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<p>We consider two-dimensional fermions on a plane with a perpendicular magnetic field, described by Landau levels. It is well-known that, semiclassically, restriction to the lowest Landau levels (LLL) amounts to imposing two constraints on a 4D phase space, which transforms the 2D coordinate space (<i>x</i>, <i>y</i>) into a 2D phase space, thanks to the non-zero Dirac bracket between <i>x</i> and <i>y</i>. A straightforward application of Dirac’s prescription of quantizing LLL in terms of <i>L</i><sup>2</sup> functions of <i>x</i> (or of <i>y</i>) fails because the wavefunctions are clearly functions of <i>x and y</i>. We find it possible, however, to construct a different 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 <i>ρ</i>(<i>x</i>, <i>y</i>) and the Wigner distribution of the 1D QM. In an appropriate large <i>N</i> limit, (a) the Wigner distribution is upper bounded by 1, reflecting the semiclassical intuition that 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) then imply an upper bound for the fermion density <i>ρ</i>(<i>x</i>, <i>y</i>) which verifies known facts from LLL physics. We also explore the entanglement entropy (EE) of subregions of the 2D noncommutative space which displays behaviour distinct from conventional 2D systems as well as from conventional 1D systems, falling somewhere between the two. The main distinguishing feature of the EE, which is directly attributable to the noncommutative nature of 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, namely 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 (see, e.g. [<a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.043610"><i>Phys. Rev. A</i> <b>98</b> (2018) 043610</a>] for a recent application).</p>

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2D or not 2D: a “holographic dictionary” for lowest Landau levels

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  • 发表日期：

    2026年7月2日

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  • Gautam Mandal,
  • Ajay Mohan,
  • Rushikesh Suroshe

摘要

We consider two-dimensional fermions on a plane with a perpendicular magnetic field, described by Landau levels. It is well-known that, semiclassically, restriction to the lowest Landau levels (LLL) amounts to imposing two constraints on a 4D phase space, which transforms the 2D coordinate space (x, y) into a 2D phase space, thanks to the non-zero Dirac bracket between x and y. A straightforward application of Dirac’s prescription of quantizing LLL in terms of L2 functions of x (or of y) fails because the wavefunctions are clearly functions of x and y. We find it possible, however, to construct a different 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 an appropriate large N limit, (a) the Wigner distribution is upper bounded by 1, reflecting the semiclassical intuition that 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) then imply an upper bound for the fermion density ρ(x, y) which verifies known facts from LLL physics. We also explore the entanglement entropy (EE) of subregions of the 2D noncommutative space which displays behaviour distinct from conventional 2D systems as well as from conventional 1D systems, falling somewhere between the two. The main distinguishing feature of the EE, which is directly attributable to the noncommutative nature of 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, namely 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 (see, e.g. [Phys. Rev. A 98 (2018) 043610] for a recent application).