Closed path

A. The Quantization of the Non-Adiabatic Coupling Matrix Along a Closed Path... 

We inb oduce a closed-path T defined by a parameter X. At the stalling point So, A, = 0 and when the path complete a full cycle, X — p(2tt, in case of circle). 

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

Let us consider a closed path L defined in terms of a continuous parameter X so that the starting point sg of the contour is at 1 = 0. Next, P is defined as the value attained by X once the contour completes a full cycle and returns to its starting point. For example, in the case of a circle, X is an angle and P = 2tu. 

Next, we refer to the requirements to be fulfilled by the matrix D, namely, that it is diagonal and that it has the diagonal numbers that are of norm 1. In order for that to happen, the veetor-funetion t(i) has to fulfill along a given (closed) path F the condition ... 

In Sections VII and Vm, it was mentioned that K yields the number of eigenfunctions that flip sign when the electronic manifold traces certain closed paths. In what follows, we shall show how this number is formed for various Nj values. 

Figure 17. The differential closed paths F and the singular point B a, b) in the (p, q) plane (a) The point B is not surrounded by F. (b) The point B is surrounded by F.

The case when one of the differential closed paths surrounds the point B[a,b) (see Fig. 17fi). Flere the derivation breaks down at the transition from Eqs. (C.5)-(C.7) and later, from Eqs. (C.7)-(C.8), because Xp and x, become infinitely large in the close vicinity of and therefore their... 

Figure 18. The closed (rectangular) path F as a sum of three partially closed paths Fi, F2, F3.

Figure 19. The closed path F as a sum of three closed paths F, Fp, F,. (a) The closed (rectangular) paths, that is, the large path F and the differential path F both surrounding the singular point B(a, b). (6) The closed path Tp that does not surround the point fi(a, b). (c) The closed path F, that does not surround the point B[a,b).

When this equation holds, each stream match (exchanger) must provide that one of the two streams involved reaches its target temperature. Such a network is called acycHc. In an acycHc network, it is not possible to trace a closed path along stream lines from exchanger to exchanger and return to the starting point without retracing some of the path. 

As seen from (3.7), the closed paths with x(to) = 3c(0) fully determine the energy spectrum of the system. Propagator K has, in terms of the energy eigenfunctions m>, the form... 

An additional complication in the PIMC simulations arises when Bose or Fermi statistics is included in the formalism. The trace in the partition function allows for paths which may end at a particle index which is different from the starting index. In this way larger, closed paths may build up which eventually spread over the entire system. All such possible paths corresponding to the exchange of indistinguishable particles have to be taken into account in the partition function. For bosons these contributions are summed up for fermions the number of permutations of particle indices involved decides whether the contribution is added (even) or subtracted (odd) in the partition function. 

Equation (1.4) states that if we add together all of the infinitesimal changes dZ over a closed path, the sum is equal to zero. This is a necessary condition for a state function. 

Figure 2.10 (a) A schematic Carnot cycle in which isotherms at empirical temperatures 6 and 62 alternate with adiabatics in a reversible closed path. The enclosed area gives the net work produced in the cycle, (b) The area enclosed by a reversible cyclic process can be approximated by the zig-zag closed path of the isothermal and adiabatic lines of many small Carnot cycles. 

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

This is true for finite changes, AZ, or for infinitesimal changes, dZ, where we replace the summation in equation (A 1.11) with an integral sign and a circle to represent an integration over a closed path. That is... 

Suppose then, we encounter a general differential expression and want to know whether it is associated with a state function. The behavior of this differential expression integrated over a closed path provides a means to answer this question. Two possibilities need be considered. 

We have already established that an integral of a differential expression associated with a state function is zero over a closed path. Now, we must consider whether the converse of that statement is true. That is, if the integral of a general differential expression over a closed path is found to be zero, this expression is the differential expression of some state function. To answer the question, let us reconsider the example described in Figure (A 1.1) and equations (A 1.13) and (A 1.14) and assume that equation (A 1.17) is true for all closed (cyclic) paths. Then, for a path 1 and a path 3 that connect the same two states, 1 and 2,... 

Thus, the assumption that an integration of a differential expression over a closed path is zero leads to a conclusion that an integration between two different, but fixed, states is independent of path. But, this property coincides with those we have ascribed to state functions. Thus, we have shown that a differential for which equation (A 1.17) is true must correspond to the differential of some state function. 

It is not convenient to test for exactness by showing that equation (Al.17) is true for all closed paths, but an easier test can be developed. Consider a general differential expression, 6Q, for a quantity Q that is associated with the variables X and Y ... 

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