A Spin Witness
A minimal picture of the system I'm trying to understand: two reservoirs (L, R) connected through a chiral molecule.
Below is a diagram of the setup. Current can flow from the left reservoir through the chiral molecule to the right reservoir.
Three possible molecule geometries between the reservoirs: a linear chain, a zig-zag chain, and a helical chain.
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The physical motivation
Overview
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The Hamiltonian and its geometry
Step 1 — The spinor ψm
The two-component object ψm = (ψm↑, ψm↓)t is a spinor: a quantity that transforms under rotations via exp(iφ n·σ/2) rather than like a vector, and famously changes sign under a 2π rotation. Everything in the Hamiltonian acts on these spinors.
- spinors
- LL3-§56
- spin-half
Check your understanding 3 questions
1. What is a spinor, and how does it differ from a vector under rotation?
2. Write the operator that describes how a rotation through angle φ around axis n acts on a spinor, and identify each piece.
3. In the paper Hamiltonian, the spin-orbit term acts on ψm via the operator vm·σ. Given what you know about spinors from LL3 §58, what is this operator physically doing to the spinor, and why does the direction of vm matter?
Step 2 — The hopping matrix Hν
The hopping matrix has two parts: scalar nearest-neighbour hopping (spin-blind), and next-nearest-neighbour spin-orbit coupling that rotates the spinor around the axis vm via the operator vm·σ.
- tight-binding
- spin-orbit
- hopping
Check your understanding 3 questions
1. The hopping matrix Hν has two terms. What does each one do, and how do you tell them apart structurally?
2. What does the index s = ±1 encode in the hopping matrix, and what happens to vm(s) when you reverse s?
3. The spin-orbit term in Hν is iλν vm(s)·σ. From what you established in Step 1 about spinors and rotation generators, what is this term doing to ψm, and why does the factor i appear?
Step 3 — The curvature vector vm(s)
The vector vm(s) = dm,s × dm+s,s is the cross product of two consecutive bond directions, a discrete curvature that encodes how the chain bends at site m, and directly determines the direction of the spin-orbit coupling via LL3 §72.
- curvature
- geometry
- LL3-§72
Check your understanding 4 questions
1. Define vm(s) precisely and explain what it is geometrically.
2. What are the three structural cases for vm(s) and what does each give?
3. LL3 §72 derives spin-orbit coupling for an electron in a self-consistent field as HSO ∝ σ·(E × p). How does this connect to the term iλ vm(s)·σ, and what plays the role of E × p?
4. Why is the longitudinal component of vm(s) specifically necessary for CISS, and why can a zig-zag chain not produce it?
Step 4 — Vibrations and reservoirs
The vibrational Hamiltonian adds harmonic molecular vibrations coupled to the electrons, with anharmonic corrections allowed by the broken inversion symmetry of the chiral structure; the reservoir coupling opens the system at both ends.
- electron-phonon
- open-system
- reservoirs
Step 5 — The full picture
The complete Hamiltonian encodes chirality entirely through vm(s): only a helical geometry gives vm a longitudinal component, which is the only component capable of generating spin polarization along the transport direction.
- chirality
- CISS
- geometry
Electron-electron interactions and the polaron transformation
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Green's function machinery
Layer 1 — The Green's function as an object
What is G, what does it contain, and why use it
- LL9 §7 — G defined
- SPT ch2 — propagators
- G<, G> — physical meaning
Layer 2 — Retarded, advanced, and spectral function
Gr, Ga, and the spectral function A; poles = excitations
- LL9 §36 — Gr and Ga
- SPT §3.2 — spectral A(ω)
- Dispersion & Kramers-Kronig
Key result: Gr − Ga = 2i Im Gr = i A(ω). Poles of Gr give the quasiparticle spectrum.
Layer 4 — Non-equilibrium: Keldysh and G<
The Keldysh relation: G< = Gr Σ< Ga. This is where LL9 ends and SPT takes over.
- SPT §3.1 — Keldysh contour
- Langreth rules → G< formula
- Σ< = i(ΓL fL + ΓR fR)
LL9 §36 eq 36.14 covers the equilibrium limit. SPT ch3 + model paper eq (16) cover the full case.
Key: in equilibrium Σ< = 2i f(ω) Im Σr; out of equilibrium each reservoir contributes separately.
MFA
Layer 1
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The physical interpretation
Step 1 — MR requires a ferromagnet, but spin polarization does not
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Step 2 — Time reversal and what it means for the molecular state (LL3 §60)
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The solution is at the center of the problem
What one has to understand is that jumping to solutions will create further problems. The solution is hidden at the most ingenious of places … The only one who finds it is the person who understands the problem. … but nobody wants to look at the problem …