Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization

We study thermalization in a one-dimensional quantum system consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. Using a density matrix renormalization group algorithm we investigate the time evolution of the observables in the chain after a quantum quench. For low densities we show that the intermediate time dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show a prethermalization for the local observables at intermediate times and a full thermalization to the grand canonical ensemble at long times. For the case of a weak bath-chain coupling we find, in particular, a Fermi momentum distribution in the chain in equilibrium in spite of the seemingly oversimplified bath in our model.

Figure 1

$C_j(t)$ from Eq. (4) for $j=0,\dots ,5$ (top to bottom) for a quench with initial state $|{\Psi }_0^{\mathrm{I}}(1,1)⟩$, and Hamiltonian $H$ with $J=1$, $\gamma =1$, and $V_s=1$ at low densities. DMRG results (symbols) are compared to the solution of Eq. (5) (lines).

Figure 2

DMRG data (symbols) for $C_j(t)$ at half filling for a quench with initial state $|{\Psi }_0^{\mathrm{II}}(1,0.6,1)⟩$, and Hamiltonian $H$ with $J=1$, $\gamma =1$, and $V_s=1$. The lines are the thermal expectation values $⟨C_j{⟩}_T$.

Figure 3

(a) $f_q(t=0)$ for $|{\Psi }_0^{\mathrm{II}}(1,0.6,1)⟩$ (circles), $f_q(t=5)$ (triangles) and Fermi function fit $⟨f_q{⟩}_{T=0.7J}$, and the extrapolated distribution $f_q(t\to \infty )$ (diamonds) with a fit $⟨f_q{⟩}_{T=J}$. (b) $f_q(t\to \infty )$ (diamonds) compared to the thermal average $\mathrm{Tr}\{f_q{e}^{-H/T}\}/Z$ (solid line) and $⟨f_q{⟩}_T$ (dashed line) where $T/J=0.54$ is fixed by Eq. (2). (c) $f_q(t)$ for $q=13\pi /52$ (solid line) and $q=16\pi /52$ (dashed line).

Figure 4

The initial distribution (circles), $f_q(t\to \infty )$ from DMRG (diamonds), the thermal distribution $\mathrm{Tr}\{f_q{e}^{-H/T}\}/Z$ (solid line), and the free fermion distribution $⟨f_q{⟩}_T$ (dashed line) with (a) $\gamma =1$, $T/J=0.19$, (b) $\gamma =0.6$, $T/J=0.18$, and (c) $\gamma =0.2$, $T/J=0.33$.