Lecture 7

03/13/23, M.

Today

Charge and magnetic impurity in Fermi liquid
Wavepacket dynamics
Semiclassical dynamics (general)

Infrared singularities: some additional interesting features of the low-energy physics of the electron gas associated with the gapless spectrum of the Landau Fermi liquid and the finite density of states for low-energy excitations.

"In physics, an infrared divergence (also IR divergence or infrared catastrophe) is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or equivalently, because of physical phenomena at very long distances."

The neutralization of the impurity charge requires electron flow from infinite distances. This means that the ground state with the impurity is orthogonal to that without, a fact called the e infrared or orthogonality “catastrophe”.
Suddenly turn on the impurity. The weight of new gs is zero. the dynamics is characterized by creating low-energy excitations in the electron gas
Energy dumped into the e-h pairs is finite, but the number is infinite. Hence the name infrared divergence or catastrophe.

1. Friedel sum rule

From the static Lindhard approximation

\begin{matrix} (1) & δ n (q) = χ^{0} (q) (v_{ext} (q) + v_{c} (q) δ n (q)) \end{matrix}

$+Ze$ $v_{\text{ext}}(q)=-Ze^2/\epsilon_0q^2$ $q\rightarrow 0$ , the divergence of the potential leads to

\begin{matrix} (2) & lim_{q \to 0} δ n (q) = + Z \end{matrix}

that is, the screening charge completely neutralizes the external charge.

In the Fermi liquid approach, we treat the particles as nearly non-interacting.

$\ell$
$\begin{matrix} (3) & ψ_{ℓ} (k r) \sim \frac{1}{k r} \sin (k r + δ_{ℓ} - \frac{ℓ π}{2}) . \end{matrix}$
$R$
$\psi_{\ell}(k R)=0$
$\begin{matrix} (4) & k_{n} R + δ_{ℓ} (k_{n}) - \frac{ℓ}{2} π = n π . \end{matrix}$
$\varepsilon_K$
$\begin{matrix} (5) & \begin{array}{r} N_{ℓ} = \sum_{n = 1}^{\infty} θ (K - k_{n}) \approx \int_{0}^{K} d k \frac{\partial n}{\partial k}, \end{array} \end{matrix}$ $\begin{matrix} (6) & \frac{\partial n}{\partial k} = \frac{1}{π} (R + \frac{\partial δ_{ℓ}}{\partial k}) \end{matrix}$
The 1st term is number of states without impurity
The change in number of states
$\begin{matrix} (7) & Δ N_{ℓ} = \int_{0}^{K} d k \frac{1}{π} \frac{d δ_{ℓ}}{d k} = \frac{1}{π} [δ_{ℓ} (K) - δ_{ℓ} (0)] = \frac{1}{π} δ_{ℓ} (k_{F}) \end{matrix}$
Since the bulk value of the Fermi energy is not shifted by the presence in the system of a single impurity, we can replace K by the Fermi wave vector
Friedel sum rule:
$\begin{matrix} (8) & 2 \sum_{ℓ = 0}^{\infty} (2 ℓ + 1) \frac{δ_{ℓ} (k_{F})}{π} = + Z \end{matrix}$
$(2 \ell + 1)$ accounts for the multiplicity of the the channel and the overall factor of 2 accounts for the spin multiplicity.

2. Kondo problem

Dilute magnetic impurity in metals

Resistivity minima in Mo-Nb alloy with traces of magnetic Fe.

The essence of the problem is the exchange coupling of a local spin with the Fermi surface

\begin{matrix} (9) & H = \sum_{k s} ε_{k} c_{k s}^{†} c_{k s} + J_{⊥} (S_{+} s_{-} + S_{-} s_{+}) + J_{z} S_{z} s_{z} \end{matrix}

$\hbar =1$

\begin{matrix} (10) & \begin{aligned} s_{+} & = \frac{1}{V} \sum_{k, k^{'}} c_{k^{'} ↑}^{†} c_{k ↓} \\ s_{z} & = \frac{1}{V} \sum_{k, k^{'}} \frac{1}{2} (c_{k^{'} ↑}^{†} c_{k ↑} - c_{k^{'} ↓}^{†} c_{k^{'} ↓}) \end{aligned} \end{matrix}

$J_\perp=J_z$
$J_\perp \neq J_z$
$J>0$ : antiferromagnetic coupling
$J<0$ : ferromagnetic coupling

We begin with a simpler problem

\begin{matrix} (11) & H = H_{0} + \frac{J}{V} \sum_{k, k^{'}} c_{k^{'}}^{†} c_{k} \end{matrix}

particle route (b)
$\begin{matrix} (12) & J (Λ - d Λ) = J (Λ) + ρ (0) d Λ \frac{J (Λ)^{2}}{- Λ} \end{matrix}$
$\rho(0)=g(0)/V$ is the density of states per unit volume.
hole route (c)
$\begin{matrix} (13) & J (Λ - d Λ) = J (Λ) + ρ (0) d Λ \frac{J (Λ)^{2}}{- Λ} (- 1)^{F} \end{matrix}$
$F=1$ accounts for the sign of fermion exchange.
So the flow equation is marginal
$\begin{matrix} (14) & \frac{d J}{d Λ} = 0 \end{matrix}$

Now we renormalize the exchange interaction.

$J_\perp=0$ $J_z$ is precisely like the above, so
$\begin{matrix} (15) & \frac{d J_{z}}{d Λ} = 0 \end{matrix}$
$J_\perp\neq 0$ , there is spin flip scattering, things get mixed up.
$J_\perp$
$\begin{matrix} (16) & \frac{J_{⊥} (Λ - d Λ)}{2} = \frac{J_{⊥} (Λ)}{2} + ρ (0) d Λ \frac{[+ \frac{1}{2} J_{⊥} (Λ)] [- \frac{1}{4} J_{z} (Λ)]}{- Λ} \end{matrix}$
$\begin{matrix} (17) & \frac{J_{⊥} (Λ - δ Λ)}{2} = \frac{J_{⊥} (Λ)}{2} + ρ (0) d Λ \frac{[+ \frac{1}{2} J_{⊥} (Λ)] [+ \frac{1}{4} J_{z} (Λ)]}{- Λ} (- 1)^{F} \end{matrix}$
So the flow equation
$\begin{matrix} (18) & \begin{aligned} \frac{d J_{⊥}}{d Λ} = - \frac{ρ (0)}{Λ} J_{⊥} J_{z} \\ \frac{d J_{z}}{d Λ} = - \frac{ρ (0)}{Λ} J_{⊥}^{2} . \end{aligned} \end{matrix}$ $\begin{matrix} (19) & J_{⊥} \frac{d J_{⊥}}{d Λ} - J_{z} \frac{d J_{z}}{d Λ} = 0 \end{matrix}$
Invariant of the flow
$\begin{matrix} (20) & J_{⊥}^{2} - J_{z}^{2} = C \end{matrix}$
$J_\perp<0$ , irrelevant, the problem reduces to a soluble model.
Antiferromagnetic coupling is relevant. quenches the Curie susceptibility, and contributes to the electrical conductance.
$SU(2)$ symmetry:
$\begin{matrix} (21) & \begin{matrix} \frac{d J}{d Λ} = - ρ (0) J^{2} / Λ \\ ρ (0) J = \frac{ρ (0) J_{0}}{1 + ρ (0) J_{0} \log (Λ / Λ_{0})} \end{matrix} \end{matrix}$
$\Lambda \rightarrow k_BT$
$\begin{matrix} (22) & ρ (0) J_{eff} = \frac{ρ (0) J_{0}}{1 + ρ (0) J_{0} \log (k_{B} T / Λ_{0})} \end{matrix}$
divergence at the Kondo temperature
$\begin{matrix} (23) & k_{B} T_{Kondo} = Λ_{0} e^{- 1 / ρ (0) J_{0}} \end{matrix}$
$\Lambda_0/k_B=T_F$ .

Kenneth Wilson’s non-perturbative numerical treatment of the flow equations (including the flow of many new coupling constants associated with new terms generated in the Hamiltonian) gives quantitatively accurate results valid down to arbitrarily low temperatures [134]. Quite remarkably, Nathan Andrei [135, 136] and Paul Wiegman [137] were later able to provide exact analytic expressions for the many-body eigenstate wave functions using Bethe ansatz methods.

3. Wavepacket in free space (1D)

4. Semiclassical dynamics

The time-dependent variational principle

\begin{matrix} (24) & L = ⟨ ψ | i ℏ \frac{d}{d t} - H | ψ ⟩ \end{matrix}

$\text{\"o}$ dinger equation.

Now we develop a generalized variational principle

\begin{matrix} (25) & ψ (t) \to ψ (λ (t)) . \end{matrix}

$\lambda ^\alpha(t)$ are time-dependent parameters.

The Lagrangian is then

\begin{matrix} (26) & L = ℏ A_{α} {\dot{λ}}^{α} - H (λ) \end{matrix}

where

\begin{matrix} (27) & \begin{aligned} A_{α} = ⟨ ψ (λ) | i \frac{\partial}{\partial λ^{α}} | ψ (λ) ⟩ \\ H (λ) \equiv ⟨ ψ (λ) | H | ψ (λ) ⟩ \end{aligned} \end{matrix}

The action

\begin{matrix} (28) & S = \int_{t_{1}}^{t_{2}} L (λ, \dot{λ}) d t \end{matrix}

The Euler-Lagrange equation of motion is

\begin{matrix} (29) & ℏ \frac{\partial A_{β}}{\partial λ^{α}} {\dot{λ}}^{β} - \frac{\partial H}{\partial λ^{α}} - ℏ \frac{d A_{α}}{d t} = 0 \end{matrix}

\begin{matrix} (30) & {\dot{λ}}^{α} = Q^{α β} \partial_{β} H \end{matrix}

where

\begin{matrix} (31) & \begin{matrix} Q^{α γ} F_{γ β} = δ_{β}^{α} ℏ \\ F_{α β} = \partial_{α} A_{β} - \partial_{β} A_{α} \end{matrix} \end{matrix}

$Q$ $F$ are antisymmetric.

\begin{matrix} (32) & \frac{d H}{d t} = \frac{\partial H}{\partial λ^{α}} {\dot{λ}}^{α} = \frac{\partial H}{\partial λ^{α}} Q^{α β} \frac{\partial H}{\partial λ^{β}} = 0 \end{matrix}

Consider an infinitesimal interval of time

\begin{matrix} (33) & \begin{matrix} t \to t + d t \\ λ \to λ^{'} \end{matrix} \end{matrix}

The phase space volume

\begin{matrix} (34) & d^{n} λ^{'} = \det [J (t^{'}, t)] d^{n} λ \end{matrix}

where the Jacobian matrix is

\begin{matrix} (35) & \begin{aligned} J_{β}^{α} (t^{'}, t) & = \frac{\partial {λ^{'}}^{α}}{\partial λ^{β}} \\ = δ_{β}^{α} + \frac{\partial Q^{α γ}}{\partial λ^{β}} \partial_{γ} H + Q^{α γ} \frac{\partial^{2} H}{\partial λ^{β} \partial λ^{γ}} \end{aligned} \end{matrix}

$J$

\begin{matrix} (36) & \begin{aligned} \log \det J & = Tr \log J \\ = (\partial_{α} Q^{α γ}) \partial_{γ} H \end{aligned} \end{matrix}

We find

\begin{matrix} (37) & d^{n} λ^{'} = [1 + (\partial_{α} Q^{α γ}) \partial_{γ} H] d^{n} λ \end{matrix}

$Q$ $F$ $\lambda$ $\text{det}J=1$

\begin{matrix} (38) & d^{n} λ^{'} = d^{n} λ \end{matrix}

phase space density is constant. This is the Liouville theorem.

$\eqref{eq:gliou}$ $\lambda(\xi)$ $Q^{(\xi)}$ $\xi$ . Then we have[1]

\begin{matrix} (39) & \sqrt{\det F^{(ξ)}} d^{n} ξ = \sqrt{\det F^{(ξ^{'})}} d^{n} ξ^{'} \end{matrix}

$\lambda$

\begin{matrix} \begin{matrix} \begin{aligned} F_{α β}^{(λ)} & = F_{δ η}^{(ξ)} \frac{\partial ξ^{δ}}{\partial λ^{α}} \frac{\partial ξ^{η}}{\partial λ^{β}} \\ \det (F^{(ξ)}) & = \det (F^{(λ)}) \det {(\frac{\partial λ^{α}}{\partial ξ^{β}})}^{2} \\ \Rightarrow \sqrt{\det F^{(λ)}} \det (\frac{\partial λ^{α}}{\partial ξ^{β}}) d^{n} ξ & = \sqrt{\det F^{' (λ)}} d^{n} λ^{'} \det (\frac{\partial {λ^{'}}^{α}}{\partial ξ^{' β}}) \end{aligned} \end{matrix} \end{matrix}

So we have

\begin{matrix} (40) & Pf [F (λ)] d^{n} λ = Pf [F (λ^{'})] d^{n} λ^{'} \end{matrix}

The genialized Liouville theorem

\begin{matrix} (41) & Pf [F (λ)] d^{n} λ \end{matrix}

Reference and notes

$F$ is antisymmetric and real valued, it can be orthogonally diagonalized with imaginary eigenvalues

\begin{matrix} (47) & F^{λ} = O K O^{T} \end{matrix}

So we can choose

\begin{matrix} (48) & J = O K^{- \frac{1}{2}} O^{T} \end{matrix}

which turns out to be real-valued.