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arxiv: 2511.01630 · v3 · submitted 2025-11-03 · ✦ hep-th · cond-mat.stat-mech· math-ph· math.MP

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

Gautam Mandal , Ajay Mohan , Rushikesh Suroshe This is my paper

Pith reviewed 2026-05-18 01:45 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechmath-phmath.MP
keywords lowest Landau levelnoncommutative geometryWigner distributionentanglement entropyrotating harmonic trapphase space hydrodynamicsholographic dictionaryfermion density bound
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The pith

A 1D quantum mechanics embedded inside 2D describes lowest Landau level physics via an exact density correspondence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the lowest Landau level physics of 2D fermions in a magnetic field can be described by a one-dimensional quantum mechanics embedded within the two-dimensional system. This is done by constructing an exact correspondence between the two-dimensional fermion density and the Wigner distribution of the one-dimensional theory. In the large particle number limit, this correspondence becomes an identity, which, together with the Pauli exclusion principle, leads to an upper bound on the fermion density in the plane. The approach also shows that entanglement entropy in the resulting noncommutative space lacks the usual logarithmic dependence on region size even with a Fermi surface, placing its scaling between standard one- and two-dimensional cases.

Core claim

We 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, the correspondence becomes an identity and implies an upper bound for 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 entangl

What carries the argument

The 1D-2D holographic dictionary that maps the 2D fermion density ρ(x,y) exactly onto the Wigner distribution of the embedded 1D quantum mechanics, becoming an identity in the large N limit.

If this is right

  • Post-quench dynamics of the LLL system can be computed more simply using established methods of 1D phase space hydrodynamics.
  • The fermion density ρ(x,y) acquires a strict upper bound in the large N limit from the Pauli principle applied to the Wigner distribution.
  • Entanglement entropy of subregions shows no logarithmic divergence with the size of the entangling region even though a Fermi surface is present.
  • The EE scaling lies between the area-law behavior of ordinary 2D systems and the logarithmic behavior of ordinary 1D systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same embedding technique may simplify calculations of dynamics and correlations in other systems whose phase space is reduced by similar noncommutative constraints.
  • Numerical many-body simulations of LLL wavefunctions can directly test whether the local density saturates the bound derived from the Wigner function.
  • The removal of the logarithmic term in EE offers a concrete signature of noncommutativity that could be looked for in cold-atom realizations of rotating traps.

Load-bearing premise

The reduction of 2D fermions in a rotating harmonic trap to the standard Landau problem preserves the essential noncommutative structure and permits direct quantization into an embedded 1D QM without additional corrections or loss of wavefunction dependence.

What would settle it

A calculation of the local fermion density in a specific many-body LLL state that exceeds the upper bound obtained when the corresponding Wigner distribution is required to stay at most one per phase space cell.

Figures

Figures reproduced from arXiv: 2511.01630 by Ajay Mohan, Gautam Mandal, Rushikesh Suroshe.

Figure 1
Figure 1. Figure 1: Landau levels in the generalized Landau problem. The energy spectra, for various values of n1 = 0, 1, 2, ..., are given by the red lines, which define the Landau levels in the generalized problem. The lowest Landau level corresponds to n1 = 0 and the black dots represent the allowed n2 values. To be contrasted with the energy levels of the original Landau problem depicted in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 2
Figure 2. 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 approxi￾mation. Panel (b) represents gauge constraints which are genuine constraints: here the entire Hilbert space is in the top left diagonal block. Dirac’s prescription for quant… view at source ↗
Figure 3
Figure 3. Figure 3: The flow chart to illustrate the quantum 1D-2D correspondence for single particle LLL wave-function. In equations, ρ(x, y) = Z dpx dpy (2πℏ) 2 u(x, y, px, py) = Z dpx dpy (2πℏ) 2 d⃗x d⃗pδ1δ2δ3δ4u˜(x1, x2, p1, p2) = Z dpx dpy (2πℏ) 2 d⃗x d⃗pδ1δ2δ3δ4u¯0(x1, p1)u(x2, p2) = mω πℏ Z dx2dp2 πℏ exp − (p2 − √ 2mωy) 2 ℏmω ! exp − ( √ 2x − x2) 2mω ℏ ! u(x2, p2) (3.14) which is the same as the equation (3.9). In the … view at source ↗
Figure 4
Figure 4. Figure 4: The wiggles in (b) are suppressed by the integral transform in (3.17) to yield (a). 3.1.2 For a time-dependent state Note that the general equation (3.9) can be obtained by taking a linear combination P n An(...) on both sides of the eigenfunction-relation (3.17) since the kernel K(x, y, x2, p2) is independent of the quantum number n. A time-dependent LLL wave-function corresponds to making An → An(t) = An… view at source ↗
Figure 5
Figure 5. Figure 5: Classical limit of ground state properties. N = 100, ℏ = 1/N, mω = 5. We give below the 3D plots corresponding to the above. (a) 3D plot of the Wigner distribution. The hor￾izontal axes represent ˜x2, p˜2. (b) 3D plot of the fermion density. The horizontal axes represent x, y [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 3D plots corresponding to [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A “band state” with fermions occupying N levels from N1 to N2. For the band state (4.26), where N1/N, N2/N are held fixed in the large N limit, both the Wigner distribution and the fermion density become step functions, related by (4.25). For N = 300, N1 = 50, N2 = 350, we obtain the following numerical plots: 16 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Classical limit of band states. N = 300, N1 = 50, N2 = 350, ℏ = 1/N, mω = 5. The large N limit is given by the following theoretical curves: u(x2, p2) = θ(˜rhigh − r˜)θ(˜r − r˜low), r˜high = p 2N2ℏ, r˜low = p 2N1ℏ (4.27) ρ(x, y) = mω πℏ (θ(rhigh − r)θ(r − rlow)), rhigh = r N2ℏ mω , rlow = r N1ℏ mω . (4.28) Note that these two functions are related by (4.25). W∞ coherent states For a time-evolving state |ψ,… view at source ↗
Figure 9
Figure 9. Figure 9: The green disc and the blue annulus represent the Wigner distributions of the ground state and a band state respectively. In the region of overlap, the net Wigner distribution is 1, and elsewhere, it is less than 1. Three Slater states Here we consider the linear combination of three distinct Slater states with insignificant overlap in their fillings. |F⟩ = α1 |F1⟩ + α2 |F2⟩ + α3 |F3⟩ (4.40) As before, the… view at source ↗
Figure 10
Figure 10. Figure 10: The red, blue and green discs represent the Wigner distributions of three distinct Slater states. The net Wigner distribution is 1 in the region where the three circles overlap and it is less than 1 everywhere else. 19 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Classical Limit of thermal states at β = 0.1, N = 100, ℏ = 1/N. “Diagonal” mixed states The above analysis quite easily extends to the class of mixed states we call “diagonal” mixed states: ρ = X {Nn} λ{Nn} |{Nn}⟩ ⟨{Nn}| (4.50) where |{Nn}⟩ are the occupation number basis states. The thermal state is just a special case of (4.50) with λ{Nn} = 1 Z exp −β X∞ n=0 ϵnNn ! (4.51) 4.4 Dynamics in the classical l… view at source ↗
Figure 12
Figure 12. Figure 12: Droplet dynamics in the semiclassical limit (see (4.53)). Since the Hamiltonian dynamics is simply a rotation, starting from any initial configuration, the dy￾namics is periodic and it cannot show thermalization [1]. In [18], we will consider an additional deformation to the dynamics to explore thermalization. The 1D-2D correspondence exists even in the situation where electrons are allowed to fill higher… view at source ↗
Figure 13
Figure 13. Figure 13: (a) In ordinary fermion systems (taken to be 1D fermions in the figure), the set of entangled phase space degrees of freedom (d.f.) — depicted in yellow — crosses the Fermi surface at the boundary of the droplet (depicted in blue) where the fermions are. The momentum integrals reach the Fermi surface and lead to a long range two-point correlator C(x, x′ ) whichcan be attributed to massless bosonic fluctua… view at source ↗
Figure 14
Figure 14. Figure 14: Landau levels of the original Landau problem The lowest band, called the lowest Landau level (LLL) consists of states |n1, n2⟩ = |0, n2⟩, which are degenerate, with energy E0,n2 = ℏω. Wavefunctions for LLL state |0, n⟩ are ψn(x, y) = ⟨x, y|0, n⟩ = e −1 2l 2 0 (x 2+y 2 ) ((x − iy)/l0) n l0 √ πn! χ˜n(x1, x2) := ⟨x1, x2|0, n⟩ = r mω πℏ e −(x 2 1+x 2 2 )/2l 2 0 √ 2 nn! Hn(x2/l0) (A.16) In the above l0 is the … view at source ↗
Figure 15
Figure 15. Figure 15: Figure illustrates the fillings defining the states |F1⟩, |F2⟩, |F3⟩ in (F.1). • Case 1: Linear combination of two Slater states (i) If we construct |F⟩gen = α1|F1⟩ + α2|F2⟩, we can easily see that such a linear combination can be rewritten as follows |F⟩ ′ =  α1c † N + α2c † N+1 c † N−1 c † N−1 ...c † 2 c † 1 |0⟩ (F.2) This is a single Slater determinant constructed from a new set of single-particle st… view at source ↗
Figure 16
Figure 16. Figure 16: Classical limit of linear combination of slater states properties in the presence of off diagonal term. N = 300, ℏ = 1/N, mω = 1. • Case 2: Linear combination of more than two Slater staes As can be seen from Figs. (17) and (18), the U˜ gen given by equation (F.7) and ρgen(x, y) given by equation (F.6) show negligible contributions from the off-diagonal terms compared to the diagonal ones. 0.5 1.0 1.5 2.0… view at source ↗
Figure 17
Figure 17. Figure 17: Classical limit of linear combination of slater states in the presence of off diagonal terms (see Section F.1). N = 300, ℏ = 1/N, mω = 1. 43 [PITH_FULL_IMAGE:figures/full_fig_p045_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Scaling of off diagonal terms with N.The y axis shows equation(F.8) and x axis is ˜r. Hence, U˜ gen ≈ X i=1,2,3 |αi | 2u (i) (x, p), ρgen(x, y) ≈ X i=1,2,3 |αi | 2 ρ (i) (x, y), (F.10) Therefore the equation (4.18) still continues to hold for equation(F.10). Note that generically, in the large N limit, U˜ gen < 1 (F.11) To see this from (F.10), suppose that each of the states |f1⟩, |f2⟩, |f3⟩, by itself, … view at source ↗
Figure 19
Figure 19. Figure 19: Entanglement entropy for N = 60.The orange dashed line shows the linear behavior of entanglement entropy in the case where subregion is smaller than the Fermi droplet. The fit to the linear regime of the plot is S = 1.86 l l0 (G.7) G.1.2 Behavior of entanglement entropy in (x1, x2) plane We will now encounter a rather different (logarithmic) behaviour of the EE when we consider a strip geometry in the (x1… view at source ↗
Figure 20
Figure 20. Figure 20: Strip in the x1-x2 plane. 45 [PITH_FULL_IMAGE:figures/full_fig_p047_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Spatial subregion A for a 1D fermi fluid. The EE is given by SA = π 2 3 log pF l ℏ  (G.18) 47 [PITH_FULL_IMAGE:figures/full_fig_p049_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Entangling region in the shape of a square for a 2D fermi fluid. The EE is given by SA ∝ pF l ℏ log pF l ℏ  (G.27) A brief derivation is given below. In this derivation we could proceed as in the 1D case, doing the momentum integrals first and arriving at (5.3). For the square droplet in [PITH_FULL_IMAGE:figures/full_fig_p051_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: On the left we consider the original droplet which has the shape of a disk of radius pF , which represents the ground state of a free fermion problem. On the right we consider a square droplet of side pF ; In the limit of pF → ∞ both shapes will fill out the plane. For large enough sizes of both figures, we expect the EE for both shapes to show similar qualitative behaviour. In such a case u(x, y, px, py)… view at source ↗
Figure 24
Figure 24. Figure 24: Spectrum for multiple filled Landau levels. The Wigner distribution function representing above state is U(x2, p2) = X n1,n2 un1,n2 (x2, p2) = NmaxX +m−1 n2=0 u0,n2 (x2, p2) + mX−1 n2=0 u1,n2 (x2, p2) (H.1) In the large N limit, the first term of the RHS of the above equation is NmaxX +m−1 n2=0 u0,n2 (x2, p2) = θ(˜r 2 N1 − r˜ 2 ), r˜N1 = p N1ℏ (H.2) The second term in the RHS of the above equation is simp… view at source ↗
Figure 25
Figure 25. Figure 25: Classical limit of ground state properties (3D plot). N = 100, ℏ = 1/N, mω = 5. 52 [PITH_FULL_IMAGE:figures/full_fig_p054_25.png] view at source ↗
read the original 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 \rho(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 \rho(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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to construct an embedded 1D quantum mechanics inside the 2D QM of fermions in a rotating harmonic trap (reducing to the Landau problem in a special limit) that captures LLL physics. It asserts an exact 1D-2D correspondence mapping the fermion density ρ(x,y) to the Wigner distribution of the 1D system; in a suitable large-N limit this becomes an identity, implying an upper bound on ρ(x,y) from the Pauli principle. The work further studies entanglement entropy of subregions in the resulting noncommutative space, finding intermediate 1D-2D behavior without logarithmic scaling, and notes simpler post-quench dynamics via 1D phase-space hydrodynamics.

Significance. If the exact correspondence holds without extra quantum corrections, the construction supplies a practical dictionary for LLL computations, enabling use of established 1D methods for dynamics and yielding a Pauli-derived density bound together with distinctive noncommutative entanglement signatures.

major comments (1)
  1. [Construction of the 1D QM via Dirac quantization (following the rotating-trap reduction)] The reduction of the rotating-trap system to the Landau problem followed by imposition of the two semiclassical constraints and Dirac quantization is presented as yielding an exact embedded 1D QM whose Wigner function maps onto ρ(x,y). No explicit finite-N verification is given that the 1D operators reproduce the matrix elements of the projected 2D density operator, leaving open uncontrolled operator-ordering or correction terms that would invalidate the claimed exact correspondence and the large-N identity (see skeptic note on quantization of Dirac brackets).
minor comments (1)
  1. [Abstract] The abstract refers to 'a suitable large N limit' without stating the precise scaling or conditions under which the correspondence becomes an identity; this should be made explicit early in the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address the major concern regarding the 1D QM construction and the exactness of the correspondence below, providing clarification on the algebraic nature of the mapping while agreeing to strengthen the presentation with additional discussion.

read point-by-point responses

  1. Referee: [Construction of the 1D QM via Dirac quantization (following the rotating-trap reduction)] The reduction of the rotating-trap system to the Landau problem followed by imposition of the two semiclassical constraints and Dirac quantization is presented as yielding an exact embedded 1D QM whose Wigner function maps onto ρ(x,y). No explicit finite-N verification is given that the 1D operators reproduce the matrix elements of the projected 2D density operator, leaving open uncontrolled operator-ordering or correction terms that would invalidate the claimed exact correspondence and the large-N identity (see skeptic note on quantization of Dirac brackets).

    Authors: We appreciate the referee highlighting this point. The rotating-trap Hamiltonian reduces exactly to the Landau problem when the trap frequency equals the cyclotron frequency. The LLL projection is then implemented by imposing the two semiclassical constraints, after which Dirac quantization is applied to the resulting noncommutative structure. This procedure yields an embedded 1D QM by construction, with the 2D density ρ(x,y) mapped to the Wigner distribution via the identification of coordinates with phase-space variables. The mapping holds at the operator level for finite N because the 1D operators are defined to reproduce the matrix elements of the projected 2D density operator; no additional corrections arise from the specific form of the constraints. The large-N identity and Pauli bound follow in the semiclassical limit where the Wigner function is bounded by 1. While explicit small-N numerical checks of matrix elements are not included in the current version, the algebraic derivation ensures exactness without uncontrolled ordering terms, as the constant Dirac bracket admits a unique quantization (with standard Weyl ordering for the Wigner function). We will add a clarifying subsection and a small-N illustrative example in the revised manuscript to make this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit quantization and correspondence

full rationale

The paper derives the 1D QM embedding from the rotating-trap reduction to the Landau problem, followed by semiclassical Dirac-bracket constraints and direct quantization into an embedded 1D system. The claimed exact 1D-2D density-Wigner correspondence is presented as a constructed output of this procedure rather than an input assumption. The large-N identity and resulting Pauli upper bound on ρ(x,y) then follow by applying the standard exclusion principle to the Wigner function once the correspondence is established; no fitted parameter is renamed as a prediction, no self-citation chain bears the central load, and no ansatz is smuggled in. The construction retains independent content from the noncommutative structure and the explicit mapping, making the overall chain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that semiclassical LLL constraints via Dirac brackets can be quantized into an embedded 1D system, plus the modeling choice that the rotating trap reduces to the Landau problem without altering the noncommutative structure.

axioms (2)
  • domain assumption Semiclassical restriction to lowest Landau levels imposes two constraints on 4D phase space that turn 2D coordinates into a noncommutative 2D phase space via nonzero Dirac bracket between x and y.
    Invoked in the opening paragraph to motivate the failure of naive L2 quantization and the need for the 1D embedding.
  • domain assumption The 2D fermions in a rotating harmonic trap reduce to the standard Landau problem in a special limit while preserving the essential LLL physics.
    Stated as the working model chosen instead of direct Landau problem.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Directly computing Wigner functions for open quantum systems

    quant-ph 2025-12 unverdicted novelty 6.0

    An expression is derived to compute time-dependent Wigner functions directly from initial values in open quantum systems of a non-relativistic particle with a general environment.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper · 7 internal anchors

  1. [1]

    Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics

    M. Kulkarni, G. Mandal and T. Morita,Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics,Phys. Rev. A98(2018) 043610, [1806.09343]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  2. [2]

    G. V. Dunne, R. Jackiw and C. A. Trugenberger,Topological (Chern-Simons) Quantum Mechanics, Phys. Rev. D41(1990) 661

    work page 1990

  3. [3]

    Testing Non-commutative QED, Constructing Non-commutative MHD

    Z. Guralnik, R. Jackiw, S. Y. Pi and A. P. Polychronakos,Testing noncommutative QED, constructing noncommutative MHD,Phys. Lett. B517(2001) 450–456, [hep-th/0106044]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  4. [4]

    P. A. M. Dirac,Generalized Hamiltonian Dynamics,Canadian Journal of Mathematics2(1950) 129–148

    work page 1950

  5. [5]

    Dirac,Lectures on Quantum Mechanics

    P. Dirac,Lectures on Quantum Mechanics. Dover Publications, New York, 1964, 2001

    work page 1964

  6. [6]

    S. R. Das, S. Hampton and S. Liu,Entanglement entropy and phase space density: lowest Landau levels and 1/2 BPS states,JHEP06(2022) 046, [2201.08330]

    work page arXiv 2022

  7. [7]

    Landau,Diamagnetismus der Metalle,Zeitschrift fur Physik64(Sept., 1930) 629–637

    L. Landau,Diamagnetismus der Metalle,Zeitschrift fur Physik64(Sept., 1930) 629–637

    work page 1930

  8. [8]

    Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions

    B. Lacroix-A-Chez-Toine, S. N. Majumdar and G. Schehr,Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance,Phys. Rev. A99(2019) 021602, [1809.05835]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  9. [9]

    Kulkarni, P

    M. Kulkarni, P. Le Doussal, S. N. Majumdar and G. Schehr,Density profile of noninteracting fermions in a rotating two-dimensional trap at finite temperature,Phys. Rev. A107(Feb, 2023) 023302

    work page 2023

  10. [10]

    Wigner,On the quantum correction for thermodynamic equilibrium,Phys

    E. Wigner,On the quantum correction for thermodynamic equilibrium,Phys. Rev.40(Jun, 1932) 749–759

    work page 1932

  11. [11]

    Klich,Lower entropy bounds and particle number fluctuations in a fermi sea,Journal of Physics A: Mathematical and General39(jan, 2006) L85

    I. Klich,Lower entropy bounds and particle number fluctuations in a fermi sea,Journal of Physics A: Mathematical and General39(jan, 2006) L85

    work page 2006

  12. [12]

    Swingle,Entanglement entropy and the fermi surface,Phys

    B. Swingle,Entanglement entropy and the fermi surface,Phys. Rev. Lett.105(Jul, 2010) 050502

    work page 2010

  13. [13]

    tong,Lectures on quantum hall effect,

    D. tong,Lectures on quantum hall effect,

    work page

  14. [14]

    R. B. Laughlin,Quantized motion of three two-dimensional electrons in a strong magnetic field,Phys. Rev. B27(Mar, 1983) 3383–3389

    work page 1983

  15. [15]

    Kulkarni, S

    M. Kulkarni, S. N. Majumdar and G. Schehr,Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap,Phys. Rev. A103(Mar, 2021) 033321

    work page 2021

  16. [16]

    Weyl,The Theory of Groups and Quantum Mechanics

    H. Weyl,The Theory of Groups and Quantum Mechanics. Dover Publications, New York, 1950. 53

    work page 1950

  17. [17]

    C. N. Pope and X. J. Wang,A Note on representations of the w(infinity) algebra, in21st Conference on Differential Geometric Methods in Theoretical Physics (XXI DGM) - 5th Nankai Workshop, 9, 1992,hep-th/9209067

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  18. [18]

    Mandal, A

    G. Mandal, A. Mohan, T. Morita and R. Suroshe,In progress.,

    work page

  19. [19]

    Tsuchiya and K

    A. Tsuchiya and K. Yamashiro,Target space entanglement in a matrix model for the bubbling geometry,Journal of High Energy Physics2022(Apr., 2022)

    work page 2022

  20. [20]

    Klich and L

    I. Klich and L. Levitov,Quantum noise as an entanglement meter,Phys. Rev. Lett.102(Mar, 2009) 100502

    work page 2009

  21. [21]

    H. F. Song, C. Flindt, S. Rachel, I. Klich and K. Le Hur,Entanglement entropy from charge statistics: Exact relations for noninteracting many-body systems,Phys. Rev. B83(Apr, 2011) 161408

    work page 2011

  22. [22]

    Calabrese, M

    P. Calabrese, M. Mintchev and E. Vicari,Entanglement entropy of one-dimensional gases,Phys. Rev. Lett.107(Jul, 2011) 020601

    work page 2011

  23. [23]

    Astrophysical Quantum Matter: Spinless charged particles on a magnetic dipole sphere

    J. Murugan, J. P. Shock and R. P. Slayen,Astrophysical Quantum Matter: Spinless charged particles on a magnetic dipole sphere,Gen. Rel. Grav.53(2021) 29, [1811.03109]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  24. [24]

    Ciftja,Detailed solution of the problem of landau states in a symmetric gauge,European Journal of Physics41(apr, 2020) 035404

    O. Ciftja,Detailed solution of the problem of landau states in a symmetric gauge,European Journal of Physics41(apr, 2020) 035404

    work page 2020

  25. [25]

    J. E. Moyal,Quantum mechanics as a statistical theory,Proc. Cambridge Phil. Soc.45(1949) 99–124

    work page 1949

  26. [26]

    S. Iso, D. Karabali and B. Sakita,One-dimensional fermions as two-dimensional droplets via Chern-Simons theory,Nucl. Phys. B388(1992) 700–714, [hep-th/9202012]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  27. [27]

    S. Iso, D. Karabali and B. Sakita,Fermions in the lowest Landau level: Bosonization, W infinity algebra, droplets, chiral bosons,Phys. Lett. B296(1992) 143–150, [hep-th/9209003]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  28. [28]

    D. S. Dean, P. Le Doussal, S. N. Majumdar and G. Schehr,Wigner function of noninteracting trapped fermions,Phys. Rev. A97(Jun, 2018) 063614

    work page 2018

  29. [29]

    De Bruyne, D

    B. De Bruyne, D. S. Dean, P. Le Doussal, S. N. Majumdar and G. Schehr,Wigner function for noninteracting fermions in hard-wall potentials,Phys. Rev. A104(Jul, 2021) 013314

    work page 2021

  30. [30]

    N. R. Smith, P. Le Doussal, S. N. Majumdar and G. Schehr,Counting statistics for noninteracting fermions in ad-dimensional potential,Phys. Rev. E103(Mar, 2021) L030105. 54

    work page 2021