貝里相位、分數電荷及手徵反常之研究

Abstract

在這份研究中，我們將深入地討論貝里相位、分數電荷及手徵反常之間的關係。由Goldstone-Wilczek 模型，我們已知，當空間中有soliton時，此時的真空會因為被該外場所極化而具有分數的電荷。且有趣的是，該分數電荷亦可由其他拓樸相關的物理概念－貝里相位及手徵反常－來解釋。如是觀之，似乎我們可以將該些物理觀念合而為一。然而，最近Fujikawa在深入研究(3+1)維系統中的pi介子衰變成2個光子的物理現象時發現，貝里相位以及手徵反常二者之間其實是具有基本差異的。且習知手徵反常是與溫度不相關的，然而分數電荷卻是會隨溫度而變的。這讓我們開始省思，在(1+1)維的Goldstone-Wilczek模型中，上述的習知將分數電荷、貝里相位及手徵反常三者物理概念解釋為相互關聯可能是有問題的。為了更清楚了解三者的異同，我們將研究Goldstone-Wilczek模型中，該三者拓樸物理量隨溫度變化的關係。 首先我們推廣Schaposnik所提出的方法來研究分數電荷以及手徵反常二者之間在有限溫度時的變化關係。我們發現，利用該方法我們可將分數電荷分解成二個項，其中一項為與溫度無關的項，此項即對應手徵反常；另外一項則為溫度修正項。所以，由此方法，我們可以清楚的看出二者的差異是來自該溫度修正項。為了進一步將本方法所得到的結果與其他方法所得到的溫度修正做比較，我們也計算了該溫度修正項。我們發現該溫度修正項即為熟知的massive Schwinger model。我們發現，其one-loop的結果即與其他方法所得到的結果有漂亮的對應。 接下來，我們發現上述方法也可以用於研究(2+1)維的另一拓樸量－induced Chern-Simons term－與手徵反常的關係。這可溯源到Fosco, Rosini, Schaposnik等人之前的研究。他們研究了一個有外加靜磁場的(2+1)維特殊系統，並發現該系統的induced Chern-Simons term的溫度修正亦可由chiral rotation所產生的Jacobian計算而得。換句話說，indueced Chern-Simons term與手徵反常具有高度相關。為了進一步了解該關係，我們試著探討另外的外場。我們選擇研究當外場是一個隨時間而變的電場時的情況。我們發現，此時induced Chern-Simons term的溫度修正並非由Jacobian所推得。有趣的是我們發現該溫度修正亦與massive Schwinger model相關。我們也計算了one-loop的溫度修正，並發現該方法所得到的溫度修正與其他方法所得到的結果相吻合。我們認為，上述induced Chern-Simons term在不同外場時的差異，是與vacuum polarization在p=0處的不可解析特性息息相關。 最後，我們再度回到(1+1)維的Goldstone-Wilczek模型，並討論其分數電荷與貝里相位的關係。我們利用dimensional reduction以及二次量子化的方法發現，在溫度為0時 ，分數電荷可連結到phase space 貝里相位。在有限溫度時，由於溫度效應，其對應關係亦不存在。

We study the connection between Berry phase, fractional charge and Chiral anomaly. Their close relationship can be most easily seen through the Goldstone-Wilczek model, whose vacuum has been known to acquire non-trivial fractional charge due to the coupling of the fermion field to a topological nontrivial soliton field, and at zero temperature this charge fractionalization has been argued to be related to both the phase-space Berry phase and chiral anomaly. However, Fujikawa recently shows that there are some basic differences between Berry phase and chiral anomaly. It suggests that the mix of the concept of fractional charge, Berry phase and Chiral anomaly together might be problematic. In this article we will try to clarify this point. To achieve this goal, we find that it is useful to study this system at finite temperature. Here we first study this problem in a framework in which the connection of charge fractionalization to the chiral anomaly is manifest. In this formalism, the fermion number can be found to be separated into two terms: one is related to chiral anomaly which is temperature independent, and the other is the temperature correction term. As a result, it becomes manifest why while fractional charge is temperature dependent but chiral anomaly is temperature independent. Moreover, the temperature correction term can be seen to be simply the massive Schwinger model. The temperature dependence of the induced charge is then solved by doing the one loop calculation of the massive Schwinger model, and is found, in consistent with the previous studies, to decrease as temperature arises and vanish at infinite temperature. Then, as a direct generalization, we restudy the induced Chern-Simons term at finite temperature. We find that, for spatially uniform but time dependent background gauge fields, the effective action is also separated into two parts: one is the chiral Jacobian which is temperature independent, and the other is related to an effective massive Schwinger model which gives the temperature correction. By explicitly calculating the one loop result of the massive Schwinger model part, we find that the well known temperature correction to the induced Chern-Simons term can be reproduced. The above two parts are based on the paper which has been accepted and will be published in Physical Review D. Finally, to see the connection of fractional charge and Berry phase, we adopt Fujikawa’s second quantization method. By explicitly studying the temperature effect on Berry phase, it is also found that the fractional charge is different to this Berry phase at finite temperature.