## Topological Photonics

**Dec. 24, 2017, 7:00pm**

This is a step-by-step guide for those of us who would like to get into the game quickly. Since Berry's phase is of central importance to the whole field, we need to get it out of the way first. So here's what you'll have to do...

**Days 1, 2:** Read (meaning, use pen and paper to reproduce, line-by-line, all of) Chapter 10 "The Adiabatic Approximation" in Griffiths, "Intro to QM" (I'm using the original 1995 edition). Topics include: the adiabatic theorem, Berry's phase, and the Aharonov-Bohm effect. Est. time required: 6-8 hours per day, which will be our ballpark estimate for everything that follows.

**Days 3, 4:** Read Supplement I, "Adiabatic Change and Geometric Phase", in Sakurai, "Modern QM".

**Day 5 (fun day):** Read "Topological physics with light", M. Hafezi and J. M. Taylor, Physics Today 67, 5, 68 (2014)

**ASIDE 1:** In order to understand Haldane's work, we need to put together several key concepts from various branches of physics and modern math. Some of them are already familiar to us, some will be new. Initially, we will be aiming at the understanding of what's known as the quantum Hall photonic topological insulators (QH-PTI) which were the first to be theoretically described and experimentally implemented (circa 2009). The corresponding 2D photonic structures were designed to operate at microwave frequencies and consisted of a gyromagnetic material subjected to an external magnetic field which induced strong gyromagnetic anisotropy and resulted in the breaking of the time-reversal symmetry. These structures supported photonic topological phases that were mathematically equivalent to the quantum Hall phase of a magnetized 2D electron gas. To arrive at the understanding of that early work, let's begin with something simple...

**Days 6, 7: Magnetism in Matter. **Forget Jackson. All you need to know is that, in the presence of a magnetic material, Ampere's Law,

∇×B = μ_{0}j,

gets modified as

∇×B = μ_{0}(j_{free}+j_{bound}),

where

j_{bound} = ∇×M,

with M being magnetization. If you want to have wave propagation, you need to throw in the time-dependent ∂D/∂t term, known as Maxwell's correction. From this you get

∇×H = j_{free} + ∂D/∂t,

where

H = B/μ_{0} - M.

In linear magnetics,

M = χ_{m}H,

and thus,

B = μH,

with

μ = μ_{0}(1 + χ_{m}),

where μ is either a scalar (not interesting), or a matrix (interesting). **The whole business of topological photonics in gyromagnetic QH-PTIs revolves around the nontrivial effects causes by the off-diagonal matrix elements of tensor μ.**

**ASIDE 2:** The literature is huge, so we have to decide which papers to read first. My take is that it doesn't really matter, as long as you **start doing something**. I have decided to start with** the easiest portions which I could actually understand** of

Wang et al, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal", PRL 100, 013905 (2008)

(specifically, re-deriving their Eq. (8), cuz I like formulas). You can see my derivation here:

AG's NOTES: Derivation of Eq. (8) in Wang, et. al., PRL 100 (2008)

Notice that I have two extra minus signs at important locations, the most troubling of which is in the definition of the wave function, *ψ.* My minus sign sits in the exponent. **So you'll have to study my derivation carefully and let me know what went wrong (if anything).** But despite the discrepancy, here's the crucial thing to consider: the role of the effective "vector potential" for photons (!) is played by the quantity

Ã ∼ **z**×∇η,

where η is the off-diagonal matrix element of the inverse matrix μ^{-1}. This is where, in the early days, the topological game was played out.

**UPDATE (Dec. 27, 2017):** Lingyu Yang is **confirming** both of my "minus" signs,

Lingyu's NOTES: Derivation of Eq. (8) in Wang, et. al., PRL 100 (2008)

**UPDATE (Dec. 30, 2017):** See also a useful paper (which elaborates on some technical details in Wang et al) by

Chong et al, "Effective theory of quadratic degeneracies," PRB 77 (2008)

In your research, you'll likely be doing similar calculations.

**ASIDE 3:** For those interested, Yiping has a large collection of papers on this subject. Ask him to grant you access to his folder (https://www.physast.uga.edu/people/yiping_zhao ). Also, if you want to learn about photonic crystals, there is a free book by the MIT group. Click here for details: http://ab-initio.mit.edu/book/

**Day 8:** The next papers to look at (which I haven't done yet, but expect lots of things to still be very fuzzy) will be the review articles by:

Khanikaev and Shvets, "Two-dimensional topological photonics," Nature Photonics** 11**, 763 (2017)

and

Lu et al, "Topological photonics," Nature Photonics **8**, 821 (2014).

**Days 9, 10, 11, 12:** By now we'd be coming to the subject of topology. That cookie is tough, so it'll occupy us for several days. I have yet to figure out which mathematical sources to use, but Prof. Sigal had suggested that Barry Simon's paper might be a good start. So when you get to this point, you may want to try:

B. Simon, "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase," PRL 51, 2167 (1983)

and then explain all that to me and everybody else. After that, the next in line (for a much more serious study!) will be the course by

J. Avron, "Adiabatic quantum transport"

In the meantime, check out

J. Avron et al, "A Topological Look at the Quantum Hall Effect," Phys. Today 56, 38 (2003)

Here's another relevant paper, which deals with Chern numbers in discretized Brillouin zone (seems important):

See also:

F. Haldane, "Berry Curvature on the Fermi Surface," PRL 93, 206602 (2004)

**Days 13, 14:** Other concepts to review: band structure, Brillouin zone, etc.

To be continued...

**DISCLAIMER:** This guide is a work in progress. Major errors are possible. People who already went through the learning curve are invited to contribute.

**Key concepts:** Berry (geometric, topological) phase; Berry connection; Berry curvature, Chern Theorem; Chern class; Chern numbers; closed manifold; Bloch's Theorem; reciprocal space; Brillouin zone; electronic band structure; photonic crystal; Quantum Hall Effect; Fractional Quantum Hall Effect; Landau levels; Fermi energy; topological insulator; surface states;

**Educational Resources:**

Paper by Chong et al, "Effective theory of quadratic degeneracies," Phys. Rev. B 77 (2008)

Paper by F. Haldane, "Berry Curvature on the Fermi Surface," PRL 93, 206602 (2004)

A course at Delft "Topology in Condensed Matter: Tying Quantum Knots" (thanks to Keenan Stone for providing the link)

The free book on photonic crystals by the MIT group: http://ab-initio.mit.edu/book/

The course by Joseph Avron, "Adiabatic quantum transport" (thanks to Prof. I. M. Sigal for recommending this reference)

J. Avron et al, "A Topological Look at the Quantum Hall Effect," Phys. Today 56, 38 (2003)

Lecture notes by David Tong, "The Quantum Hall Effect"

**What others are doing:**

Gennady Shvets' Group at Cornell

Alexander Khanikaev at CCNY: See, e.g., the following paper with co-authors,

Slobozhanyuk et al, "Three-dimensional all-dielectric photonic topological insulator," Nature Photonics 11, 130 (2017)

Yuri Kivshar at ANU

Charles Kane at UPenn

Hafezi Group at UMD

**NOTES:**

**Lingyu Yang**: "Ch. 10 Griffiths QM: Adiabatic Theorem, Berry's Phase, Aharonov-Bohm Effect", "Chern Numbers in Quantum Mechanics & QHE",

AG (2017, Dec. 26): "Ch. 10 Griffiths QM: Adiabatic Theorem, Berry's Phase, Spin-1/2 in a precessing magnetic field"

AG (2017, Dec. 27): "Ch. 10 Griffiths QM: Aharonov-Bohm Effect"

AG (2017, Dec. 29): "Landau Levels"; (2017, Dec. 31): "Hall Conductance", ...

AG (2018, Jan. 05): "Photonic Crystals: Inner product of fields & Hermiticity of the master operator (NOTE: here bi-anisotropy is NOT considered, so this is applicable to Wang et al, but not Haldane)"

Table of Fourier Transforms & Trigonometric Identities

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