[Paper Review] Spectral Gaps of Quantum Hall Systems with Interactions

Tohru Koma|arXiv (Cornell University)|Sep 17, 1998

Quantum and electron transport phenomena3 references3 citations

TL;DR

This paper investigates spectral gaps in two-dimensional quantum Hall systems with finite-range two-body interactions using the Lieb-Schultz-Mattis method. It proves that for non-integer filling factors, a non-zero excitation gap implies either spontaneous translational symmetry breaking or a unique gapless ground state, and that rational filling factors are necessary for a gapped, translationally invariant ground state, explaining the preference for odd-denominator filling fractions in the quantum Hall effect.

ABSTRACT

A two-dimensional quantum Hall system without disorder for a wide class of interactions including any two-body interaction with finite range is studied by using the Lieb-Schultz-Mattis method [{\it Ann. Phys. (N.Y.)} {\bf 16}: 407 (1961)]. The model is defined on an infinitely long strip with a fixed large width, and the Hilbert space is restricted to the lowest $(n_{ m max}+1)$ Landau levels with a large integer $n_{ m max}$. We proved that, for a non-integer filling $ν$ of the Landau levels, either (i) there is a symmetry breaking at zero temperature or (ii) there is only one infinite-volume ground state with a gapless excitation. We also proved the following two theorems: (a) If a pure infinite-volume ground state has a non-zero excitation gap for a non-integer filling $ν$, then a translational symmetry breaking occurs at zero temperature. (b) Suppose that there is no non-translationally invariant infinite-volume ground state. Then, if a pure infinite-volume ground state has a non-zero excitation gap, the filling factor $ν$ must be equal to a rational number. Here the ground state is allowed to have a periodic structure which is a consequence of the translational symmetry breaking. We also discuss the relation between our results and the quantized Hall conductance, and phenomenologically explain why odd denominators of filling fractions $ν$ giving the quantized Hall conductance, are favored exclusively.

Motivation & Objective

• To understand the conditions under which spectral gaps can exist in interacting quantum Hall systems with non-integer filling factors.
• To clarify the role of translational symmetry breaking in the emergence of gapped ground states.
• To establish a rigorous connection between the existence of a non-zero excitation gap and the rationality of the filling factor ν.
• To explain why odd-denominator filling fractions are favored in the quantized Hall conductance, based on symmetry and spectral properties.

Proposed method

• The Lieb-Schultz-Mattis method is applied to a two-dimensional quantum Hall system on an infinitely long strip with fixed width and restricted to the lowest (n_max + 1) Landau levels.
• The system is analyzed in the infinite-volume limit with a large but finite n_max, ensuring a well-defined Hilbert space for analysis.
• The analysis distinguishes between pure infinite-volume ground states with and without translational symmetry breaking.
• Theorems are derived by examining the implications of a non-zero excitation gap on the structure of the ground state and the value of the filling factor ν.
• The model includes a wide class of finite-range two-body interactions, generalizing beyond short-range or Coulomb interactions.
• The study incorporates periodic potentials and energy/space cutoffs to ensure mathematical rigor and physical relevance.

Experimental results

Research questions

• RQ1 Under what conditions can a gapped ground state exist in an interacting quantum Hall system with non-integer filling ν?
• RQ2 Does the existence of a non-zero excitation gap necessarily imply translational symmetry breaking for non-integer ν?
• RQ3 Can a gapped, translationally invariant ground state exist for irrational filling factors ν?
• RQ4 Why are odd-denominator filling fractions ν = p/q (with p, q odd integers) favored in the quantum Hall effect?
• RQ5 What is the relationship between spectral gaps and the rationality of the filling factor ν in the absence of symmetry breaking?

Key findings

• For non-integer filling ν, a non-zero excitation gap in a pure infinite-volume ground state implies that translational symmetry must be broken at zero temperature.
• If no non-translationally invariant ground state exists, then a gapped ground state can only occur when the filling factor ν is rational.
• The filling factor ν must be rational for a gapped, translationally invariant ground state, even if the ground state has a periodic structure due to symmetry breaking.
• The results provide a theoretical explanation for the experimental preference of odd-denominator filling fractions in the quantum Hall effect.
• The analysis confirms that finite-range two-body interactions do not alter the fundamental constraints on spectral gaps and filling factors.
• The theorems hold under general conditions, including periodic potentials and energy/space cutoffs, ensuring broad physical applicability.