New surprises in the Quantum Hall Effect

The quantum Hall effect (QHE) is one of the most unexpected phenomena emerging from condensed matter since the discovery of superconductivity. Forty years after von Klitzing was awarded the Nobel prize for experimentally realizing the effect, scientists have amassed an enormous body of research on the subject. For example, it is well known that the Hall electrons behave as a liquid. However, in my work with Barry Bradlyn and Gustavo Monteiro, we find that it is possible to induce anisotropy in such a manner as to throw this and our understanding of electron hydrodynamics into question.

Despite the counterintuitive phenomena quantum Hall systems exhibit, they are surprisingly straightforward to set up. All one needs is a strong magnetic field oriented perpendicular to a flat conductor. Measure the transverse (Hall) resistance and one finds a staircase of perfectly flat plateaus whose heights are exact integer multiples of fundamental constants (see figure 1). This is a remarkable result as it violates the notion that classical formulations of physics should be accurate on macroscopic scales aside from small quantum corrections. This is known as the correspondence principle. We now know that the correspondence principle does not apply to the QHE because of collective effects. Crucial to the QHE is the fact that a macroscopic number of electrons — one hundred billion for a fingernail-sized conductor in a magnetic field of 1T — all have the same energy. In other words, instead of quantum effects averaging out over many particles, they are enhanced by the combined effort of billions of electrons behaving cohesively. This property implies that the quantum Hall fluid (QHF) is a perfect candidate to feature unusual fluid mechanics since hydrodynamic quantities are also derived from averaging out the motion of many particles. Thus, the quantum effects of the QHF can easily infiltrate the hydrodynamic equations of motion.

A particularly important property of classical fluids is isotropy: they do not possess a preferred direction. If such a direction existed, e.g., in the stress, then it would imply a pressure gradient and particles would freely flow until all forces were balanced and the stress became isotropic once again. We show in our paper, however, that anisotropy can be induced in the QHF by simply tilting the magnetic field relative to the normal of the plane of the material.

To understand this result, we used the quantization method pioneered by Bohr and Sommerfeld in the early days of quantum mechanics: take the classical trajectory of a particle in phase space and restrict the enclosed area by Planck’s constant times a positive half-integer: ℏ(n+ 1/2 ) . Ordinarily, the Bohr–Sommerfeld method gives only an approximation of the true quantum result, but luckily this approach is exact for the QHE.

Figure 2 shows an example of a classical trajectory. Along with the magnetic field (orange vector), we must also take into account the force that confines the particles to a plane. Figures 2.a and 2.b show different perspectives of the seemingly chaotic path as the particle orbits the magnetic flux lines and is pushed toward the x − y plane by the confining force. These trajectories, however, are not chaotic; as shown in figure 2(c) and 2(d) the motion of the particle decouples into two ellipses. The total motion can be understood as the sum of the red and purple trajectories, i.e. the purple ellipse orbits the red ellipse with frequency ω₁ while the particle orbits the purple ellipse with frequency ω₂.

How are these trajectories and the stress related? Notice the red and purple ellipses are stretched in different directions in figure 2.c. We show that the motion of e.g. the red trajectory, which is stretched in the x−direction, induces a greater stress in that direction. Likewise, the purple trajectory contributes more to the y component of the stress. The sum of these two effects yields the total stress. In a classical material containing many electrons, each particle follows a trajectory in which the area is statistically sampled and the average stress perfectly balances in both directions resulting in isotropy. However, the Bohr-Sommerfeld quantization scheme imposes a fundamental limitation on this averaging process: not every possible ellipse is physically realizable. In fact, there is a minimum area for both trajectories. Crucially, for particles constrained to these minimal-area trajectories, the anisotropic contributions to the stress no longer cancel. Since there are billions of these electrons, each in the minimal orbit, the hydrostatic stress acquires significant anisotropy.

In summary, using a realistic model of the QHF in a tilted magnetic field, we predict the hydrostatic stress tensor picks up an anisotropic characteristic that can only arise in liquids as a quantum effect. This discovery opens the door to the study of a new type of material: the quantum strange liquid, where particles flow to maintain an asymmetric equilibrium.