Engineering:The almost anti-symmetric gauge

The almost anti-symmetric gauge (AAG) is a gauge for the magnetic vector potential that is designed for tight-binding models with periodic boundary conditions.

Introduction

Periodic boundary conditions in condensed matter systems

Exploring the properties of condensed matter systems in torus geometry is of particular interest for a few reasons. First, field theoretical calculations often make use of periodic boundary conditions that maintain translation invariance. Second, numerical procedures with periodic boundary conditions avoid the occurrence of edge effects that may obscure the physical properties related to the bulk system. Third, in some cases, there is a definite procedure (based on scaling arguments) to deduce the properties of an infinite system from those of a finite system with periodic boundary conditions. These kind of procedures, which are valid only when the system’s correlation lengths do not exceed its dimensions, allow us to study systems in the thermodynamic limit. Fourth, some physical phenomena depend on the real-space topology, e.g., in topological phases of matter the multiplicity of the degenerate ground state depends on the boundary.

Gauges in non-relativistic quantum mechanics

In non-relativistic quantum mechanics, the electromagnetic field is regarded as external to the quantum system and is described by an electric scalar potential [math]\displaystyle{ \varphi }[/math] and a magnetic vector potential [math]\displaystyle{ \mathbf{A} }[/math]. The electromagnetic potentials are not defined uniquely by the electromagnetic field and each particular choice, or specification, of [math]\displaystyle{ \varphi }[/math] and [math]\displaystyle{ \mathbf{A} }[/math] is referred as a gauge. Although all the physical quantities are gauge-invariant, choosing a convenient gauge may simplify Schrödinger's equation, thus allowing to obtain the wave-functions. For example, the physics of the Landau levels is elegantly elucidated by exploiting the symmetric gauge.

Tight-binding models with periodic boundary conditions

Within the tight-binding framework, an approach to the calculation of electronic band structure, the magnetic field is incorporated by the Peierls substitution. Under this method, when a system is subjected to a magnetic field, [math]\displaystyle{ \mathbf{B} = \nabla \times \mathbf{A} }[/math], the hopping terms in the Hamiltonian acquire a phase [math]\displaystyle{ |j\rangle\langle i| \rightarrow |j\rangle\langle i|e^{i\theta_{j,i}} }[/math]. Here [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] are indices that represent the atomic sites [math]\displaystyle{ \mathbf{r}_i }[/math] and [math]\displaystyle{ \mathbf{r}_j }[/math], in which the electron may be localized. In addition, the Peierls phase are defined as [math]\displaystyle{ \theta_{j,i} \equiv \frac{q}{\hbar}\int_{\mathbf{r}_i}^{\mathbf{r}_j} \mathbf{A}(\mathbf{r})\cdot\text{d}\mathbf{r} }[/math]. When periodic boundary conditions are imposed on a 1D lattice with [math]\displaystyle{ q }[/math] atomic sites, than the Peierls phase is required to satisfy the relation, [math]\displaystyle{ \theta_{j,i} = \theta_{j+q,i} = \theta_{j,i+q} \ (\text{mod}~2\pi) }[/math] and the generalization to the case of 2D is straight forward. Thus, this condition imposes another constrain on the vector potential, [math]\displaystyle{ \mathbf{A} }[/math].

Introducing the almost anti-symmetric gauge

The almost anti-symmetric gauge (AGG) is designed to produce a homogeneous magnetic field over a 2D parallelogrammic lattice with [math]\displaystyle{ q \times (q+1) }[/math] atomic sites and periodic boundary conditions. For the simple case of a 2D square lattice with unity lattice constant the AAG is [math]\displaystyle{ \mathbf{A}(\mathbf{r})=2 p \Phi_0 \left( \frac{y}{q+1},\frac{x}{q} \right), }[/math] where [math]\displaystyle{ \Phi_0=\frac{h}{2e} }[/math] is a magnetic flux quantum and [math]\displaystyle{ p=1,2,\ldots,q(q+1) }[/math]. Moreover, under this gauge [math]\displaystyle{ \partial_x A_y }[/math] is slightly greater than [math]\displaystyle{ \partial_y A_x }[/math] and their contributions to the magnetic field are counter-oriented. The flux per unit cell is then [math]\displaystyle{ \frac{2p\Phi_0}{ q(q+1)} }[/math], and thus the flux through the entire 2D system is [math]\displaystyle{ 2p\Phi_0 }[/math]. In the standard procedure using the Landau gauge, the flux through the entire 2D area can only take values from a narrow and sparse range, [math]\displaystyle{ 2pq\Phi_0 }[/math] with [math]\displaystyle{ p=1,2,\ldots,q+1 }[/math].

The almost anti-symmetric gauge

The AAG that is designed to generate a homogeneous magnetic field in two-dimensional tight-binding models with [math]\displaystyle{ q \times (q+1) }[/math] sites and periodic boundary conditions, [math]\displaystyle{ \mathbf{A}=\frac{2\Phi_0 p}{a_1 a_2 \sin ^2(\alpha_1 - \alpha_2 )} \left[\frac{(\boldsymbol{r} \times \hat{\boldsymbol{\tau}}_1) \times \hat{\boldsymbol{\tau}}_2}{ q + 1}+\frac{ (\boldsymbol{r} \times \hat{\boldsymbol{\tau}}_2) \times \hat{\boldsymbol{\tau}}_1}{q}\right], }[/math] where [math]\displaystyle{ a_i }[/math] and [math]\displaystyle{ \alpha_i }[/math] are the magnitude and orientation of the lattice vectors [math]\displaystyle{ \mathbf{a}_i \equiv a_i \boldsymbol{\hat{\tau_i}} }[/math], respectively. Moreover, [math]\displaystyle{ \boldsymbol{\tau}_1=q a_1 \boldsymbol{\hat{\tau_1}} }[/math] and [math]\displaystyle{ \boldsymbol{\tau}_2=(q+1) a_2 \boldsymbol{\hat{\tau_2}} }[/math] span the entire system. The flux through the entire system is [math]\displaystyle{ 2p\Phi_0 }[/math], where [math]\displaystyle{ p=1,2,\ldots,q(q+1) }[/math] and [math]\displaystyle{ \Phi_0 = h/(2e) }[/math] is a magnetic flux quanta. Moreover, it produces a homogeneous magnetic field, [math]\displaystyle{ \nabla \times \mathbf{A} = const }[/math] and results single-valued phase factor in the Peierls substitution method.

Discussion

Although the AAG, [math]\displaystyle{ \mathbf{A}(\mathbf{r})=2 p \Phi_0 \left( \frac{y}{q+1},\frac{x}{q} \right), }[/math] requires a slight deviation of the geometry from a perfect square system, it has a major advantage over the Landau gauge, [math]\displaystyle{ \boldsymbol{A}(\boldsymbol{r})=-2\Phi_0 p(\frac{y}{q+1},0) }[/math]. The latter supports only flux quanta of [math]\displaystyle{ \Phi=2\Phi_0 q p }[/math] with [math]\displaystyle{ p\in(1,\ldots,q+1) }[/math] through the entire physical system and in many cases this range too sparse and narrow. In the other hand the AAG, supports flux quanta of [math]\displaystyle{ \Phi=2\Phi_0 p }[/math] with [math]\displaystyle{ p\in(1,\ldots,q(q+1)) }[/math]. Thus, the AAG allows the highest flux resolution study of a problem with periodic boundary conditions. A simple demonstration of the gauge significance is to consider the Hofstadter butterfly spectrum in the presence of disorders, where translation invariant is lost and it can not be obtained by solving the Harper equation.