Possible restoration of particle-hole symmetry in the 52 Quantized Hall State at small magnetic eld Lo c Herviou1and Fr ed eric Mila1

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Possible restoration of particle-hole symmetry in the 5/2 Quantized Hall State at
small magnetic field
Lo¨ıc Herviou1and Fr´ed´eric Mila1
1Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
(Dated: October 19, 2022)
Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling
ν= 5/2, a result incompatible with both the Pfaffian and the Antipfaffian states, we have pushed
the expansion of the effective Hamiltonian of the 5/2 quantized Hall state to third-order in the
parameter κ=Ec/~ωc∝1/√Bcontrolling the Landau level mixing , where Ecis the Coulomb
energy and ωcthe cyclotron frequency. Exact diagonalizations of this effective Hamiltonian show
that the difference in overlap with the Pfaffian and the AntiPfaffian induced at second-order is
reduced by third-order corrections and disappears around κ= 0.4, suggesting that these states are
much closer in energy at smaller magnetic field than previously anticipated. Furthermore, we show
that in this range of κthe finite-size spectrum is typical of a quantum phase transition, with a
strong reduction of the energy gap and with level crossings between excited states. These results
point to the possibility of a quantum phase transition at smaller magnetic field into a phase with
an emergent particle-hole symmetry that would explain the measured 5/2 thermal conductance of
the 5/2 quantized Hall state.
I. INTRODUCTION
Following up on Laughlin’s pioneering explanation of
the 1/3 and 1/5 plateaus1in terms of wave-functions,
the theory of the Fractional Quantum Hall Effect
(FQHE) has relied to a large extent on variational
wave-functions, with considerable success thanks in
particular to Jain’s composite fermion theory2–5. It was
completed by its extension to the Pfaffian (Pf) state
by Moore and Read6,7, and its particle-hole conjugate,
the AntiPfaffian8,9(APf) state in order to explain the
plateau observed at filling 5/210,11. This approach has
explained many plateaus and has led to the highly
nontrivial prediction that the system is gapless at
filling 1/212–14. It seemed that all states observed in
the FQHE could be explained in terms of variational
wave-functions. The alternative approach in terms of
an effective Hamiltonian to describe the degeneracy
lifting by Coulomb repulsion in a partially filled Landau
level has nevertheless proven to be extremely useful.
The variational wave-functions were shown to be the
ground states of parent Hamiltonians that are truncated
versions of the effective model2,15–18. Effective Hamil-
tonians have also played a crucial role in discussing the
competition between the Pf and the APf states for the
5/2 quantized Hall state, with the conclusion that, to
second order in κ=Ec/~ωc, where Ecis the Coulomb
energy and ωc∝Bthe cyclotron frequency, the APf
state is favored when the effect of the empty Landau
levels are taken into account19–26. It thus came as a
big surprise when the quantized thermal conductance
was found to be equal to 5/2 in the 5/2 plateau27–29,
a value in contradiction with both the APf (3/2) and
the Pf (7/2) but consistent with particle-hole symmetry.
This restored symmetry would be consistent with the
Particle-Hole Pfaffian (PHPf)30,31, another candidate
for the 5/2 plateau, but this state is usually believed to
be gapless and energetically unfavored32–35 except in a
few field-theoretical works36,37. Explanations in terms
of domains or of lack of equilibration have been put
forward38–51, but as of today there is no consensus on
the resolution of this discrepancy.
In view of the impressive corpus of theory, one may
wonder if there is still room for the identification of
a particle-hole symmetric ground state that would
have been missed so far. Our results point towards
such an unlikely conclusion. The starting point of our
approach is the observation that the effective model
is an expansion in κ∝1/√B, hence a high-field
expansion. In experimental conditions, the field is
around 4 T, and κ'1.3811. This large value raises a
natural question: Is κtoo large in experiments for the
second-order expansion to be justified? The only way to
answer that question is to push the expansion to higher-
order in κ, something that has not been attempted so far.
In the present work, we have pushed this expansion to
the next order in κ. Although this simply relies on third-
order perturbation theory in the Coulomb repulsion, this
turned out to be a rather formidable task that could only
be carried out with the help of computer-aided formal
calculations. As we shall see, the third order changes the
physics qualitatively already around κ'0.3−0.5, show-
ing that relying on second-order perturbation theory is
definitely not justified for κ'1.38. The most remark-
able effect is that the lifting of the degeneracy in favour
of the APf is counter-acted by the third-order term, and
that the degeneracy is restored at κ'0.4. Moreover,
and maybe more importantly, the excitation spectrum of
finite-size systems has all the characteristics of a quantum
phase transition which, in view of the apparent restora-
tion of particle-hole symmetry between the Pf and the
APf, might lead to a phase with an emergent particle-
arXiv:2210.04925v2 [cond-mat.str-el] 18 Oct 2022
2
hole symmetry. Let’s already emphasize however that, if
there is indeed a quantum phase transition, the physics
beyond the phase transition cannot be reached by per-
turbation theory, and any attempt at discussing it on the
basis of a truncated perturbative Hamiltonian, as done
previously for the second-order model, is prone to fail.
Non-perturbative approaches will have to be employed
to study that problem.
The paper is organized as follows. In Section II, we
explain the main ideas of the algorithm we have used
to derive the third-order Hamiltonian. In Section III-A,
we compare the results we have obtained at second-order
with previous results, and in Section III-B we present
and discuss the central results of this paper obtained at
third-order. Section IV is devoted to two related models
that help assessing the validity of our approach, a model
that only includes the first two Landau levels (IV-A),
and a model that assumes full polarization of the low-
est Landau level (IV-B). Finally, the implications for the
5/2 quantum Hall state are discussed in Section V. De-
tails about all aspects of this study can be found in the
Appendices.
II. DERIVATION OF THE EFFECTIVE
HAMILTONIAN
We consider two-dimensional electrons on a square
torus in a normal magnetic field and in the presence of
Coulomb interaction. Up to a constant, the Hamiltonian
can be formulated in a second-quantization formalism as
Hexact =H0+H1(1)
H0=~ωcX
l
lNl−gB X
σ
σNσ(2)
H1=EcX
~m,~n,~
l,~σ
A~
l,~σ
~m,~nc†
m1,lm1,σm1c†
m2,lm2,σm2
×cn2,ln2,σn2cn1,ln1,σn1.(3)
ωcis the cyclotron frequency, Ec=e2
εlBthe Coulomb
energy, lBthe magnetic length, Bthe magnetic field
and gthe electronic magnetic moment. In the rest of
the article, we set both lBand Ecto 1 for simplicity,
and generally neglect the Zeeman splitting given its
magnitude. l∈Ndenotes the Landau level, σ=±1
its spin flavour, Nlthe number of electrons in a given
Landau level and Nσthe number of electrons with spin
σ. We also denote by Lthe total number of Landau
levels we consider (the spin degeneracy is not included in
the counting). We will show results with up to L= 11,
although our results depend very little on Las soon as
L≥3, i.e. when we take into account the influence of
the empty Landau levels. Finally, the operator c†
m,l,σ
creates an electron in the mth orbital of the lth Landau
level with spin σ. On a torus, m∈[0, Nφ−1] with
Nφthe number of elementary magnetic fluxes through
the torus. The interaction coefficient A~
l,~σ
~m,~n can be
straightforwardly obtained for any translation-invariant
interaction as detailed in App. B. For a Coulomb-like
interaction (central and spin-diagonal), symmetry en-
forces m1+m2=n1+n2[Nφ] and σm1+σm2=σn1+σn2.
In the rest of this article, we focus on the physics of the
5/2 filled Landau levels. In a strong magnetic field, the
splitting of the Landau levels dominates, and it appears
reasonnable to project the Hamiltonian on its low-energy
sector (depending on the filling). Despite the weak Zee-
man effect, numerical simulations seem to indicate that
the half-filled Landau level is spin polarized. We there-
fore introduce P0, the projector on the subspace where
the 0th Landau level is fully occupied for both spin fla-
vors, and where the 1st Landau level with spin +1 is
half-filled. We project the Hamiltonian Hexact on this
subspace and define:
H0=P0H0P0=E0Id with E0= (~ωc−gB)Nφ
2(4)
H1=P0H1P0=EcX
~m,~n
A~
l,~σ
~m,~nc†
m1c†
m2cn2cn1+E0,1
C,(5)
with cn=cn,1,+1 and E0,1
Cthe static energy due to
the presence of the filled Landau levels. Note that this
energy constant plays a crucial role in the third-order
expansion and cannot be neglected. This projected
Hamiltonian with no Landau-level mixing has been
extensively studied15,52–56. On the square torus, it
admits six quasi-degenerate ground states in six different
translation sectors corresponding to the six-fold topolog-
ical degeneracy of the Pf or APf state. Unless specified,
we show results in the (π, 0) sector — our results are
largely independent of this choice. This Hamiltonian
is particle-hole symmetric at half-filling. Due to this
symmetry, it cannot discriminate between the Pf and
APf phases. The ground state in a given sector is unique
and has equal overlap with both Pf and APf states.
In order to reach the experimentally relevant regime,
we compute perturbatively the effect of the presence of
the empty and occupied Landau levels using κ=e2
ε~lBωc
as a small parameter. Following Rezayi26, who per-
formed a calculation to second order, we compute directly
the effective Hamiltonian without attempting to project
onto pseudo-potentials. The development in pseudo-
potentials is not convenient on a finite torus due to the
periodicity, and the complexity of the higher-body terms
rises quickly. The second-order and third-order terms of
the degenerate perturbative expansion are given by
H2=−P0H1G0H1P0
H3=P0H1G0H1G0H1P0−1
2{H1, P0H1G2
0H1P0}(6)
where
G0=Id −P0
H0−E0
.(7)
3
H2includes two- and three-body terms as discussed
in previous works. H3also includes an additional
four-body term. The five-body terms generated by
P0H1G0H1G0H1P0are exactly cancelled by the anticom-
mutator.
We perform the numerical computation of the effective
Hamiltonians directly at the operator level. The details
of our algorithm can be found in App. B. As a quick sum-
mary, our computational process consists of four parts:
1. Computation of the effective interaction in the Lan-
dau basis.
2. Derivation of all the Feynman diagrams corre-
sponding to H2and H3.
3. Exact summation of all processes corresponding to
a given diagram.
4. Computation of the effective many-body Hamilto-
nian and diagonalization.
The latter two steps are the most computationaly ex-
pensive. The complexity of the third step scales as
O(max(5!LN7
φ,(2L)4N4
φ)), depending on the diagrams
considered. The fourth step has the standard exponen-
tial complexity of exact diagonalization, but with an
additional difficulty: the effective Hamiltonian consists
of a sum of several millions of n-body operators. To
give a concrete illustration, for Nφ= 28, although the
symmetry-resolved Hilbert space is only of dimension
≈105, it includes ≈1.5×107operators (≈106if we take
into account translation invariance). Even if we were to
discard coefficients below 10−6, we would need to apply
several millions of operators to each basis element. Con-
sequently, it is not surprising that the effective Hamil-
tonians themselves are also very dense (approximately a
quarter of the matrix elements are non-zero for Nφ= 28).
This density limits the practically achievable sizes signifi-
cantly: both the memory cost to store the matrix and the
cost of applying it to a state become quickly prohibitive.
III. PERTURBATIVE EXPANSION
A. Second-order expansion
We start by a brief discussion of the second-order
expansion of Hexact as a benchmark of our approach.
This computation has been previously done on the
sphere19–22,24–26,57 and on the hexagonal torus24,26
and we verify that we qualitatively and quantitatively
recover known results.
Concretely, we work with the Hamiltonian
H(2) =H1+κH2(8)
and compute its ground state. In Fig. 1(a-d), we show
the low-energy spectrum of H(2). The color represents
0 0.5 1 1.5
0.00
0.05
0.10
0.15
0.20
Energy levels
a) Nφ=16, L = 11
−0.2
−0.1
0.0
0.1
0.2
0 0.5 1 1.5
0.00
0.05
0.10
0.15
b) Nφ=20, L = 11
0 0.5 1 1.5
κ
0.00
0.05
0.10
Energy levels
c) Nφ=24, L = 11
0 0.5 1 1.5
κ
0.00
0.02
0.04
0.06
d) Nφ=28, L = 4
0 0.25 0.5 0.75 1 1.25 1.5
κ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
|hΨref|Ψ2i|
e)
Sector (π,0)
Nφ=20
Nφ=24
Nφ=28
APf
Pf
H3b
2
−V1
Figure 1. (a-d) Low-energy spectrum in the sector (π, 0) at
second order. The color code is given in Eq. (9): positive
(resp. negative) numbers mark that the Pf (resp. APf) state
is favored. (e) Overlap between the ground state and several
reference states for different system sizes. Here the overlap
is not corrected as in Eq. (9) for simplicity. Below κ≈0.8,
the APf is favored. We then observe a collapse of all low-
energy levels. After a small transitional regime, two consecu-
tive phases open. The large κ≥1.4 regime is strongly gapped
in the translation invariant sector.
the difference in overlaps between the Pf and APf. More
precisely, the color is a measure of
|hΨPf |ΨEDic| − |hΨAPf |ΨEDic|(9)
with hΨPf |ΨEDic
hΨAPf |ΨEDic=M−1hΨPf |ΨEDi
hΨAPf |ΨEDi(10)
and M=hΨPf |ΨPf i hΨPf |ΨAPf i
hΨAPf |ΨPf i hΨAPf |ΨAPf i.(11)
摘要：

Possiblerestorationofparticle-holesymmetryinthe5/2QuantizedHallStateatsmallmagnetic eldLocHerviou1andFredericMila11InstituteofPhysics,EcolePolytechniqueFederaledeLausanne(EPFL),CH-1015Lausanne,Switzerland(Dated:October19,2022)Motivatedbytheexperimentalobservationofaquantized5/2thermalconductan...

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