Are Bloch Bands at Finite Field Adiabatically Connected to Those at Zero Field?
Gregory H. Wannier, J. P. Van Dyke · Journal of Mathematical Physics · 1968
It has been shown previously that if a potential consists of a superposition of a periodic part and a uniform electric field, then for a particle moving in this field there are Bloch bands closed in time. The present paper addresses itself to the question whether these bands may be identified with the field-free bands. The most natural thing is to expect that the bands are slightly field dependent, but converge toward the field-free bands as E goes to zero. Bands for which this is true are said to be adiabatically connected to corresponding bands at zero field. In Sec. 2 of the paper, two model cases are given for which this adiabatic connection pertains. Section 3 is the central part of the paper and provides the conclusion that the answer to the question in the title is almost always negative. In this proof the positive cases serve an essential function. It is shown that the parameters of the periodic potential must obey at least one supplementary condition to allow adiabatic connection, and that the collected cases precisely obey this condition. Adiabatic connection is thus generally not possible. Section 4 provides an explicitly soluble case which does not allow adiabatic connection. An infinite number of field values E converging toward zero are found at each of which the two bands under consideration switch identity (hyperbolic rather than linear connection at energy crossings). The connection postulated in the effective-mass approximation must therefore be of a nonadiabatic nature. It probably involves the ``sudden'' approximation of quantum theory.