Geometric Engineering and Almost Mathieu Operator

TL;DR

This paper explores the spectral properties of a geometric engineering model derived from string theory, revealing phase transitions in the spectrum related to the almost Mathieu operator and quantum Hall phenomena.

Contribution

It connects string theory geometric models with spectral analysis of the almost Mathieu operator, identifying phase transitions in the spectrum.

Findings

Spectrum is absolutely continuous for R^2<1

Spectrum is singular continuous for 1≤R^2<e^β

Spectrum is pure point for R^2>e^β

Abstract

The type IIA string theory on a non-compact Calabi-Yau geometry known as the local $\mathbb{P}^{1} \times \mathbb{P}^{1}$ gives rise to five-dimensional N =1 supersymmetric SU(2) gauge theory compactified on a circle, known as geometric engineering. So it is necessary to study the $\mathbb{P}^{1} \times \mathbb{P}^{1}$ in details. Since the spectrum of the local $\mathbb{P}^{1} \times \mathbb{P}^{1}$ can be written as $E=R^{2}\left(\mathrm{e}^{p}+\mathrm{e}^{-p}\right)+\mathrm{e}^{x}+\mathrm{e}^{-x}$, then by the result of almost Mathieu operator, we show that: (1) when $R^{2}<1$, the spectrum is absolutely continuous which meanings the medium is conductor. (2) when $1\le R^{2}<e^{\beta}$, the spectrum is singular continuous known as quantum Hall effect. (3) when $R^{2}>e^{\beta}$, the spectrum is almost surely pure point and exhibits Anderson localization. In other words, there are two phase transition points which one is $R^{2}=1$ and the other one is $R^{2}=e^{\beta}$.

Equations

$$H_{\alpha ,\lambda ,\theta }u(n)=u(n+1)+u(n-1)+2\lambda \cos (2\pi \alpha n+\theta )u(n)$$

$$-y^{\prime \prime }(x)+\lambda \cos (x)y(x)=Ey(x)$$

$$H={\mathrm{e}}^{\mathrm{i}x}+{\mathrm{e}}^{-\mathrm{i}x}+{\mathrm{e}}^{\mathrm{i}p}+{\mathrm{e}}^{-\mathrm{i}p},$$

$$H={\mathrm{e}}^x+{\mathrm{e}}^{-x}+{\mathrm{e}}^p+{\mathrm{e}}^{-p}$$

$${\mathrm{e}}^x+z_1{\mathrm{e}}^{-x}+{\mathrm{e}}^p+z_2{\mathrm{e}}^{-p}=1$$

$$x\to x-\log E,p\to p+2\log R-\log E$$

$$z_1=\frac{1}{E^2},z_2=\frac{R^4}{E^2}$$

$$R^2\left({\mathrm{e}}^p+{\mathrm{e}}^{-p}\right)+{\mathrm{e}}^x+{\mathrm{e}}^{-x}=E$$

$$H_{\alpha ,\lambda ,\theta }u(n)=u(n+1)+u(n-1)+2\lambda \cos (2\pi \alpha n+\theta )u(n)$$

$$R^2\left({\mathrm{e}}^p+{\mathrm{e}}^{-p}\right)+{\mathrm{e}}^x+{\mathrm{e}}^{-x}=E$$

Full text

I Introduction

One of the most powerful consequences of our deep understanding of the string theory has been the appreciation of the fact that the gauge theory can be encoded geoemetrically in the structure of compactifications of type II string theory Kachru:1995wm ; Bershadsky:1995sp . Gauge groups arise through ADE singularities of geometry Klemm:1995tj , however matter arises as loci of enhanced singularities. It not only leads to a unified description of gravitational and gauge theory, but it also leads directly to a deeper understanding of gauge dynamics, even in the limit of turning off gravitational effects Klemm:1996bj ; Kachru:1995fv . The basic idea is to geometrically engineer the gauge symmetry and matter content, and then study the corresponding theory using string techniques. As we know, the type IIA string theory on local Calabi-Yau geometry is known to give rise to five-dimensional $N=1$ supersymmetric $SU(2)$ gauge theory compactified on a circle, and as its mirror, the curve knows the non-perturbative information on this gauge theory. This curve reduces to the celebrated Seiberg-Witten Seiberg:1994rs curve encoding the information on instantons in $N=2$ supersymmetric pure $SU(2)$ gauge theory. So it is natural to study the property of the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$, which is known as the non-compact Calabi-Yau geometry. In fact, we argue that there exist metal-insulator phase transition in the $\mathbb{P}^{1}\times\mathbb{P}^{1}$. It seem that there is no relation between string theory and metal-insulator transition. However, the relation really exists.

In Ref. Vafa:2015euh , Vafa suggested a new model for fractional quantum Hall effect which connected it to the recent developments in string theory and led to new predictions for the principal series of fractional quantum Hall effect with filling fraction. And he found that the six dimensional $(2,0)$ super-conformal field theory Witten:2009at on the Riemann surface provided a realization of fractional quantum Hall effect in the $M$-theory. Moreover, he proposed that the $q$-deformed version of Chern-Simons theories were related to the anisotropic limit of fractional quantum Hall systems which split the zeroes of the Laughlin wave function. And he also suggested that the non-abelian topologically twisted Yang-Mills theory Vafa:1994tf could realize as topological insulators. Now we can see the condense matter theory has been applied in string theory. So, we want to say that the string theory may have deep relation with the metal-insulator transition.

The almost Mathieu operator Jitomirskaya is the discrete one dimensional Schrödinger operator (acting on $\ell^{2}(Z)$),

[TABLE]

where $\alpha,\lambda,\theta\in R$ are parameters, called the coupling, frequency, and phase. The name of the operator comes from the similarity to the Mathieu equation,

[TABLE]

$H_{\alpha,\lambda,\theta}$ is a tight-binding model for the Hamiltonian of an electron in a one- or two-dimensional lattice, subjecting to a perpendicular magnetic field. This model also describes a square lattice with anisotropic nearest neighbor coupling and isotropic next nearest neighbor coupling. The almost Mathieu operator plays an important role in the study of fundamental problems related to Bloch electrons in magnetic fields. In particular, it plays a major role in the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) Thouless:1982zz theory of the integer quantum Hall effect, where it gives rise to a rich set of possible integer Hall conductances. It is interesting to note that the TKNN theory which reflect the relevance of the almost Mathieu operator for describing electrons in magnetic fields has been verified experimentally Albrecht .

In this work, we focus on the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and almost Mathieu operator. And we hope that it may shed light on the issue of geometric engineering. The local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model is briefly introduced in Section II. And Section III is about almost Mathieu operators. A brief summary is given in the last section.

II local $\mathbb{P}^{1}\times\mathbb{P}^{1}$

Firstly, Let us consider the two-dimensional motion of electrons in the periodic potential and the magnetic field perpendicular to the two-dimensional plane. Then, in suitable limits, the Hamiltonian of the system is described by,

[TABLE]

The spectrum is an intricate pattern which study by Hofstadter, known as Hofstadter s butterfly. And this system was later used as a model system where the topological numbers determine the quantum Hall conductance. Actually, in high energy physics, a similar equation has been widely studied Hatsuda:2016mdw ,

[TABLE]

When $x$ and $p$ are restricted to be purely imaginary, then this equation reduces to Hofstadter s Hamiltonian. In fact, this equation arises when geometric engineering is applied to a non-compact Calabi-Yau geometry known as the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ geometry. The Kahler structure of the original Calabi-Yau is mapped to the complex structure of the mirror Calabi-Yau, and vice versa. The mirror Calabi-Yau to local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is described by the following equation Huang:2014nwa ; Huang:2014eha ,

[TABLE]

where $z_{1}$ and $z_{2}$ are the complex moduli of the mirror Calabi-Yau space. The mirror curve has many information to construct the all-genus free energy of the topological string theory Marino:2006hs ; Marino:2009jd ; Kozcaz:2010af . For our purpose, we have,

[TABLE]

and to set,

[TABLE]

Then the Eq. (5) can be rewritten as,

[TABLE]

So, we can write the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ as Eq. (8)

III Almost Mathieu Operator

As we know, the origin almost Mathieu operator is,

[TABLE]

For our purpose, we can rewrite it as,

[TABLE]

where $R^{2}=\lambda$. So, it is the same with local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model. And one can find that when $R=1$, this equation reduces to Hofstadter’s Hamiltonian.

In fact, the spectrum of $E$ is widely researched in mathematics. And one can find that the spectrum is depending on the coupling parameter $\lambda$. Actually, It is known that ,

(1) For $0<\lambda<1$, $E$ has surely purely absolutely continuous spectrum Avila . In physics, we can regard the particle ”travels freely” in the medium. In this case, the medium is conductor.

(2) For $1\leq\lambda<e^{\beta}$, $E$ has almost surely purely singular continuous spectrum. It is the quantum Hall effect.

(3) For $\lambda>e^{\beta}$, $E$ has almost surely pure point spectrum youjiangong and exhibits Anderson localization. Or in other words, the medium is insulator

Anderson localization Anderson:1958vr ; Pixley:2015xla can be understood as an interference phenomenon. In this case, the materials can undergo a phase transition from conductor to insulator. In the original tight-binding model formulated by Anderson, the electrons are able to tunnel between neighbouring lattice sites. However, at high enough disorder in the lattice, the quantum amplitudes associated with tunnelling paths cancel each other, resulting in a localized wave function, and in the way, the scattered wavelets interfere destructively in the forward direction, causing the wave to decay exponentially.

Now, one may find that the property of local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ highly depends on coupling parameter $\lambda$. When the coupling parameter is small, the medium is in order, then the particle travels freely. And when the coupling parameter is large, the medium behaves as insulator, then the particle do not travel.

IV Summary

We introduce the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model and almost Mathieu operators. Then by the result of almost Mathieu operator, we show that there exist metal-instructor transition in the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model. And there are two phase transition points which one is $R^{2}=1$ and another is $R^{2}=e^{\beta}$ in the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model.

It is interesting that Vafa suggests the potential experimental realization of topologically twisted non-abelian Yang-Mills theories as effective descriptions in the bulk of topological insulators in lab by a suitable model. And Vafa and his coauthor are currently studying potential applications of this to condensed matter systems.

So, in our case, we argue that there may exist metal-insulator transition in the geometric engineering. And we hope that our comment will provide a new view for the studying of string theory

Acknowledgment

Part of the work was done when Jing Zhou visited the Yau Mathematical Science Center. Jing Zhou thanks for discussing with An Qi. This work is partly supported by the National Science Foundation of China under Contract Nos. 11775118, 11535005.