Time Scale for Adiabaticity Breakdown in Driven Many-Body Systems and Orthogonality Catastrophe

The adiabatic theorem is a fundamental result in quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what “slowly enough” means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely, many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter-dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving nonautonomous time evolution in a large Hilbert space.

Figure 1

Triangle inequality resulting in the estimate Eq. (8). States are shown as points in the projective Hilbert space. The red trajectory shows the evolution of the instantaneous ground state, Eq. (1), while the blue trajectory corresponds to the physical evolution given in Eq. (2). The length of side $b$ is bounded by the quantum speed limit, while the length of side $c$ approaches the maximally possible distance of 1 in the large $N$ limit.

Figure 2

Adiabaticity, orthogonality catastrophe, and transport in a Thouless pump described by the Hamiltonian (12) with $N=1000$ particles. (a) Illustration of the inequality (8). Solid blue—the orthogonality overlap $\mathcal{C}(\lambda )$. The shaded region is the one which has to contain the $\mathcal{F}(\lambda )$ curve due to the inequality (8). This bound is tight enough to guarantee the adiabaticity breakdown for the chosen set of parameters, $J=0.4E_R$, $U=0.4E_R$, $\mathrm{\Delta }=\lambda E_R$ with $\lambda =\mathrm{\Gamma }t$ and $\mathrm{\Gamma }=0.7E_R$. For $E_R=6.4{\mathrm{ms}}^{-1}$ these parameters coincide with those of the effective Hamiltonian describing the optical lattice in the experiment Ref. [43] at the $\mathrm{\Delta }=0$ point of the pumping cycle. Remarkably, true adiabatic fidelity $\mathcal{F}(\lambda )$ calculated numerically follows $\mathcal{C}(\lambda )$ even closer than what can be expected from the bound (8), so that the two curves look indistinguishable in the figure. The latter fact indicates that sides $b$ and $c$ of the triangle depicted in Fig. 1 are in fact nearly orthogonal in the present case. (b) Operation of the Thouless pump in the first cycle as compared to a steady-state regime, for different values of the driving rate, $\mathrm{\Gamma }$. The cycle is given by $\mathrm{\Delta }=(J/2)\sin \lambda$, $U=(J/2)\cos \lambda$ with $\lambda =\mathrm{\Gamma }t$. Initially the system is in equilibrium. Dash-dotted gray—charge transferred in the first cycle. Dashed red—the adiabatic fidelity $\mathcal{F}$ at the end of the first cycle. Solid black—charge transferred per cycle in a continuous regime after the stationary state is reached. One can see that the charge transferred in the first cycle is quantized even when the many-body adiabaticity has gone completely ($\mathcal{F}\simeq 0$), while the quantization of the charge in the continuous regime disappears as soon as the many-body adiabaticity is broken.