theorem (Cauchy):

$G$ a finite group and $p$ a prime. Then if $p\big| |G|$ $\Rightarrow$ $\exists g\in G$ s.t. $g$‘s order is $p$.

Proof:

(i)When $G$ is abelian: do induction on $|G|$. For $|G|=n$: consider $a \in G$, $a\neq e$. If $p\big |\langle a\rangle|$, done; otherwise $p\big| [G:\langle a\rangle]$, by inductive hypothesis $\exists x\langle a\rangle$ of order $p$ in $G/\langle a\rangle$ for some $x\in G$. If $m$ is the order of $x$ in $G$, then $(x\langle a\rangle)^m=\langle a\rangle\Rightarrow p|m$, done.

(ii)For a general $G$: also do induction on $|G|$. for $|G|=n$: $Z(G)$ is abelian. If $p\big| |Z(G)|$, done by (i); otherwise from conjugacy class equation, $\exists \text{Cl}(a)$, where $a\notin Z(G)$, s.t. $p\nmid |\text{Cl}(a)|=[G:C_G(a)]\Rightarrow p\big||C_G(a)|, done by induction.

Hilbert’s fifth problem and Gleason metrics

The easy version of this question is: when a topological group is a Lie group?

Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

Theorem 1 (Hilbert’s fifth problem) Let $latex {G}&fg=000000$ be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then $latex {G}&fg=000000$ is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem) Let $latex {G}&fg=000000$ be a locally compact group, and let $latex {U}&fg=000000$ be an open neighbourhood of the identity in \$latex…

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Notes on Group Extension

Here is notes on group extension I wrote, both for the modern algebra class and for my research:

220-notes

The group extension section follows closely section 9.1 of An Introduction to Homological Algebra by Jeseph Rotman.

By the time you see it, I may have added a correspondence between group extension and the PSG used in classifying U(1) Z2 spin liquids.