Structure and dynamics of electron-phonon coupled systems using neural quantum states

AM, Robinson, Lee, Reichman arXiv:2405.08701

Ankit Mahajan

Introduction

V_{e-\text{lattice}}= \sum_n V(r_e-R_n-\delta R_n)\approx \sum_n V(r_e-R_n)-\delta R_n.\nabla V(r_e-R_n)

H_{\text{lattice}} \approx -\sum_n \frac{\nabla_n^2}{2} + \sum_{n,m}\frac{1}{2}\delta R_n. D_{mn}.\delta R_m

Quantum effects:

Riley, et al. Nature Communications (2018)

Nan, et al. Physical Review B (2009)

Electron-lattice interaction using linear approximation

Lattice energy within harmonic approximation

Outline

Model Hamiltonians

NQS and variational Monte Carlo (VMC)

Results

Calculation of dynamical correlation functions

Results

Ground state properties

Excited states and dynamical properties

Model Hamiltonians

H_{\text{eph}}=\sum_n\delta R_n.\nabla V(r_e-R_n)

Using a Bloch basis

H_{\text{eph}} = g_{kq}c_{k+q}^{\dagger}c_{k}\left(b^{\dagger}_{-q} + b_{q} \right)

H_e = \epsilon_{k}c_{k}^{\dagger}c_{k}\qquad H_{\text{ph}} = \omega_{q}b_{q}^{\dagger}b_{q}

General linear coupling

Local lattice Hamiltonians (in the site basis)

Holstein: density coupling

H_{\text{Holstein}} = -t_{\langle ij\rangle}(c_i^{\dagger}c_j+\text{h.c.}) + \omega_0b_i^{\dagger}b_i - g c^{\dagger}_ic_i(b_i^{\dagger}+b_i)

electron coupling to intramolecular vibrations in molecular crystals

Model Hamiltonians

SSH / Peierls: phonon modulated hopping

H_{\text{SSH}}^{\text{eph}} = -g \sum_{\langle ij\rangle} (c_i^{\dagger}c_j + \text{h.c.})(x_i-x_j)

Bond model: phonons on bonds

H_{\text{Bond}}^{\text{eph}} = - g \sum_{\langle ij\rangle}(c_i^{\dagger}c_j + \text{h.c.})x_{\langle ij\rangle}

Fillings:

Dilute limit: polaron and bipolaron effects, relevant in lightly doped or photoexcited carriers in semiconducting systems
Dense limit: interplay of e-e and eph interactions

H_{\text{Hubbard}} = Un_{i\uparrow}n_{i\downarrow}

Outline

Model Hamiltonians

NQS and variational Monte Carlo (VMC)

Results

Calculation of dynamical correlation functions

Results

Ground state properties

Excited states and dynamical properties

Neural quantum states

x_{i+1, j} = \text{ReLU}(W_{i,j,k}\circ x_{i,k}+b_{i,j})

NN(n_e, n_{\nu})

Occupation numbers as inputs to a fully-connected feedforward network

n_e,n_{\nu}

|\psi_{\text{NQS}}\rangle = \sum_{n} \frac{\exp{(f(n))}}{\sqrt{\prod_in_{\nu}^i!}} |n\rangle

f(n) = NN_r(n) + iNN_{\phi}(n)

Symmetry projection (site basis): drastically improves performance

For dense systems, we use a GHF reference:

\langle n|\psi\rangle = \langle n|\psi_{\text{NQS}}\rangle \langle n_e|\psi_{\text{GHF}}\rangle

\psi^S(n)=\frac{1}{N}\sum_g c_g \psi(g.n)

With translational symmetry, polaron NQS with one hidden neuron is equivalent to the Toyozawa wave function

|\psi_{\text{Toyozawa}}^k\rangle=P_k\left(\sum_i \phi_i c_i^{\dagger}\right) \exp \left(-\sum_{\nu}\xi_{\nu}b_{\nu}^{\dagger} - \xi_{\nu}^*b_{\nu}\right)\ket{0}

e-e and e-ph Jastrows can also be efficiently represented using NQS

Variational Monte Carlo:

\frac{\langle \psi|O|\psi\rangle}{\langle \psi|\psi\rangle} = \sum_{w} \frac{|\langle\psi|w\rangle|^2}{\langle\psi|\psi\rangle}\frac{\langle w|O|\psi\rangle}{\langle w|\psi\rangle}

\frac{\langle w|O|\psi\rangle}{\langle w|\psi\rangle} = \sum_{w'}\langle w|O|w'\rangle\frac{\langle w'|\psi\rangle}{\langle w|\psi\rangle}

For polarons and bipolarons, cost of energy calculation with translational symmetry ~ cost in the momentum basis

Rejection free sampling, AMSGrad optimization