Quantum Spin Pump in S=1/2 antiferromagnetic chains -Holonomy of phase operators in sine-Gordon theory-

In this paper, we propose the quantum spin pumping in quantum spin systems where an applied electric field ($E$) and magnetic field ($H$) cause a finite spin gap to its critical ground state. When these systems are subject to alternating electromangetic fields; $(E,H)=(\sin\frac{2\pi t}{T},\cos\frac{2\pi t}{T})$andtravelalongtheloop$\Gamma_{\rm{loop}}$ which encloses their critical ground state in this $E$-$H$ phase diagram, the locking potential in the sine-Gordon model slides and changes its minimum. As a result, the phase operator acquires $2\pi$ holonomy during one cycle along ${\mathrm{\Gamma }}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$, which means that the quantized spin current has been transported through the bulk systems during this adiabatic process. The relevance to real systems such as Cu-benzoate and ${\mathrm{Y}\mathrm{b}}_4{\mathrm{A}\mathrm{s}}_3$ is also discussed.