附录
A Brief Note on the Fractional Quantum Hall Effect
B. G. Sidharth
Received: 6 April 2014 / Accepted: 9 December 2014 / Published online: 19 February 2015 ? Springer Science+Business Media New York 2015
Abstract ：We explain features of two dimensional structures in a precise way, in terms of the non-commutative space which defines these structures. The novel feature here is a fundamental explanation for the FQHE.
Keywords ： Hall·Effect·Quantum
1 Two Dimensional Considerations
The Quantum Hall Effect was discovered experimentally in the 1980s [1]. In this case, as is by now well known, for a two dimensional system of electrons, the Hall conductivity is found to be of the form
G=λ.e2/h (1) where λ takes on values
λ=m/n （2） m and n being integers. In particular we have the case that λ is an integer, called the integral Quantum Hall Effect [2].
The important thing is that (1) and ( 2) play a very precise role in metrology. There have been attempts to explain this phenomenon theoretically in particular by invoking gauge invariance [3]. However there have been some persisting puzzles.
We will now look at the Fractional Quantum Hall Effect (FQHE) from a completely novel perspective, for example in the case of graphene. As is well known graphene is a single layer graphite with many interesting properties. Some of these have been predicted by the author starting 1995 [4–7]. The important point here is that graphene has a honeycomb like lattice structure so that the space of graphene resembles a chessboard with “holes” in space itself [8].This means that there is a fundamental minimum length underpinning the system.
This fundamental length L leads to a non-commutative geometry [9]. This was pointed out a long time ago in the context of Quantum Electrodynamics [10]. This means that if (x,y) are the coordinates, xy = yx. Indeed, as pointed out, graphene therefore provides a test bed for these principles of physics which play a role in Quantum Gravity approaches. In such a situation it has been shown by the author and independently Saito [11-13] that there is a strong magnetic field. Further, the author showed that this field is given by
BL2=hc/e (3) L2 defines a Quantum of area exactly as in Quantum Gravity approaches [14, 15]. This as is well known is the area of individual lattices, in our case.
To elaborate, the author had argued that (3) holds in the case of a noncommutative geom- etry. This happens when there is a fundamental length L which acts as a minimum length of the system. In this latter case Snyder had shown that commutation relations like
[x,y]=(iL2)Lxect. (4) hold good. In the modern context L is the Compton (or Planck) length and we have Lx in (4) that reduces the Quantum mechanical spin value /2 (Cf. [13, 16]). Moreover Saito had shown [12] from a slightly different point of view that
[x,y]=ih/eB (5) (c = 1)
where B is the magnetic field that is generated due to this noncommutative geometry. Fol- lowing from (4) and ( 5), as pointed out by the author, equation (3) follows. From here we get equations like
y = ic/eBP
which were deduced by Landau much earlier, but without realizing the novel noncommuta- tive feature [17, 18].
In these considerations for graphene as is very well known, the Fermi velocity νF replaces the velocity of light. So we have for the electron mobility and conductivity μ=vF/|E| (6) σ=(n/A)e·vF/|E|,A～L2 (7) where A, as in the usual theory is the area and n is the number of electrons. In our case as noted above A, the area is made up of a number of honeycomb lattice areas, each with area ～ L2, that is
A = mL2 （8） where m is an integer.
We also note that the electric field strength E equals the magnetic field strength B for two dimensional structures.
To see this we invoke the well known fact that the electric and magnetic fields are given by [19]
/3E（x）=p(x?x)dx?
(9)
In the case of Graphene or more generally other two dimensional structures c as noted
is replaced by the Fermi velocity νF while the Fermions (quasi particles) are massless as such luminal, that is like neutrinos in three dimensions, they move with velocity νF. So specializing to the 2D case the current J in (9) is given by
J?pvF (10) Substitution of (10) in ( 9) leads to the equality
E?B (11) which we have noted above. That is the magnetic fields are much stronger than in the usual case. Using these inputs in (7) we get
u=n/m.evF/BL2 (12) If we now use (3) in (12) (with νF replacing c) we get for the conductance
σ=n/m.e2h (13)
which defines the fractional Quantum Hall Effect. It must be mentioned that equation (3) takes care of any factor in the formula A ～ L2.
Earlier the author had shown that it is this non-commutative space feature in two dimen- sional structures that explains also Landau levels [18] or the minimum conductivity that exists in graphene even when there are practically no electrons at the Dirac points [19]. In other words several supposedly diverse phenomena arise from the non-commutative space of these two dimensional structures.
2 Discussion
It must be mentioned that the idea of trying to consider graphene from the perspective of its noncommutative space has been studied by several authors [20–23]. Some of the approaches were motivated by the Quantum Electrodynamics in the spirit of Wilson’s Lattice Gauge Theory. We would like to point out that unlike in the other approaches we have added two new inputs not used earlier which have lead to the rather comprehensive and neat deduction of (13). These are firstly (3) which was deduced several years ago in the context of high energy physics and quantum gravity and secondly (11), which as shown applies to two dimensional systems.
It may be mentioned that over the years the Integral Hall Effect and Fractional Quantum Hall Effects have not only been observed experimentally but several excellent simulations exist [24]. Furthermore while it has been known that both the integral and the fractional effects may be qualitatively related, exact theories to explain this have not been fully devel- oped. It must be mentioned that there is another approach that of composite Fermions which could potentially provide a unified description, for example that of the approach of J. Jain of Pensylvania State University. To put it briefly a composite Fermion describes an electron together with an even number of vortices.
Finally it may be mentioned that the work of the author and Saito briefly described above and which shows the production of a magnetic field due to noncommutative space, provides an explanation for the Quantum Anomalous Hall Effect which takes place in the absence of an external magnetic field. This effect was observed recently [25].
References
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4. Sidharth,B.G.:ANoteonTwoDimensionalFermions in BSC-CAMCS-TR-95-04-01 (1995)
5. Sidharth, B.G.: Low Dimensional Electrons in Solid State Physics. In: Mukhopadhyay, R., et al. (eds.), vol. 41, p. 331. Universities Press, Hyderabad (1999) 6. Sidharth, B.G.: arXiv:9506002
7. Sidharth, B.G.: J. Stat. Phys. 95(3/4), 775–784 (1999)
8. Mecklenburg, M., Regan, R.C.: Phy. Rev. Lett. 106, 116803 (2011) 9. Sidharth, B.G.: Int. J. Mod. Phys. E 25(23), 5 (2014) 10. Snyder, H.S.: Phys. Rev. 1(1), 68–71 (1947)
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分数量子霍尔得到了短暂的注意
B. G. Sidharth
Received: 6 April 2014 / Accepted: 9 December 2014 / Published online: 19 February 2015 ? Springer Science+Business Media New York 2015
摘要
我们抽象的解释二维结构的功能通过一种精确的方式,在非交化换空间的条件下定义这些结构。这是一个小说的特征对FQHE基本解释。
关键词:霍尔效应；量子
1二维的考虑
在1980年代，量子霍尔效应是通过实验发现。在这种情况下,因为现在是众所周知的,对于一个二维系统的电子表中，霍尔电导发现：
G??.e2/h (1) 当把λ代入得到
??m/n (2) 其中m和n是整数，特别是我们的λ是分数,因此称为分数量子霍尔效应[2]。 其中(1)和(2)在一个非常精确的计量中占有重要地位。有人试图解释这一现象，尤其是从理论上通过调用计的不变性[3]。然而，有一些坚持拼图。
现在我们看分数量子霍尔效应(FQHE)是从一个完全的小说角度来看,例如对于石墨烯，众所周知石墨烯是一个单层石墨，具有许多有趣的特性，作者对其中的一些已经预测1995年[4 - 7]。重要的是,石墨烯具有一个蜂巢晶格结构,石墨烯类似棋盘的空间空间的“洞”[8]。这意味着有一个基本的支撑系统的最小长度的体系。
这个基本长度导致非交换几何[9]。这是指出很久以前在量子电动力学所提到的[10]。这意味着,如果(x,y)坐标,存在xy 不等于yx。事实上,正如所指出的,因此石墨烯提供测试床这些物理原理方法同样在量子引力中发挥作用。在这种情况下,它已被证明的作者和独立齐藤(11 - 13)是一个强大的磁场。进一步,作者证实这个领域是由
BL2?hc/e (3)
L2定义了一个量子量子引力区，根据完全一样的方法(14、15)。在我们案例中，这是是众所周知的面积是单独的晶格。