2D or not 2D: a "holographic dictionary'' for Lowest Landau Levels

TL;DR

This work reveals that restricting fermions to the lowest Landau level induces a noncommutative 2D plane that can be holographically encoded by a 1D quantum system embedded within the full 2D Hilbert space. A precise 1D-2D dictionary is established, linking the 2D fermion density to the 1D Wigner distribution via a universal kernel, with the large-N limit turning this into an identity and yielding a Pauli-density bound ρ(x,y) ≤ ρ_max. The analysis extends to N-particle states, including Slater, thermal, and coherent states, and demonstrates that dynamics in the LLL can be captured by 1D phase-space hydrodynamics. Most notably, the entanglement entropy for disks in the noncommutative LLL plane scales linearly with region size (S_A ∝ l/l_0) and lacks the logarithmic enhancement seen in ordinary 2D Fermi systems, though a 1D strip geometry recovers logarithmic behavior, reflecting a rich, dimensionally hybrid EE structure. Overall, the holographic dictionary clarifies how 1D methods encode 2D LLL physics and offers tractable approaches to post-quench dynamics and EE in noncommutative spaces.”,

Abstract

We consider 2D fermions on a plane with a perpendicular magnetic field, described by Landau levels. It is wellknown that, semiclassically, restriction to the lowest Landau levels (LLL) implies two constraints on a 4D phase space, that transforms the 2D coordinate space (x,y) into a 2D phase space, thanks to the non-zero Dirac bracket between x and y. A naive application of Dirac's prescription of quantizing LLL in terms of L2 functions of x (or of y) fails because the wavefunctions are functions of x and y. We are able, however, to construct a 1D QM, sitting differently inside the 2D QM, which describes the LLL physics. The construction includes an exact 1D-2D correspondence between the fermion density ρ(x,y) and the Wigner distribution of the 1D QM. In a suitable large N limit, (a) the Wigner distribution is upper bounded by 1, since a phase space cell can have at most one fermion (Pauli exclusion principle) and (b) the 1D-2D correspondence becomes an identity transformation. (a) and (b) imply an upper bound for the fermion density ρ(x,y). We also explore the entanglement entropy (EE) of subregions of the 2D noncommutative space. It behaves differently from conventional 2D systems as well as conventional 1D systems, falling somewhere between the two. The main new feature of the EE, directly attributable to the noncommutative space, is the absence of a logarithmic dependence on the size of the entangling region, even though there is a Fermi surface. In this paper, instead of working directly with the Landau problem, we consider a more general problem, of 2D fermions in a rotating harmonic trap, which reduces to the Landau problem in a special limit. Among other consequences of the emergent 1D physics, we find that post-quench dynamics of the (generalized) LLL system is computed more simply in 1D terms, which is described by well-developed methods of 2D phase space hydrodynamics.

Figures (22)

• Figure 1: Landau levels in the generalized Landau problem. The energy spectra, for various values of $n_1=0,1,2,...$, are given by the red lines, which define the Landau levels in the generalized problem. The lowest Landau level corresponds to $n_1=0$ and the black dots represent the allowed $n_2$ values. To be contrasted with the energy levels of the original Landau problem depicted in Figure \ref{['fig:ll-band']}.
• Figure 2: Symbolic representation of constrained Hilbert spaces in terms of projection operators (which are the top left diagonal blocks). Panel (a) represents the LLL constraints which are effective constraints arising from a low energy approximation. Panel (b) represents gauge constraints which are genuine constraints: here the entire Hilbert space is in the top left diagonal block.
• Figure 3: The flow chart to illustrate the quantum 1D-2D correspondence for single particle LLL wave-function.
• Figure 4: The wiggles in (b) are suppressed by the integral transform in (\ref{['un-rhon']}) to yield (a).
• Figure 5: Classical limit of ground state properties. $N=100, \hbar=1/N, m\omega=5$.