Variational description of electron dynamics
Why study dynamics?
- Directly related to experiments: electronic response to experimental probes
- At finite temperatures, electron dynamics are always present
- Goal: Develop a set of ab initio methods to study electron dynamics in strongly correlated molecular and condensed systems
Damped harmonic oscillator
x¨+γx˙+ωo2x=F(t)
\ddot{x}+\gamma\dot{x}+\omega_o^2x =F(t)
Response to an external time-dependent force F(t)
x(t)=∫−∞∞dt′χ(t−t′)F(t′)Fx(ω)=χ(ω)F(ω)
x(t) = \int_{-\infty}^{\infty} dt' \chi(t-t')F(t') \xrightarrow{\mathcal{F}} x(\omega) = \chi(\omega) F(\omega)
χ(ω)=−ω2+iγω+ω021
\chi(\omega) = \dfrac{1}{-\omega^2+i\gamma\omega+\omega_0^2}
green's function, susceptibility, response function, ...
χ(τ)∝Θ(τ)e−γτ/2
\chi(\tau)\propto\Theta(\tau)e^{-\gamma\tau/2}
Imχ(ω)=(ω02−ω2)2+γ2ω2ωγ
\text{Im}\chi(\omega) = \dfrac{\omega\gamma}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}
Spectral function ∝ dissipation (γ)
ω±=−2iγ±ω02−γ2/4
\omega_{\pm}=-\dfrac{i\gamma}{2}\pm\sqrt{\omega_0^2-\gamma^2/4}
Poles:
Linear response functions in QM
Suppose a system is perturbed as Hext=F(t)B^, then the response of ⟨A^⟩ to linear order is given by
δ⟨A^⟩(t)=∫−∞∞χAB(t−t′)F(t′)dt′
\delta\langle\hat{A}\rangle(t) = \int_{-\infty}^{\infty}\chi_{AB}(t-t')F(t')dt'
Kubo formula:
χAB(t−t′)=−iΘ(t−t′)⟨[A^(t),B^(t′)]⟩0
\chi_{AB}(t-t') =-i\Theta(t-t')\langle[\hat{A}(t),\hat{B}(t')]\rangle_0
Spectral representation:
χAB(ω)=∑m=0ω−(Em−E0)+iη⟨ψ0∣A^∣ψm⟩⟨ψm∣B^∣ψ0⟩−ω−(E0−Em)+iη⟨ψ0∣B^∣ψm⟩⟨ψm∣A^∣ψ0⟩
\chi_{AB}(\omega) = \sum_{m\neq0}\dfrac{\langle\psi_0|\hat{A}|\psi_m\rangle\langle\psi_m|\hat{B}|\psi_0\rangle}{\omega-(E_m-E_0)+i\eta} -
\dfrac{\langle\psi_0|\hat{B}|\psi_m\rangle\langle\psi_m|\hat{A}|\psi_0\rangle}{\omega-(E_0-E_m)+i\eta}
For the case A^=B^, dissipation ∝ fluctuations
Imχ(ω)=−π∑m=0∣⟨ψ0∣A^∣ψm⟩∣2δ(ω−(En−E0))∝⟨A^(ω)A^(0)⟩0
\text{Im}\chi(\omega)=-\pi\sum_{m\neq0}|\langle\psi_0|\hat{A}|\psi_m\rangle|^2\delta(\omega-(E_n-E_0))\propto\langle \hat{A}(\omega)\hat{A}(0)\rangle_0
Fermi's golden rule: scattering amplitude for probe particle, could be a photon, neutron, electron, etc.
Different spectroscopies couple to different operators A^
- Photoemission/attachment: c^k,c^k†
- X-ray scattering: ρ(x)
- neutorn scattering: ρσ(x)
Adiabatic theorem: Elementary excitations of the electron Fermi liquid
- Model condensed system: interacting electron liquid, compensating uniform background
- In the noninteracting limit, reduces to free Fermi gas with excitations described by free electrons and holes
- Adiabatic theorem: If you imagine turning on the interaction slowly, these simple states evolve into quasiparticles
How can we describe particle-hole excitations?
- Add excitations to interacting ground-state:
∣ψq⟩=ck+q†ck∣ψ0⟩
|\psi_{\mathbf{q}}\rangle=c_{\mathbf{k}+\mathbf{q}}^{\dagger}c_{\mathbf{k}}|\psi_0\rangle
- Construct effective Hamiltonian for these states that encodes interactions between them
Hqq′eff=⟨ψq∣H∣ψq′⟩
H^{\text{eff}}_{\mathbf{q}\mathbf{q'}}=\langle\psi_{\mathbf{q}}|H|\psi_{\mathbf{q'}}\rangle
- Find low-lying eigenvalues of this effective Hamiltonian to find excitation energies
- This approach is very general and used in traditional theories like equation of motion CCSD
Strong interactions destroy quasiparticles
- Hubbard model: undergoes Mott transition at critical U, even at zero temperature!
H=∑⟨ij⟩,σ(ciσ†cjσ+h.c.)+∑iUni↑nj↓
H = \sum_{\langle ij\rangle,\sigma}(c_{i\sigma}^{\dagger}c_{j\sigma}+\text{h.c.}) +\sum_{i}Un_{i\uparrow}n_{j\downarrow}
- For small U, behaves like a weakly interacting Fermi liquid
- As U is increased, electrons become localized and the system becomes an insulator, no explanation in terms of quasiparticles possible
Describing correlated excitations
- Same strategy as before, act excitations onto the ground state, now in terms of local orbitals
∣ψij⟩=ai†aj∣ψ0⟩
|\psi_{ij}\rangle=a_{i}^{\dagger}a_{j}|\psi_0\rangle
- Construct an effective Hamiltonian for these states and find low lying states
- Numerically, deterministic algorithms require construction of high-rank reduced density matrices to accomplish this
∣ψ0⟩=J^P^∣ϕ⟩
|\psi_0\rangle=\hat{\mathcal{J}}\hat{P}|\phi\rangle
- We use Jastrow symmetry projected mean field states
A way to avoid RDM's: use stochastic sampling
⟨ψs∣ψs⟩⟨ψμ∣H∣ψν⟩=∑n⟨ψs∣ψs⟩∣⟨n∣ψs⟩∣2⟨ψs∣n⟩⟨ψμ∣n⟩⟨n∣ψs⟩⟨n∣H∣ψν⟩
\dfrac{\langle\psi_{\mu}|H|\psi_{\nu}\rangle}{\langle{\psi_s}|{\psi_s}\rangle} = \sum_n\dfrac{|\langle n|\psi_s\rangle|^2}{\langle\psi_s|\psi_s\rangle}\dfrac{\langle\psi_{\mu}|n\rangle}{\langle\psi_{s}|n\rangle}\dfrac{\langle n|H|\psi_{\nu}\rangle}{\langle n|\psi_s\rangle}
walker
Generate Hamiltonian excitations from the walker using heat-bath screened sampling
We implemented this for multireference configuration interaction calculations which require similar Hamiltonian construction
Preliminary results for excited states
Improvements: need to include more excitation classes, e.g. cofermions
Hydrogen chain of 10 atoms, near equilibrium geometry
ziσ=ciσniσˉ
z_{i\sigma}=c_{i\sigma}n_{i\bar{\sigma}}
Time-dependent variational principle
A variational principle for the time-dependent Schrodinger equation!
⟨δψ∣(H−i∂t)∣ψ⟩=0
\langle\delta\psi|\left(H-i\partial_t\right)|\psi\rangle = 0
Geometrically:
Expand wavefunction linearly in parameters at ∣ψ0⟩, and diagonalize the effective Hamiltonian in the tangent space:
Another approach to get excited states
Numerically, all gradients can be evaluated at the same cost as the function!
Thank you!
Variational description of electron dynamics
super_group2
By Ankit Mahajan
super_group2
- 247
