# An Anomalous Reentrant 5/2 Quantum Hall Phase at Moderate Landau-Level-
Mixing Strength

Sudipto Das  Sahana Das  Sudhansu S. Mandal  Department of Physics, Indian
Institute of Technology, Kharagpur, West Bengal 721302, India

###### Abstract

A successful probing of neutral Majorana mode in recent thermal Hall
conductivity measurements opines in favor of the particle-hole symmetric
Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of
Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous
quantized phase which is found to be gapped in the thermodynamic limit,
distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an
intermediate strength of Landau level mixing. Our proposed wave function
consistent with the PH-Pf shift in spherical geometry rightly captures the
topological order of this phase, as its overlap with the exact ground state is
very high and it reproduces low-lying entanglement spectra. A unique
topological order, irrespective of the flux shifts, found for this phase
possibly corroborates the experimentally found topological order.

††preprint: APS/123-QED

The discovery Willett  _et al._ (1987) of fractional quantum Hall effect in an
even denominator filling factor 5/2 way back in 1987 had surprised the entire
physics community. The first significant understanding of the possibility of
this state had been put forward by Moore and Read (MR) through a unique
proposal Moore and Read (1991) of the Pfaffian (Pf) wave function, which was
later interpreted Read and Green (2000) as a chiral p-wave pairing of the
composite fermions Jain (1989, 2007) owing to their effective attractive
interactions Scarola  _et al._ (2000) in the second Landau level.
Consequently, a flurry of new experimental techniques (See Ref.Feldman and
Halperin (2021) for a review) were developed for realizing exotic properties
of this state such as quasiparticle charge, non-Abelian braiding statistics of
the quasiparticles, and Majorana edge modes. On the other hand, subsequent
proposals Lee  _et al._ (2007); Levin  _et al._ (2007) of the anti-Pfaffian
(A-Pf) which is topologically distinct from Pf yet degenerate for any two-body
interaction makes the issue intriguing for understanding the true nature of
the state. However, the Landau-level mixing (LLM), which is important for the
second and higher Landau level quantum Hall states, generates three-body
interaction Bishara and Nayak (2009); Peterson and Nayak (2013); Simon and
Rezayi (2013); Sodemann and MacDonald (2013) that breaks Peterson  _et al._
(2008) this degeneracy. A topologically distinct phase, namely, the particle-
hole symmetric Pfaffian (PH-Pf) as the s-wave pairing of the Dirac composite
fermions is also proposed Son (2015), giving rise to yet another competing
non-Abelian topological phase in the list for the same state. A PH-Pf-like
instability in favor of the Dirac composite fermions with a greater mass
arising due to LLM has also been proposed Antonić  _et al._ (2018) in a gauge
theory.

The verdict of the numerical studies Rezayi and Simon (2011); Zaletel  _et
al._ (2015); Pakrouski  _et al._ (2015); Rezayi (2017); Simon  _et al._ (2020)
based on the LLM is possibly in favor of the A-Pf phase over the Pf phase.
Both these topological phases and also the PH-Pf phase host the Majorana edge
modes whose presence can be probed as half-integral thermal Hall conductance
along the edge. These phases are, however, distinguishable because the
respective predicted values of thermal Hall conductances Simon (2018);
Banerjee  _et al._ (2018) are $3/2$, $7/2$ and $5/2$ in the unit of
$G_{0}=\pi^{2}k_{B}^{2}T/(3h)$. In contrast to the theoretical expectation
Rezayi and Simon (2011); Zaletel  _et al._ (2015); Pakrouski  _et al._ (2015);
Rezayi (2017); Simon  _et al._ (2020); Simon (2018), the recent thermal Hall
conductance and shot-noise measurements Banerjee  _et al._ (2018); Dutta  _et
al._ (2022) are consistent with the PH-Pf phase. For reconciling this, several
probable scenarios have been proposed in the literature such as non-
equilibration Simon (2018) of the thermal Majorana mode and subsequent
proposals of partial-equilibration of anti-Pfaffian edge Ma and Feldman
(2019); Simon and Rosenow (2020); Asasi and Mulligan (2020), formation of
puddles of Pf and A-Pf phases Mross  _et al._ (2018); Zhu and Sheng (2019);
Zhu  _et al._ (2020), and stabilization of PH-Pf due to disorder Lian and Wang
(2018); Mross  _et al._ (2018); Wang  _et al._ (2018); Fulga  _et al._ (2020).
However, no general consensus has yet been achieved and thus the 5/2 state
remains enigmatic. Moreover, whereas the 5/2 state in GaAs is typically
observed Willett  _et al._ (1987); Bishara and Nayak (2009); Peterson and
Nayak (2013); Banerjee  _et al._ (2018); Dutta  _et al._ (2022); Dean  _et
al._ (2008a, b); Zhang  _et al._ (2010); Pan  _et al._ (2001, 2014);
Samkharadze  _et al._ (2017) in the range of $12$–$1$ Tesla magnetic field
which amounts to the LLM parameter $\kappa=0.7$–$2.5$, the theoretical studies
Rezayi and Simon (2011); Zaletel  _et al._ (2015); Pakrouski  _et al._ (2015);
Rezayi (2017) have been performed for $\kappa\lesssim 1$ only. Also, a
topological phase transition from a quantum Hall state to an unquantized state
in the vicinity of $\kappa\sim 0.7$–1 followed by a hint of a new quantized
phase for Pf shift has been found Pakrouski  _et al._ (2015) by the numerical
calculation of the lowest excitation energies and entanglement entropy.
However, the latter phase has not been explored further due to the lack of its
understanding in terms of the MR wave function Moore and Read (1991)
$\Psi_{{\rm MR}}$ or its particle-hole conjugate Lee  _et al._ (2007); Levin
_et al._ (2007) wave function $\Psi_{{\rm MR}}^{{\rm phc}}$.

Here we perform exact diagonalization of the Coulomb Hamiltonian corrected
with LLM of strength $\kappa$ separately at Pf, A-Pf, and PH-Pf flux shifts
for few-electron systems (up to $N=16$) in spherical geometry and determine
overlaps of the exact ground states at different $\kappa$ values if the
corresponding ground states are found at total angular momentum $L=0$. It
amazingly shows (generic to all the shifts) clear segregation of two distinct
quantum Hall phases formed at low and a moderate range of $\kappa$ separated
by an unquantized (ground state at $L\neq 0$) regime in the vicinity of
$\kappa\sim 0.7$. Whereas the overlaps of the ground states for intra-phase
$\kappa$ values are nearly unity, it is negligible for inter-phase $\kappa$
values. A finite excitation gap for a pair of quasiparticle and quasihole in
the thermodynamic limit for the phase at moderate-$\kappa$ suggests it to be a
quantized phase. The entanglement spectra (ES) for this quantized phase show
an entanglement gap with the number of edge state counting as 1-1-2-2 (up to
the resolution obtained in finite systems). Further, the ES for all the three
flux shifts at this moderate-$\kappa$ anomalous phase ($\mathcal{A}$-phase)
has a high degree of resemblance, signifying a unique topological order which
is independent of these shifts. We propose a trial wave function
$\Psi_{\mathcal{A}}$ for this anomalous $\mathcal{A}$-phase. As its flux-shift
matches with PH-Pf shift, we determine the overlap of it with the exact ground
state at PH-Pf shift and found to be very high. Moreover, the low-lying ES for
$\Psi_{\mathcal{A}}$ is consistent with the same for the exact state and hence
can be considered as representative for the true topological order of the
$\mathcal{A}$-phase. We further analyze the topological properties of
$\Psi_{\mathcal{A}}$ including the Majorana mode governing $2.5\,G_{0}$
thermal Hall conductance. Therefore, the $\mathcal{A}$-phase can possibly be
attributed to the experimentally observed phase Banerjee  _et al._ (2018);
Dutta  _et al._ (2022) at moderate $\kappa$.

Figure 1: (color online) (a) Overlaps (shown as color map) of the exact ground
states (found at $L=0$) of the Hamiltonian $\hat{H}_{\text{eff}}$ in Eq. (1)
with LLM-corrected pseudopotentials pse  at Pf flux shift ($N=14$ and
$N_{\phi}=25$) for varying $\kappa$. Two topologically distinct phases are
found as the overlaps of the ground states at different $\kappa$ values
belonging to the same phase are closer to unity and two different phases are
closer to zero. These two phases are intermediated by an unquantized regime
(shown as crossed marks) of $\kappa$ where the ground state is not found at
$L=0$. The gray zones indicate no overlap has been shown as one of the ground
states corresponds to the unquantized regime. (b) Same as (a) but for A-Pf
flux shift ($N=12$ and $N_{\phi}=25$). (c) Same as (a) but for PH-Pf flux
shift ($N=14$ and $N_{\phi}=27$).

The effective interaction between spin-polarized electrons in the second
Landau level with the consideration of LLM available in the literature Bishara
and Nayak (2009); Peterson and Nayak (2013) as,

$\displaystyle\hat{H}_{\text{eff}}(\kappa)$ $\displaystyle=$
$\displaystyle\sum_{m\,\text{odd}}\left[V_{m}^{(2)}+\kappa\,\delta
V_{m}^{(2)}\right]\sum_{i<j}\hat{P}_{ij}(m)$ (1)
$\displaystyle+\sum_{m\geqslant
3}\kappa\,V_{m}^{(3)}\sum_{i<j<k}\hat{P}_{ijk}(m)\,,$

where $\hat{P}_{ij}(m)$ and $\hat{P}_{ijk}(m)$ are two- and three-body
projection operators respectively onto pairs or triplets of electrons with
relative angular momentum $m$. Here $V_{m}^{(2)}$ represents two-body Coulomb
pseudo-potential in the second Landau level and $\delta V_{m}^{(2)}$ is its
correction due to LLM and $V_{m}^{(3)}$ is three-body pseudo-potential arising
due to LLM whose strength is defined by
$\kappa=(e^{2}/\epsilon\ell_{0})/\hbar\omega_{c}$ that is the ratio between
Coulomb energy and cyclotron energy scales. We henceforth (unless mentioned
otherwise) perform exact diagonalization of $\hat{H}_{\text{eff}}$ with
pseudo-potentials pse  listed in Ref. Peterson and Nayak, 2013 for GaAs in
spherical geometry for the Pf, A-Pf, and PH-Pf shifts, i.e., the respective
number of flux quanta $N_{\phi}=2N-3$, $N_{\phi}=2N+1$ and $N_{\phi}=2N-1$
with the shifts from $2N$, for different values of $\kappa$. We determine
overlaps of the exact ground states when found at $L=0$ for different values
of $\kappa$, i.e., ${\cal
O}_{ij}=\langle\Psi_{\text{gs}}(\kappa_{i})|\Psi_{\text{gs}}(\kappa_{j})\rangle$.

Figure 2: (color online) Scaling of Gap for a pair of quasiparticle-quasihole
excitation at PH-Pf flux shift in the $\mathcal{A}$-phase for $\kappa=1.1$.
and 1.2 with $1/N$.

In Fig. 1(a)–(c), we show ${\cal O}_{ij}$ for Pf, A-Pf, and PH-Pf shifts
respectively. The color mapping clearly shows two distinct topological phases
separated by an unquantized phase in the vicinity of an intermediate value
$\kappa_{c}\sim 0.7$ for all the three shifts. The transition between two
quantum Hall phases is sharp (see supplemental material sup ) even for the
finite width of the quantum wells, although the unquantized zone shrinks to
zero at larger widths for finite size systems. Whereas ${\cal O}_{ij}$ is
nearly unity when both $\kappa_{i}$ and $\kappa_{j}$ belong to the same phase,
it is vanishingly small when $\kappa_{i}$ and $\kappa_{j}$ belong to two
different phases. We find (see supplemental material sup ) that the
$\mathcal{A}$-phase at moderate-$\kappa$ is present even in the absence of all
$V_{m}^{(3)}$ but the presence pse  of all $\delta V_{m}^{(2)}$. Although a
step-by-step addition of $V_{3}^{(3)}$, up to $V_{3}^{(8)}$, and finally up to
$V_{3}^{(9)}$ reduces the range of the phase, it gets sharpened and the range
reduction is compensated with the increase in $N$. Although the values of the
pseudopotentials are estimated Peterson and Nayak (2013); Rezayi (2017)
perturbatively with perturbation parameter $\kappa$, because the
$\mathcal{A}$-phase is robust against all these variations of
pseudopotentials, we believe that the phase will be restored even for improved
pseudopotentials at moderate-$\kappa$; the only quantitative change will be
expected in terms of the change in the range of the phase in $\kappa$-space.

We further calculate the energy for creating a pair of quasiparticle and
quasihole by taking an average Rezayi  _et al._ (2021) of $E_{\rm
qh}=E(N,2N)-E(N,2N-1)$ and $E_{\rm qp}=E(N,2N-2)-E(N,2N-1)$, i.e.,
$\Delta_{c}=(E_{\rm qh}+E_{\rm qp})/2$, where $E(N,N_{\Phi})$ denotes net
ground state energy (after subtraction of the background energy of
$N^{2}/\sqrt{2N_{\Phi}}$) of the system of $N$ electrons with $N_{\Phi}$
number of flux quanta. In Fig. 2, we show $\Delta_{c}$ at different values of
$N$ for $\kappa=1.1$ and $1.2$ belonging to $\mathcal{A}$-phase. Although each
of these $\Delta_{c}$ is negative, their scaling with $1/N$ provides its
thermodynamically extrapolated value $\Delta_{c}\sim
0.002$–$0.003\,e^{2}/(\epsilon\ell_{0})$ which amounts to $\sim
200$–$300\,\text{mK}$ (in the same ballpark of the charged gap found in the
experiments) for 5 Tesla magnetic field, where $\epsilon\simeq 13$ is the
dielectric constant of the host of the electron gas. The neutral excitations
are also gapped (see supplemental material sup ). Therefore the
$\mathcal{A}$-phase is incompressible and quantized.

Figure 3: (color online) (a) ES for PH-Pf flux shift with $N_{\phi}=27$ at a
$\kappa$ belonging to $\mathcal{A}$-phase. $L_{z}^{A}$ represents the sum of
the azimuthal components of angular momenta occupied by the particles in the A
part of the partition sup . An equal number of electrons ($N_{A}=N_{B}=7$) in
both partitions are considered for computing the corresponding ES. Here $\xi$
represents entanglement energy in an arbitrary unit. The entanglement gap is
shown by a line with an arrow-headed top and bottom, and the counting of edge
states is marked. Inset: Overlaps of the ground state of PH-Pf flux
($N_{\phi}=27$) at $\kappa=1.2$ with the ground states at Pf flux
($N_{\phi}=25$) and A-Pf flux ($N_{\phi}=29$) for different values of $\kappa$
in the range of $\mathcal{A}$-phase for $N=14$. The respective overlaps are
$\mathcal{O}_{\rm PH-Pf}^{\rm Pf}$ (circles) and $\mathcal{O}_{\rm PH-Pf}^{\rm
A-Pf}$ (triangles). (b) The ES for Pf flux shift with $N_{\phi}=25$,
$N_{A}=N_{B}=7$, and $\kappa=1.2$. (c) Same as (b) but for A-Pf flux shift
with $N_{\phi}=29$.

We show (Fig. 3) the ES Li and Haldane (2008); Chandran  _et al._ (2011);
Sterdyniak  _et al._ (2012) for $\mathcal{A}$-phase for all three flux shifts
(see supplemental material sup  for other phases). The low-lying ES for all
the three shifts are broadly similar (Fig. 3(a), (b) and (c)) in nature,
suggesting its unique topological order. It does not belong to any of the
conventional topological sectors, namely Pfaffian, anti-Pfaffian or particle-
hole symmetric Pfaffian. The Hilbert space for PH-Pf (Pf) flux is a subspace
of A-Pf (PH-Pf) flux (see supplemental material sup ) for a fixed $N$. The
similar ES occurs because the ground-state at PH-Pf flux which is in between
Pf and A-Pf fluxes for a fixed $N$ has counter-intuitively (see supplemental
material sup ) sizable overlap (inset Fig. 3(a)) with the same for the latter
two fluxes. The ES is gapped and it displays the counting of edge states as
1-1-2-2-$\cdots$.

We propose a trial ground state wave function for the $\mathcal{A}$-phase of
5/2 state in the spherical geometry as

$\displaystyle\Psi_{\mathcal{A}}(\\{u_{i},v_{i}\\})=\prod_{i<j}^{N}(u_{i}v_{j}-u_{j}v_{i})$
$\displaystyle\,\,\,\,\times{\cal S}\left[\prod_{1\leqslant k,l\leqslant
N/2}\left(u_{k}v_{N/2+l}-u_{N/2+l}v_{k}\right)^{2}\right]$ (2)

where $u_{j}=\cos(\theta_{j}/2)e^{i\phi_{j}/2}$ and
$v_{j}=\sin(\theta_{j}/2)e^{-i\phi_{j}/2}$ are the spherical spinors in terms
of spherical angles $0\leq\theta_{j}\leq\pi$ and $0\leq\phi_{j}\leq 2\pi$, and
${\cal S}$ represents symmetrization in particle indices. This wave function
corresponds to $N_{\phi}=2N-1$, i.e., PH-Pf flux shift a la the previously
proposed PH-Pf wave function Zucker and Feldman (2016), although the former
doesn’t have particle-hole symmetry. Owing to higher three-body interaction in
the $\mathcal{A}$-phase, the system loses the particle-hole symmetry anyway.
The wave function $\Psi_{\mathcal{A}}$ in Eq. (2) may be interpreted as two
separate condensates of two-flavored composite bosons (electrons attached with
one unit of flux quantum) with strong inter-flavored repulsive correlation. It
further signifies that the bosonic wave function (ignoring the ubiquitous
Jastrow factor required for Pauli exclusion principle for fermions) will not
vanish even if the macroscopic $N/2$ bosons coincide. As per other known wave
functions Moore and Read (1991); Read and Rezayi (1999) with the possibility
of coinciding two or more bosons supporting non-Abelian quasiparticles, the
wave function $\Psi_{\mathcal{A}}$ in Eq. (2) is likely to support non-Abelian
quasiparticles. The wave function in Eq. (2) may also be regarded as a fully
antisymmetrized 113 Halperin wave function Halperin (1983).

Figure 4: (color online) Overlaps of $\Psi_{\rm MR}$, $\Psi_{\rm MR}^{\rm
phc}$ (particle-hole conjugate of $\Psi_{\rm MR}$), and $\Psi_{\mathcal{A}}$
in Eq. (2) with the corresponding exact ground states for
$(N=14,\,N_{\phi}=25)$, $(N=12,\,N_{\phi}=25)$, and $(N=14,\,N_{\phi}=27)$
respectively versus $\kappa$ at zero quantum well width. The unconnected zones
refer to the unquantized regimes between two distinct topological quantized
phases for each case.

In Fig. 5, we show the overlap of $\Psi_{\rm MR}$, $\Psi_{\rm MR}^{\rm phc}$,
and $\Psi_{\mathcal{A}}$ with the corresponding exact ground states of
$\hat{H}_{\text{eff}}$ at their respective flux-shifts. While the former two
overlaps decrease with the increase of $\kappa$ and those are exceedingly low
at moderate-$\kappa$ regime, the latter has, in contrast, very high overlap
tab  in the latter regime which coincides with the experimental regime of
$\kappa=0.8$–1.8. We find, beyond this regime, another transition to an
unquantized phase. However, as pointed out in Ref. Pakrouski  _et al._ , 2015,
the perturbatively obtained $\hat{H}_{\text{eff}}$ (Eq. 1) may not be reliable
at high $\kappa$ and thus such transition may defer to higher $\kappa$.
Contrary to $\Psi_{\rm MR}$ and $\Psi_{\rm MR}^{\rm phc}$ wave functions, the
wave function $\Psi_{\mathcal{A}}$ in Eq. (2) with the exact ground states
even at Pf and A-Pf fluxes in the $\mathcal{A}$-phase has a sizable overlap
(see supplemental material sup ). Therefore, the $\mathcal{A}$-phase appears
as independent of the flux shifts and is well characterized by the wave
function $\Psi_{\mathcal{A}}$. Further, the low-lying ES corresponding to
$\Psi_{\mathcal{A}}$ nicely resembles (see Fig. 5) with the same for the exact
ground state. Therefore, $\Psi_{\mathcal{A}}$ seems representing similar
topological order as the exact ground state has for the 5/2
$\mathcal{A}$-phase.

Figure 5: (color online) ES for $N=10$ and $N_{\phi}=19$ for the exact ground
state (red dashed) and for $\Psi_{\mathcal{A}}$ (blue diamonds) at
$\kappa=0.8$ belonging to quantized $\mathcal{A}$-phase. Equal number of
particles ($N_{A}=5,\,N_{B}=5$) have been considered for both the partitions
for calculating the corresponding ES.

The topological properties of the $\mathcal{A}$-phase can be extracted by
exploiting the two-component structure of $\Psi_{\mathcal{A}}$. The
corresponding low-energy effective Lagrangian density Wen and Zee (1992) is
given by

$\mathcal{L}=-\frac{1}{4\pi}\epsilon^{\alpha\beta\gamma}\sum_{I,J=1}^{2}K_{IJ}a_{\alpha}^{I}\partial_{\beta}a_{\gamma}^{J}-\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}\sum_{I=1}^{2}t_{I}A_{\alpha}\partial_{\beta}a_{\gamma}^{I}$
(3)

with $K_{11}=K_{22}=1,\,K_{12}=K_{21}=3$ and $t_{1}=t_{2}=1$. Here
$a_{\alpha}^{1}$ and $a_{\alpha}^{2}$ represent two components of Chern-Simons
gauge fields, $A_{\alpha}$ is the external electromagnetic field, and
$\epsilon^{\alpha\beta\gamma}$ is the antisymmetric Levi-Cevita tensor.
Further introducing quasiparticle vector $l^{T}=(1,0)$ and spin vector
$s^{T}=(1/2,1/2)$ and following Ref. Wen and Zee, 1992, we find topological
properties such as filling factor $\nu=t^{T}K^{-1}t=1/2$, quasiparticle charge
$q=e\,l^{T}K^{-1}t=e/4$, topological shift $S=(2/\nu)t^{T}K^{-1}s=1$ relevant
for Hall viscosity, and the ground state degeneracy $D=|{\rm
Det}(K)|^{g}=8^{g}$ where $g$ is the genus of the geometry of the system. As
the two eigenvalues of $K$ are opposite in sign, there will be one downstream
charge mode with charge $q=e/4$ and one upstream neutral mode. The wave
function $\Psi_{\mathcal{A}}$ has got hidden $\mathbb{Z}_{2}$ symmetry because
the composite bosons are divided into two groups and up to $N/2$ composite
bosons can occupy the same position. A similar hidden $\mathbb{Z}_{2}$
symmetry is present Cappelli  _et al._ (2001) in the reformulated form of the
Read-Rezayi Read and Rezayi (1999) wave function for $5/2$ state. Because of
the $\mathbb{Z}_{2}$ symmetry, one downstream neutral Majorana mode will also
be present. Therefore, the net thermal Hall conductance becomes $2.5G_{0}$ as
the fully filled lowest Landau level will provide the contribution of
$2G_{0}$.

In this article, we show an anomalous topological 5/2 quantized fractional
quantum Hall phase at a moderate strength of the Landau-level-mixing that is
in the ballpark of the typical GaAs systems. We have proposed a wave function
which turns out to be excellent for describing the ground state of this phase.
Because the topological properties provided by this wave function are
consistent with the enigmatic 5/2 state, we attribute the experimentally
observed Banerjee  _et al._ (2018) $2.5$ unit of thermal Hall conductance (not
expected from the theoretical predictions for the conventional phases) in the
system with $\kappa\sim 1.1$ to this phase. However, this assertion will be
strengthened by the identification of appropriate conformal field operators
Moore and Read (1991); Hansson  _et al._ (2017) whose correlation can
determine a wave function with a topological order that should be
adiabatically connected to our proposed wave function and at the same time
explains Majorana mode along with other charge modes for the desired thermal
Hall conductance.

Our work opens an avenue for exploring further characteristics of the
$\mathcal{A}$-phase that is possibly the cause of the recent experimentally
observed Banerjee  _et al._ (2018); Dutta  _et al._ (2022) unusual value of
thermal Hall conductance at 5/2 quantum Hall state. The role of Landau-level-
mixing in the fractional quantum Hall states in graphene Lin  _et al._ (2014);
Li  _et al._ (2017) and ZnO-based systems Falson  _et al._ (2018); Luo  _et
al._ (2021), where it is stronger, should also be an interesting direction of
future work; whether or not the predicted $\mathcal{A}$-phase occurs in such
systems. Our results suggest a new approach for uncovering phases relevant in
the experimental domain for other fractional quantum Hall states in the second
Landau level, which are mostly enigmatic.

We thank Sutirtha Mukherjee for making useful comments on an early version of
the manuscript. We thank the developers of the DiagHam package for keeping it
open access. We acknowledge the Param Shakti (IIT Kharagpur) – a National
Supercomputing Mission, Government of India for providing computational
resources. S.S.M. is supported by Science and Engineering Research Board
(Government of India) through Grant No. MTR/2019/000546.

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Supplemental Material for “An Anomalous Reentrant 5/2 Quantum Hall Phase at
Moderate Landau-Level-Mixing Strength”

This supplemental material consists of six sections. In Section S1 , we
determine phase diagrams by calculating overlaps between exact ground states
at PH-Pf flux by varying LLM strength $\kappa$ for different widths of the
quantum wells. In Section S2, we obtain phase diagrams with step-by-step
inclusion of three-body pseudopotentials of different relative angular
momentum for zero width for confirming no adverse effect of any
pseudopotentials that may change our conclusion in the main paper. In Section
S3, we show the neutral excitation gap with the variation of $\kappa$ at zero
as well as finite widths of the quantum wells. In Section S4, we review the
procedure of calculating ES and entanglement entropy, and show ES for all the
three flux shifts for low-$\kappa$ phase and the subsequent unquantized phase.
In Section S5, we determine the overlap of $\Psi_{\mathcal{A}}$ with the exact
ground states at three different fluxes for a fixed $N$. In Section S6, we
show the overlap of $\Psi_{\mathcal{A}}$ in Eq. (2) with the exact ground
states at PH-Pf flux for different values of $\kappa$ and $N$.

FIG. S 1: (color online) (a) The overlap matrix (color map) of exact ground
states for different $\kappa$ of the effective Hamiltonian
$\hat{H}_{\text{eff}}$ in Eq. (1) at zero width, i.e. $w/\ell_{0}=0$ at PH-Pf
flux for $N=14$ particles. (b) Same as (a) but for $w/\ell_{0}=1$. (c) Same as
(a) but for $w/\ell_{0}=2$. (d) Same as (a) but for $w/\ell_{0}=3$. The
pseudopotentials used for this calculation is considered as tabulated in Ref.
11.

##  S1 Phase Diagrams for Finite Quantum Well Width

In Fig. S1(a)-(d), we show ${\cal O}_{ij}$ for the exact ground states of the
effective Hamiltonian, $\hat{H}_{\text{eff}}$ in Eq. (1) corresponding to
$N=14$ at PH-Pf flux considering finite width of $w/\ell_{0}=0,1,2$, and 3
respectively. The two-body and three-body (up to $V_{8}^{(3)}$)
pseudopotentials for the systems with finite widths have been taken from Ref.
11 for exact diagonalizations. The $\mathcal{A}$-phase appears at the
moderate-$\kappa$ and its transition from conventional low-$\kappa$ phase is
sharp even when the effects of finite width is considered. However, the regime
in which $\mathcal{A}$-phase appears shifts toward higher-$\kappa$ as width
increases. Also, the unquantized phase disappears for higher widths in finite
systems.

| $w/\ell_{0}$
---|---
$\quad\kappa\quad$ | 0 | 1 | 2 | 3
0 | 0.001031 | 0.003069 | 0.000106 | 0.000258
0.1 | 0.000650 | 0.003149 | 0.000194 | 0.000154
0.2 | 0.000884 | 0.003246 | 0.000299 | 0.000040
0.3 | 0.001009 | 0.003361 | 0.000423 | 0.000086
0.4 | 0.001159 | 0.003494 | 0.000569 | 0.000221
0.5 | 0.001442 | 0.003627 | 0.000735 | 0.000363
0.6 | 0.000570 | 0.003728 | 0.000912 | 0.000509
0.7 | - | 0.003767 | 0.001082 | 0.000654
0.8 | - | 0.002774 | 0.001214 | 0.000798
0.9 | 0.993464 | - | 0.001244 | 0.000940
1 | 0.967333 | 0.968478 | 0.001041 | 0.001085
1.1 | 0.946944 | 0.942558 | 0.851365 | 0.001252
1.2 | 0.933147 | 0.927919 | 0.90826 | 0.001524
1.3 | 0.924722 | 0.918359 | 0.905918 | 0.071239
1.4 | 0.918953 | 0.911577 | 0.901242 | 0.842247
1.5 | 0.912824 | 0.906495 | 0.897057 | 0.865946
1.6 | 0.908678 | 0.902535 | 0.893576 | 0.871898
1.7 | - | - | 0.890697 | 0.873432
1.8 | - | - | - | 0.873493
1.9 | - | - | - | -
TABLE S 1: Overlap of $\Psi_{\mathcal{A}}$ in Eq. (2) with the exact ground
state of Hamiltonian $\hat{H}_{\text{eff}}$ in Eq. (1) with variation of
$\kappa$ at different quantum well widths, $w/\ell_{0}$. Cells with dashed
mark are unquantized points.

In Table S1, the overlaps of $\Psi_{\mathcal{A}}$ with exact ground states of
the effective Hamiltonian, $\hat{H}_{\text{eff}}$ for widths
$w/\ell_{0}=0,1,2,3$ are shown. Even when the effect of finite width is
considered, $\Psi_{\mathcal{A}}$ still posses very high overlap with the exact
ground states in the $\mathcal{A}$-phase.

##  S2 Phase Diagrams for Different Effective Pseudo-Potentials

A detailed account of the effective Hamiltonian by incorporating the effect of
LLM consisting of two-body and three-body pseudo-potentials (up to
$V_{8}^{(3)}$) corrections are available in Ref. 11. The importance of three-
body pseudopotential $V_{9}^{(3)}=-0.0072\,e^{2}/(\epsilon\ell_{0})$ has
subsequently been shown [20] for switching over the topological order from Pf
to A-Pf in the region of low-$\kappa$.

FIG. S 2: (color online) (a) The overlap matrix (shown as color map) of the exact ground states for different $\kappa$ of the effective Hamiltonian, Eq. (S1) incorporating 2-body pseudo-potential corrections [11] only at PH-Pf flux for $N=10$ particles. The unquantized points are marked with red cross and the unquantized regime is grayed out. (b) Same as (a) but for effective Hamiltonian in Eq. (S2), $\hat{H}_{\text{eff}}^{\prime}(\kappa,3)$ i.e. 3-body $V_{3}^{(3)}$ correction [11] only along with 2-body corrections. (c) Same as (a) but for effective Hamiltonian in Eq. (S2), $\hat{H}_{\text{eff}}^{\prime}(\kappa,8)$ i.e. 3-body corrections upto $V_{8}^{(3)}$ [11] along with 2-body corrections. (d) Same as (a) but for effective Hamiltonian in Eq. (S2), $\hat{H}_{\text{eff}}^{\prime}(\kappa,9)$ i.e. 3-body corrections upto $V_{9}^{(3)}$ [11, 20] along with 2-body corrections.  $\quad\kappa\quad$ | $\langle\Psi_{\mathcal{A}}\mid\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{\rm 2b}(\kappa)}\rangle\quad$ | $\langle\Psi_{\mathcal{A}}+\Psi^{{\rm phc}}_{\mathcal{A}}\mid\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{\rm 2b}(\kappa)}\rangle\quad$ | $\langle\Psi_{\mathcal{A}}\mid\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,3)}\rangle\quad$ | $\langle\Psi_{\mathcal{A}}\mid\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,8)}\rangle\quad$ | $\langle\Psi_{\mathcal{A}}\mid\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,9)}\rangle\quad$
---|---|---|---|---|---
0 | - | - | - | - | -
0.1 | - | - | - | - | -
0.2 | - | - | - | - | -
0.3 | - | - | - | - | -
0.4 | - | - | - | - | -
0.5 | - | - | - | - | -
0.6 | - | - | - | - | -
0.7 | - | - | - | - | -
0.8 | - | - | 0.98490 | 0.99397 | 0.96590
0.9 | 0.70097 | 0.99132 | 0.98005 | 0.96463 | 0.96501
1 | 0.66771 | 0.94428 | 0.95079 | 0.94240 | 0.93139
1.1 | 0.63184 | 0.89356 | 0.92946 | 0.92629 | 0.90654
1.2 | 0.61315 | 0.86711 | 0.91315 | 0.91403 | 0.88753
1.3 | 0.60206 | 0.85144 | 0.90005 | 0.90429 | -
1.4 | 0.59478 | 0.84114 | 0.88920 | - | -
1.5 | 0.58963 | 0.83386 | - | - | -
1.6 | - | - | - | - | -
TABLE S 2: Overlap of $\Psi_{\mathcal{A}}$ with the exact ground states of
different effective Hamiltonians for $N=10$ particle system at PH-Pf flux:
$\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{2b}(\kappa)}$ for
$\hat{H}_{\text{eff}}^{2b}(\kappa)$, $\Psi_{\rm
gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,3)}$ for
$\hat{H}_{\text{eff}}^{\prime}(\kappa,3)$, $\Psi_{\rm
gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,8)}$ for
$\hat{H}_{\text{eff}}^{\prime}(\kappa,8)$, and $\Psi_{\rm
gs}^{\hat{H}_{\text{eff}}^{\prime}(\kappa,9)}$ for
$\hat{H}_{\text{eff}}^{\prime}(\kappa,9)$. In the third column, overlap of the
ground state $\Psi_{\rm gs}^{\hat{H}_{\text{eff}}^{2b}(\kappa)}$ and $1:1$
linear combination of $\Psi_{\mathcal{A}}$ and it’s particle-hole-conjugate,
$\Psi_{\mathcal{A}}^{{\rm phc}}$ is shown. The dashed marks indicate the exact
ground states are not found at total angular momentum $L=0$. The conventional
low-$\kappa$ phase is absent for $N=10$ at PH-Pf flux.

In Fig. S2, we systematically show the effect of two-body and three-body
pseudo-potential corrections to the phase diagram in ${\cal O}_{ij}$ for the
system of zero width. Incorporating only the two-body pseudo-potential
corrections [11] to effective Coulomb pseudopotentials for the second Landau
level i.e,

$\displaystyle\hat{H}_{\text{eff}}^{2b}(\kappa)$ $\displaystyle=$
$\displaystyle\sum_{m\,\text{odd}}\left[V_{m}^{(2)}+\kappa\,\delta
V_{m}^{(2)}\right]\sum_{i<j}\hat{P}_{ij}(m)\,,$ (S1)

the overlap matrix for the exact ground states with the variation of $\kappa$
at PH-Pf flux for $N=10$ particles is shown in Fig. S2(a). We note that there
is no low-$\kappa$ quantized phase for 10 particle system at PH-Pf flux.
Nevertheless, the $\mathcal{A}$-phase emerges at moderate-$\kappa$ just for
LLM two-body corrections. However, this phase will be particle-hole symmetric
as any three-body pseudopotentials have not been considered in this case. We
then add the three-body pseudo-potentials up to relative angular momentum
$m=M$ along with two-body pseudo-potentials as described in the Hamiltonian,

$\displaystyle\hat{H}_{\text{eff}}^{\prime}(\kappa,M)=\hat{H}_{\text{eff}}^{2b}(\kappa)+\sum_{m=3}^{M}\kappa\,V_{m}^{(3)}\sum_{i<j<k}\hat{P}_{ijk}(m)\qquad$
(S2)

In Fig. S2(b)–(c) we show the phase diagrams respectively for $M=3$, 8. The
addition of more and more three-body pseudopotentials make the
$\mathcal{A}$-phase sharper ($\mathcal{O}_{ij}$ is very close to unity for all
$\kappa_{i}$ and $\kappa_{j}$). Although the range of $\mathcal{A}$ phase
shrinks with more and more addition of pseudopotentials, it increases (Table
S4) with the increase in $N$.

As there are doubly degenerate quantum states for three-body relative angular
momentum $m=9$, namely, $|0,3\rangle$ and $|3,1\rangle$ where $m=2p+3q$ for a
state $|p,q\rangle$ [13, SLaughlin, 1983], the corresponding Hamiltonian is
given by [20],

$\displaystyle H|_{m=9}$ $\displaystyle=$ $\displaystyle-0.0088\mid
0,3\rangle\langle 0,3\mid+0.0033\mid 3,1\rangle\langle 3,1\mid\quad$ (S3)
$\displaystyle+0.0007\left[~{}\mid 0,3\rangle\langle 3,1\mid+\mid
3,1\rangle\langle 0,3\mid~{}\right]$

However, as pointed out in Ref. 20, Eq. (S3) can be well approximated by the
dominant first term only with effective coefficient $-0.0072$, i.e.,
$V_{9}^{(3)}=-0.0072\,e^{2}/(\epsilon\ell_{0})$. We then obtain phase diagram
(Fig. S2(d)) by diagonalizing $\hat{H}_{\text{eff}}^{\prime}(\kappa,9)$ in Eq.
(S2). As per Fig. S2, the $\mathcal{A}$-phase is robust with respect to the
degree of pseudo-potential corrections. Although the pseudo-potentials that
have been used for obtaining phase diagrams in Fig. 1 of the main paper are
obtained in Ref. 11 perturbatively which also possess good agreement with the
other studies [12, 13], this robustness against change in pseudopotentials
indicates that the $\mathcal{A}$-phase is inevitable even if a better
estimation of pseudopotentials is used in the moderate regime of $\kappa\sim
1.1$.

In Table S2, we show the overlap of $\Psi_{\mathcal{A}}$ in Eq. (2) with the
exact ground states for different Hamiltonian,
$\hat{H}_{\text{eff}}^{2b}(\kappa),\hat{H}_{\text{eff}}^{\prime}(\kappa,3),\hat{H}_{\text{eff}}^{\prime}(\kappa,8)$,
and $\hat{H}_{\text{eff}}^{\prime}(\kappa,9)$ for $N=10$ particles. The
overlaps for all the three Hamiltonian for which certain three-body potentials
are considered are very high. In case of two-body only
$\hat{H}_{\text{eff}}^{2b}(\kappa)$, the overlap is moderate, but that is
expected because $\Psi_{\mathcal{A}}$ is not particle-hole symmetric whereas
the exact ground state for $\hat{H}_{\text{eff}}^{2b}(\kappa)$, $\Psi_{\rm
gs}^{\hat{H}_{\text{eff}}^{2b}(\kappa)}$ is particle-hole symmetric. However,
$1:1$ linear combination of $\Psi_{\mathcal{A}}$ with its particle-hole
conjugate $\Psi_{\mathcal{A}}^{\rm phc}$ provides high overlap (Table S2) with
the corresponding exact ground state.

##  S3 Neutral Excitation Gap

The neutral excitation gap $\Delta$, the energy difference of the minimum
energy state at total angular momentum $L=0$ and the minimum energy found
amongst all other $L$, is shown in Fig. S3.

FIG. S 3: (color online) (a) The neutral excitation gap at $N=14$ and
$N_{\phi}=25$ for quantum well widths $w=0$, 1, 2, and 3 in the unit of
$\ell_{0}$. (b) Same as (a) but for A-Pf shift with $N=12$ and $N_{\phi}=25$.
(c) Same as (a) but for PH-Pf shift with $N=14$ and $N_{\phi}=27$.

As $\kappa$ increases, $\Delta$ decreases for all three fluxes and in the
$\mathcal{A}$-phase saturates to nearly $10^{-3}~{}e^{2}/(\epsilon\ell_{0})$.
The neutral excitation gap, $\Delta$ remains finite in the $\mathcal{A}$-phase
when the effect of finite width [11] is also included. Although when the
finite width is considered, the states don’t go through an unquantized phase
(ground state at $L\neq 0$ thus negative $\Delta$) which might be restored
through further improvements of the LLM pseudo-potential corrections; however,
the occurrence of the $\mathcal{A}$-phase can not be hindered through such
small perturbations.

##  S4 Entanglement entropy and entanglement spectra

The topological properties of a system can be unfolded with the study of
quantum entanglement. Usually, the entanglement of a quantum state is
quantified by the entanglement entropy (EE) [SZozulya  _et al._ , 2007, SHaque
_et al._ , 2007], called the von Neumann entropy. However, by the advent of a
method of finding entanglement spectra (ES) of an FQHE state by Li and Haldane
[43] and later by others [44, 45], one obtains more topological information of
a quantum state in comparison to EE which is an integrated number. To get the
information about the entanglement of the state through EE and ES, the whole
system is partitioned into two blocks, and thus the Fock space of the full
Hamiltonian $\mathcal{H}$ is divided into two parts:
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\mathcal{H}_{A}$
and $\mathcal{H}_{B}$ respectively represent the Hamiltonian of parts A and B.
Here we adopt the method in Ref. 44 by making an orbital partition for the
particles on a sphere whose center possesses magnetic monopole charge $Q$ for
generating $2Q$ number of magnetic flux.

FIG. S 4: (a) ES for Pf flux shift with $N_{\phi}=25$ at a $\kappa$
belonging to low-$\kappa$ phase. $L^{A}_{z}$ represents the sum of the
azimuthal components of angular momenta occupied by the particles in A part of
the partition. Equal number of electrons ($N_{A}=N_{B}=7$) in both the
partitions are considered for computing the corresponding ES. Here $\xi$
represents entanglement energy in an arbitrary unit. (b) Same as (a) but for
A-Pf flux shift with $N_{\phi}=29$. (c) Same as (a) but for PH-Pf flux shift
with $N_{\phi}=27$. (d) ES for Pf flux shift ($N_{\phi}=25$) at a $\kappa$
belonging to unquantized phase. (e) Same as (d) but for A-Pf flux shift
($N_{\phi}=29$). (f) Same as (d) but for PH-Pf flux shift ($N_{\phi}=27$). (g)
Entanglement entropy versus $\kappa$ for Pf flux shift ($N_{\phi}=25$). (h)
Entanglement entropy versus $\kappa$ for A-Pf flux shift ($N_{\phi}=29$). (i)
Entanglement entropy versus $\kappa$ for PH-Pf flux shift ($N_{\phi}=27$). The
shaded regions in (g)-(i) denote the A-phase.

Here the set of single-particle orbitals $\lambda=\\{-Q,-Q+1,...,Q-1,Q\\}$ is
divided into two disjoint sets $\mu=\\{-Q,...,Q-l_{A}-1\\}$ and
$\nu=\\{Q-l_{A},...Q-1,Q\\}$ respectively for block-B and block-A. If there
are $N_{A}$ and $N_{B}$ number of electrons in the block-A and -B
respectively, one finds the corresponding basis states $|\nu_{i}\rangle$ and
$|\mu_{i}\rangle$ while the basis states for the whole system are denoted by
$|\lambda_{i}\rangle$. Therefore, the ground state (or the trial state) may be
expressed as

$|\psi\rangle=\sum_{\\{\lambda_{i}\\}}c_{\lambda_{i}}|\lambda_{i}\rangle=\sum_{\\{\mu_{i}\\},\\{\nu_{j}\\}}(O_{f})_{ij}\text{
}|\mu_{i}\rangle\otimes|\nu_{j}\rangle\,,$ (S4)

where the component of entanglement matrix $O_{f}$ is the coefficient of
$|\mu_{i},\nu_{j}\rangle$ basis state of $\psi$, i.e.,
$(O_{f})_{ij}=c_{(\mu_{i},\nu_{j})}$. This $O_{f}$ matrix is block diagonal
where each block is labeled by $N_{A}$ and
$L_{z}^{A}=\sum_{i=1}^{N_{A}}\nu_{i}$.

While the reduced density matrix for the A-block may be expressed as
$\rho_{A}=O_{f}O_{f}^{\dagger}$, it is $\rho_{B}=O_{f}^{\dagger}O_{f}$ for the
block-B. Consequently, EE of the state $|\psi\rangle$ is given by
$S=S_{A}=-\text{Tr}[\rho_{A}\text{ ln}\rho_{A}]=-\text{Tr}[\rho_{B}\text{
ln}\rho_{B}]=S_{B}$.

For a given $N_{A}$ and $L_{z}^{A}$, the singular value decomposition of the
matrix $O_{f}$ will give rise to

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{\\{\mu_{i}\\},\\{\nu_{j}\\}}(O_{f})_{ij}\text{
}|\mu_{i}\rangle\otimes|\nu_{j}\rangle$ (S5) $\displaystyle=$
$\displaystyle\sum_{i=1}^{\text{Rank}(O_{f})}e^{-\xi_{i}/2}|\psi_{B}^{i}\rangle\otimes|\psi^{i}_{A}\rangle\,,$

where $e^{-(1/2)\xi_{i}}\geq 0,\quad|\psi_{A}^{i}\rangle\in A\text{ block
},\quad|\psi^{i}_{B}\rangle\in B\text{ block }$ and
$\langle\psi_{A}^{i}|\psi_{A}^{j}\rangle=\langle\psi_{B}^{i}|\psi_{B}^{j}\rangle=\delta_{ij}$.
$\xi_{i}$ and $|\psi^{i}_{A}\rangle$ (or $|\psi^{i}_{B}\rangle$) respectively
are the eigenvalues and the eigenstates of the reduced density matrix
$\rho_{A}$ (or $\rho_{B}$) of the considered block. If we construct a
fictitious Hamiltonian $H_{A}$ using reduced density matrix $\rho_{A}$ such
that $H_{A}=-\text{log }\rho_{A}$, then, $\xi_{i}$ and $|\psi^{i}_{A}\rangle$
are the eigenvalues and eigenstates of this Hamiltonian. In the calculation of
this ES with eigen values $\xi_{i}$ for orbital partitioning, we have
considered the orbitals in two blocks given in sets $\mu$ and $\nu$ with
$l_{A}=(2Q+1)/2$, and equal number of particles $(N_{A}=N/2=N_{B})$ in both
the blocks.

In Fig. S4(a)-(c) ES is shown at Pf flux shift of $N_{\phi}=25$, A-Pf flux
shift of $N_{\phi}=29$, and PH-Pf flux shift of $N_{\phi}=27$ respectively at
the Coulomb limit i.e. $\kappa=0.0$ and well gapped ES can be seen for all
those cases as expected. But there was no gap observed in the ES in the
unquantized regime of $\kappa=0.7$ for the Pf, A-Pf, and PH-Pf flux shifts,
Fig. S4(d)-(f). The Entanglement entropy with the variation of $\kappa$ for
the corresponding Pf, A-Pf, and PH-Pf flux shifts are shown in Fig. S4(g)-(i).
A peak is observed in the entanglement entropy when the system undergoes the
phase transition through the unquantized regime at around $\kappa\simeq 0.7$
and again attains a stable value in the $\mathcal{A}$-phase.

##  S5 Overlap of $\Psi_{\mathcal{A}}$ with the exact ground states for three
different flux-shifts

For a fractional quantum Hall system in spherical geometry with a given number
of flux quanta $N_{\phi}=2Q$ where $Q$ is the magnetic monopole charge placed
at the center of the sphere, the $z$-component of orbital angular momentum is
a good quantum number and the allowed values (in the lowest Landau level) are,

$l_{z}=-Q,-(Q-1),...,Q-1,Q$ (S6)

with degeneracy $2Q+1$. $N<2Q+1$ electrons can occupy such quantum states in
several possible ways with the criteria that the sum of the occupied $l_{z}$
must be zero $(\sum_{i=1}^{N}l^{i}_{z}=0)$ if we fix the $z$-component of
total angular momentum, $L_{z}$ to be zero. We consider $L_{z}=0$ so that we
do not lose any angular momentum $L$. Now, for another system with
$Q^{\prime}>Q$, such that $Q^{\prime}=Q+q$ ($q$ integer), $z$-components of
angular momenta span as,

$l_{z}^{\prime}=-Q^{\prime},-(Q^{\prime}-1),....,Q^{\prime}-1,Q^{\prime}$ (S7)

Therefore, $N$-particle Hilbert space for $Q$ is a subspace of the same for
$Q^{\prime}$. As an example, for $N=4$ particles the single-particle eigen
basis at Pf flux are, {$-$2.5, $-$1.5, 1.5, 2.5}, {$-$2.5, $-$0.5, 0.5, 2.5},
{$-$1.5, $-$0.5, 0.5, 1.5} and at PH-Pf flux are, {$-$3.5, $-$2.5, 2.5, 3.5},
{$-$3.5, $-$1.5, 1.5, 3.5}, {$-$3.5, $-$0.5, 0.5, 3.5}, {$-$2.5, $-$1.5, 0.5,
3.5}, {$-$3.5, $-$0.5, 1.5, 2.5}, {$-$2.5, $-$1.5, 1.5, 2.5}, {$-$2.5, $-$0.5,
0.5, 2.5}, {$-$1.5, $-$0.5, 0.5, 1.5}. Clearly, the Hilbert space for the Pf
shift is the subspace of the Hilbert space of the PH-Pf shift, and similarly
it can be shown that the Hilbert space for the PH-Pf shift is the subspace of
the Hilbert space of the A-Pf shift. Although our proposed wave function in
Eq. (2) corresponds to the PH-Pf flux shift, our hunch is that it will possess
sizable overlap in the other two Hilbert spaces of the Pf and A-Pf as well, as
the ES in the $\mathcal{A}$-phase is independent of these shifts. We calculate
the overlap of $\Psi_{\mathcal{A}}$ in Eq. (2) for 14 particles with the exact
ground state at the above three flux shifts for different values of $\kappa$
using the Monte Carlo method. In the low-$\kappa$ phase which is identified as
the Pf or A-Pf phase, the overlap, as expected, is negligibly small (Fig. S5)
for all the three flux shifts. On the other hand, $\Psi_{\cal A}$ in Eq. (2)
not only has a high overlap (Fig. S5) in the $\mathcal{A}$-phase with the PH-
Pf shift, it has an appreciable ($\sim 40-50\%$) overlap for the Pf and A-Pf
shifts as well. This is quite unusual because $\Psi_{\mathcal{A}}$ corresponds
to a flux shift that neither agrees with the Pf nor A-Pf shifts, although they
share some part of the Hilbert space. The significant overlaps between two
ground states at different flux shifts occur because the corresponding weight
factors of Fock-space are correlated in sign for the common Hilbert sub-space.

FIG. S 5: (color online) Overlap of $\Psi_{\mathcal{A}}$ in Eq. (2) shown
for $N=14$ particles with the exact ground states at Pf flux$=25$, A-Pf
flux$=29$, and PH-Pf flux$=27$ with varying Landau level mixing strength,
$\kappa$.

In contrary to the above findings, the overlap of the ground states at two
different filling factors in the lowest Landau level at two different flux
shifts but with same number of electrons is extremely low as shown in Table
S3, despite the sharing of common single-particle basis states as described
above. As these two quantum Hall states are of different topologies, the
weight factors of the common Hilbert subspace have uncorrelated signs and thus
producing low overlap between the two states.

$\quad N\quad$ | $\nu,N_{\phi}$ | $\nu^{\prime},N_{\phi}^{\prime}$ | $\langle\Psi_{\text{gs}}^{(\nu,N_{\phi})}\mid\Psi_{\text{gs}}^{(\nu^{\prime},N_{\phi}^{\prime})}\rangle$
---|---|---|---
8 | 2/5, 16 | 2/3, 12 | 0.060856
10 | 2/5, 21 | 2/3, 15 | 0.002706
10 | 1/3, 27 | 2/5, 21 | 0.000213
TABLE S 3: Overlap of exact ground states of different fluxes but same number
of particles in the lowest Landau level.

##  S6 Overlap of $\Psi_{\mathcal{A}}$ with the variation of $\kappa$ at PH-Pf
flux

The overlaps of $\Psi_{\mathcal{A}}$ in Eq. (2) with the exact ground state
corresponding to PH-Pf flux by varying $\kappa$ for different number of
particle systems are tabulated in Table S4. The range of ${\cal A}$-phase
appears to be increasing with $N$.

| $N$
---|---
$\quad\kappa\quad$ | 8 | 10 | 12 | 14 | 16
0.0 | 0.09797 | - | 0.0004(1) | 0.0010(2) | -
0.1 | 0.10267 | - | 0.0004(1) | 0.0006(3) | -
0.2 | 0.10929 | - | 0.0005(1) | 0.0009(3) | -
0.3 | 0.11933 | - | 0.0006(3) | 0.0010(3) | -
0.4 | 0.13652 | - | 0.0007(2) | 0.0012(5) | -
0.5 | 0.17442 | - | 0.0009(2) | 0.0014(6) | -
0.6 | - | - | 0.0015(2) | 0.0006(2) | 0.003(1)
0.7 | 0.95379 | - | 0.0028(2) | - | -
0.8 | 0.99029 | 0.99397 | 0.9761(1) | - | -
0.9 | 0.96157 | 0.96463 | 0.9778(1) | 0.9935(0) | 0.37(1)
1.0 | 0.93345 | 0.94240 | 0.9517(1) | 0.9673(3) | 0.989(0)
1.1 | 0.90923 | 0.92629 | 0.9349(2) | 0.9470(2) | 0.966(1)
1.2 | - | 0.91403 | 0.9232(2) | 0.9331(2) | 0.950(1)
1.3 | - | 0.90429 | 0.9147(2) | 0.9247(4) | 0.939(1)
1.4 | - | - | 0.9082(3) | 0.9190(5) | 0.932(1)
1.5 | - | - | - | 0.9128(2) | 0.926(1)
1.6 | - | - | - | 0.9087(5) | 0.922(1)
1.7 | - | - | - | - | 0.919(1)
1.8 | - | - | - | - | -
1.9 | - | - | - | - | -
TABLE S 4:  Overlaps of $\Psi_{\mathcal{A}}$ in Eq. (2) with the exact ground
states of $\hat{H}_{\text{eff}}$ in Eq. (1) at PH-Pf shift for different
values of $\kappa$ and for $N=8$, 10, 12, 14 and 16 electrons. The dashed mark
shells represent unquantized phase between two quantized phases: low-$\kappa$
conventional phase and the ${\cal A}$-phase at moderate-$\kappa$. There is no
low-$\kappa$ phase at PH-Pf flux for $N=10$ and 16. The numbers within
parentheses indicate uncertainty in the Monte Carlo evaluation. For $N=8$ and
$10$ we get the exact overlap value through decomposition [SDas  _et al._ ,
2021] into single particle eigen basis.

## References

  * Laughlin (1983) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * Zozulya  _et al._ (2007) O. S. Zozulya, M. Haque, K. Schoutens, and E. H. Rezayi, Phys. Rev. B 76, 125310 (2007).
  * Haque  _et al._ (2007) M. Haque, O. Zozulya, and K. Schoutens, Phys. Rev. Lett. 98, 060401 (2007).
  * Das  _et al._ (2021) S. Das, S. Das, and S. S. Mandal, Phys. Rev. B 103, 075304 (2021).


# Self-consistent equation for torsion arising as a consequence of the Dirac
sea quantum fluctuations in external classical electromagnetic and
gravitational fields

S.N<EMAIL_ADDRESS>Landau Institute for Theoretical
Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432
Russia and Moscow Institute of Physics and Technology, Department of
Theoretical Physics, Dolgoprudnyj, Moskow region, 141707 Russia

###### Abstract

The quantum fluctuations of the Dirac field in external classical
gravitational and electromagnetic fields are studied. A self-consistent
equation for torsion is calculated, which is obtained using one-loop fermion
diagrams.

###### pacs:

04.62.+v

##  I Introduction

In this paper, we study the theory of gravity, minimally related to the
massive Dirac field, formulated in the form of Cartan-Palatini. The Dirac
field is considered to be quantized, while the gravitational (and
electromagnetic) fields are classical. Since the gauge fields are classical,
the mean values of bilinear forms constructed using Dirac fields are exhausted
by one-loop diagrams. The torsion tensor appears to be one of such mean value.
This tensor appears as the fermion part of one of the equations of motion (see
the next Section, Eq. (5)):

$\displaystyle e^{\mu}_{b}e^{\nu}_{c}\left({\cal D}_{\mu}e^{a}_{\nu}-{\cal
D}_{\nu}e^{a}_{\mu}\right)=i\frac{l_{P}^{2}}{4}\overline{\Psi}(\gamma^{a}\sigma_{bc}+\sigma_{bc}\gamma^{a})\Psi$
(1)

By definition, the left-hand side of the last equation is the torsion tensor
$\mathop{{\mathfrak{T}}}\nolimits_{bc}^{a}$.

In this work, the mean value of the right-hand side of Eq. (1) is calculated
explicitly. Only divergent contributions (in momentum space) are taken into
account. Therefore, the result can depend only on local geometric quantities
(curvature and torsion tensors and their covariant derivatives). As a result
of these calculations, the right-hand side of Eq. (1) turns out to be a local
function of the curvature and torsion tensors and their covariant derivatives.
If we assume that the curvature tensor is given, then in this way a self-
consistent equation for the torsion tensor arises. This equation is one of the
equations of motion in the studied model.

The self-consistent equation for torsion tensor (33) may be interesting for
the following reasons. Even a superficial study of this equation shows that in
the present cosmological epoch the torsion tensor generated by the Dirac field
must be zero. This fact is consistent with experimental data. However,
conditions near the Big Bang may occur when the torsion tensor turns out to be
nonzero. To study the cosmological consequences arising from this, separate
studies are needed.

In this paper, the question of the contribution to the right-hand side of Eq.
(1) from the Weyl fermion fields (neutrinos) remained open. This contribution
must be calculated since Weyl fermions exist. However, the corresponding
computational procedure must be somewhat modified, since the Weyl fields are
either massless or their masses are extremely small.

There is a good reason to consider gravitational fields as classical. The
quantum theory of gravity is a non-renormalizable theory. In particular, this
means that quantum fluctuations of gravitational dynamical variables (tetrads
and connections) are large on ultra-small scales of the order of the Planck
length Vergeles (2021). But on the scales much larger than the Planck scale,
these fluctuations decrease rapidly (according to a power law). Therefore, we
will assume that when considering physics on scales that are much larger than
Planck’s, fluctuations of the gravitational degrees of freedom are
insignificant, that is, these degrees of freedom are described by classical
fields. That’s why the gravitational fields $e^{a}_{\mu}$ and
$\omega^{ab}_{\mu}$ are considered as classical. However, the quantum
fluctuations of the Dirac field are important for wavelengths shorter than the
Compton wavelength of the Dirac particle.

Recently, the problems of the dynamics of the Dirac and Weyl spin current in
gravitational and electromagnetic fields have been actively studied. For
example, in the work de Camargo et al. (2021) the master equations governing
the interaction between gravitational fields and gauge invariant spin-current
is derived. Note that the canonical spin current (7) is proportional to the
torsion tensor, and therefore such a spin current in an external
electromagnetic field is considered here. In the paper Chu and Leung (2021) a
variant of the spin current constructed with the help of the spin operator in
the Pauli-Lubanski form is studied. In this paper, it is shown that the vacuum
expectation value of such a spin in an external electromagnetic field is
nonzero. It is shown below that the same vacuum expectation value of the spin
current (7) is zero. We refer the reader for the interesting results and a
significant number of citations on this topic to the review Cranganore (2021).
The one-loop calculation of the effective action arising due to integration
over the Dirac field in the same model (2) is contained in Nascimento et al.
(2022).

We also point to the review Hehl et al. (1976), entitled ”General relativity
with spin and torsion”, which may be useful in getting acquainted with the
problem under study here.

This article is structured as follows. Section II defines the model under
study and introduces notation. Section III presents the results of
calculations and writes out a self-consistent equation for the torsion tensor.
Then comes the Conclusion.

##  II Definition of the model and technical means

Let us write out the action of gravity coupled with a massive Dirac field in
external classical electromagnetic and gravitational fields:

$\displaystyle{\mathop{{\mathfrak{A}}}\nolimits}={\mathop{{\mathfrak{A}}}\nolimits}_{g}+{\mathop{{\mathfrak{A}}}\nolimits}_{\Psi},$
$\displaystyle{\mathop{{\mathfrak{A}}}\nolimits}_{g}=-\frac{1}{4l_{P}^{2}}\varepsilon_{abcd}\varepsilon^{\mu\nu\lambda\rho}\int\mathop{{\mathfrak{R}}}\nolimits^{ab}_{\mu\nu}e^{c}_{\lambda}e^{d}_{\rho}\mathop{\rm
d}\nolimits^{4}x,$
$\displaystyle\mathop{{\mathfrak{R}}}\nolimits^{ab}_{\mu\nu}=\partial_{\mu}\omega^{ab}_{\nu}-\partial_{\nu}\omega^{ab}_{\mu}+\omega^{a}_{c\mu}\omega^{cb}_{\nu}-\omega^{a}_{c\nu}\omega^{cb}_{\mu}=-\mathop{{\mathfrak{R}}}\nolimits^{ba}_{\mu\nu},$
$\displaystyle{\mathop{{\mathfrak{A}}}\nolimits}_{\Psi}=\int\mathop{\rm
d}\nolimits^{4}x\sqrt{-g}\left\\{\frac{i}{2}\tilde{e}^{\mu}_{a}\left(\overline{\Psi}\gamma^{a}{\cal
D}_{\mu}\Psi-\overline{{\cal
D}_{\mu}\Psi}\gamma^{a}\Psi\right)-m\overline{\Psi}\Psi\right\\}.$ (2)

Here

$\displaystyle{\cal
D}_{\mu}\Psi=\left(\partial_{\mu}-ieA_{\mu}+\omega_{\mu}\right)\Psi,$
$\displaystyle\omega_{\mu}\equiv\frac{1}{2}\omega_{ab\mu}\sigma^{ab},\quad\sigma^{ab}=\frac{1}{4}[\gamma^{a},\gamma^{b}],$
(3)

The vector fields $\\{\tilde{e}^{\mu}_{a}\\},\,a=0,1,2,3$ form an local
orthonormal basis, so that

$\displaystyle
g_{\mu\nu}\tilde{e}^{\mu}_{a}\tilde{e}^{\nu}_{b}=\eta_{ab}=\mathop{\rm
diag}\nolimits(1,-1,-1,-1),$ (4)

and $\tilde{e}^{\mu}_{a}e^{b}_{\mu}=\delta^{b}_{a}$. Levi-Civita symbols are
equal to units if their indices are ordered as $(0123)$.

We regard the lattice theory of gravity Vergeles (2021) as a regularization of
the continuum theory (2). Thus, divergent integrals in the calculation of
fermionic loops should be cut off at lattice (or Planck) scales.

Equation $\delta{\mathop{{\mathfrak{A}}}\nolimits}/\delta\omega^{ab}_{\mu}=0$
gives the definition of torsion:

$\displaystyle\mathop{{\mathfrak{T}}}\nolimits^{a}_{\mu\nu}\equiv{\cal
D}_{\mu}e^{a}_{\nu}-{\cal
D}_{\nu}e^{a}_{\mu}=\frac{il^{2}_{P}}{4}\overline{\Psi}\left(\gamma^{a}\sigma_{bc}+\sigma_{bc}\gamma^{a}\right)\Psi
e^{b}_{\mu}e^{c}_{\nu}.$ (5)

Or, equivalently

$\displaystyle\mathop{{\mathfrak{T}}}\nolimits^{a}_{bc}=\frac{l^{2}_{P}}{4}\varepsilon^{a}_{\,\,\,bcd}\overline{\Psi}\gamma^{5}\gamma^{d}\Psi.$
(6)

In the process of transforming the right side of Eq. (5) we used the equality

$\displaystyle\gamma_{a}\sigma_{bc}+\sigma_{bc}\gamma_{a}=-i\varepsilon_{abcd}\gamma^{5}\gamma^{d}.$

Variation of action ${\mathop{{\mathfrak{A}}}\nolimits}_{\Psi}$ with respect
to connection $\omega_{\mu}^{bc}$ determines the expression for the Dirac spin
current ${\mathop{{\mathfrak{S}}}\nolimits}^{a}_{bc}$:

$\displaystyle\delta_{\omega}{\mathop{{\mathfrak{A}}}\nolimits}_{\Psi}\equiv-\frac{1}{2}\int\mathop{\rm
d}\nolimits^{4}x\sqrt{-g}{\mathop{{\mathfrak{S}}}\nolimits}^{\mu}_{bc}\delta\omega_{\mu}^{bc},$
$\displaystyle{\mathop{{\mathfrak{S}}}\nolimits}^{a}_{bc}=-\frac{1}{2}\varepsilon^{a}_{\,\,\,bcd}\overline{\Psi}\gamma^{5}\gamma^{d}\Psi.$
(7)

Comparing (6) and (7), we find:

$\displaystyle\mathop{{\mathfrak{T}}}\nolimits^{a}_{bc}=-\frac{l^{2}_{P}}{2}\mathop{{\mathfrak{S}}}\nolimits^{a}_{bc}.$
(8)

Thus, the torsion tensor and the Dirac spin current are proportional in the
considered model.

Hereinafter, $\langle...\rangle$ means averaging over the fermionic vacuum at
fixed external classical electromagnetic and gravitational fields.

The Dirac causal propagator is usually denoted as $S_{c}(x,y)$. However, in
order to avoid misunderstandings and confusion with the lower Latin indices of
numerous tensors that are present in the formulas along with the causal
propagator, we will everywhere denote the causal propagator without an index:
$S(x,y)$. Thus for $y^{0}=x^{0}+0$ we have (see Eq. (6))

$\displaystyle\big{\langle}\overline{\Psi}_{\beta}(x+0)\Psi_{\alpha}(x)\big{\rangle}=-iS(x,x+0)_{\alpha\beta},$
$\displaystyle\mathop{{\mathfrak{T}}}\nolimits_{abc}=-\frac{il^{2}_{P}}{4}\varepsilon_{abcd}\mathop{\rm
tr}\nolimits\gamma^{5}\gamma^{d}S(x,x+0).$ (9)

Since geometric quantities are assumed to be classical, we always assume
$\mathop{{\mathfrak{T}}}\nolimits_{abc}=\langle\mathop{{\mathfrak{T}}}\nolimits_{abc}\rangle$.

Further, to simplify calculations, we use normal coordinates $x^{\mu}$
centered at point $p$, so that $x^{\mu}(p)=0$ and in the vicinity of this
point

$\displaystyle
e^{a}_{\mu}(x)=\delta^{a}_{\mu}+\frac{1}{2}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\nu\mu}(p)x^{\nu}$
$\displaystyle+\left(\frac{1}{6}{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\nu\lambda\mu}+\frac{1}{6}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\nu
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\lambda\mu}+\frac{1}{3}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\nu\mu;\lambda}\right)(p)x^{\nu}x^{\lambda}$
$\displaystyle+\left(\frac{1}{24}\left[{\mathop{{\mathfrak{R}}}\nolimits}^{c}_{\nu\lambda\mu}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\rho
c}+{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\rho\nu
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\lambda\mu}+{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\rho
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\nu
d}{\mathop{{\mathfrak{T}}}\nolimits}^{d}_{\lambda\mu}\right]+\frac{1}{12}{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\rho\nu\mu;\lambda}\right.$
$\displaystyle\left.+\frac{1}{12}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\rho
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\nu\mu;\lambda}+\frac{1}{8}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\rho
c;\lambda}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\nu\mu}+\frac{1}{8}{\mathop{{\mathfrak{T}}}\nolimits}^{a}_{\rho\mu;\nu;\lambda}\right)(p)x^{\nu}x^{\lambda}x^{\rho},$
(10)
$\displaystyle\omega_{ab\mu}(x)=\frac{1}{2}\mathop{{\mathfrak{R}}}\nolimits_{ab\nu\mu}(p)x^{\nu}+\left(\frac{1}{6}\mathop{{\mathfrak{R}}}\nolimits_{ab\nu
f}\mathop{{\mathfrak{T}}}\nolimits^{f}_{\lambda\mu}+\frac{1}{3}\mathop{{\mathfrak{R}}}\nolimits_{ab\nu\mu;\lambda}\right)(p)x^{\nu}x^{\lambda}.$
(11)

Using the formula (10) we get:

$\displaystyle\tilde{e}^{\mu}_{a}(x)=\delta^{\mu}_{a}+\delta\tilde{e}^{\mu}_{a}(x),$
$\displaystyle\delta\tilde{e}^{\mu}_{a}(x)=-\frac{1}{2}\mathop{{\mathfrak{T}}}\nolimits^{\mu}_{\nu
a}(p)x^{\nu}+{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda
a}x^{\nu}x^{\lambda}+{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda\rho
a}x^{\nu}x^{\lambda}x^{\rho},$
$\displaystyle{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda
a}=\left(-\frac{1}{6}\mathop{{\mathfrak{R}}}\nolimits^{\mu}_{\nu\lambda
a}+\frac{1}{12}\mathop{{\mathfrak{T}}}\nolimits^{\mu}_{\nu
c}\mathop{{\mathfrak{T}}}\nolimits^{c}_{\lambda
a}-\frac{1}{3}\mathop{{\mathfrak{T}}}\nolimits^{\mu}_{\nu
a;\lambda}\right)(p),$
$\displaystyle{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda\rho
a}=\left(\frac{1}{24}\mathop{{\mathfrak{R}}}\nolimits^{c}_{\nu\lambda
a}{\mathop{{\mathfrak{T}}}\nolimits}^{\mu}_{\rho
c}+\frac{1}{24}\mathop{{\mathfrak{R}}}\nolimits^{\mu}_{\nu\lambda
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\rho
a}-\frac{1}{12}\mathop{{\mathfrak{R}}}\nolimits^{\mu}_{\nu\lambda
a;\rho}\right.$
$\displaystyle\left.+\frac{1}{12}{\mathop{{\mathfrak{T}}}\nolimits}^{\mu}_{\nu
c}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\lambda
a;\rho}+\frac{1}{24}{\mathop{{\mathfrak{T}}}\nolimits}^{\mu}_{\nu
c;\rho}{\mathop{{\mathfrak{T}}}\nolimits}^{c}_{\lambda
a}-\frac{1}{8}{\mathop{{\mathfrak{T}}}\nolimits}^{\mu}_{\nu
a;\lambda;\rho}\right)(p).$ (12)

In quantities (12)-(11), the terms of a higher degree relative to the
coordinates $x^{\mu}$ will not be needed further. Note that at the point $p$
Latin indices $a,b,\ldots$ and Greek indices $\mu,\nu,\ldots$ are
indistinguishable. Further, the argument $(p)$ for geometric quantities is
omitted, since this does not lead to misunderstandings.

We will calculate the Dirac causal propagator in (9) using perturbation
theory. Thus222Strictly speaking,
$S(0,y)=\left\\{\sqrt{-g}\left(i\tilde{e}^{\mu}_{a}\gamma^{a}{\cal
D}_{\mu}-m\right)\right\\}^{-1}_{0,y}=\left\\{i\tilde{e}^{\mu}_{a}\gamma^{a}{\cal
D}_{\mu}-m\right\\}^{-1}_{0,y}(-g)^{-1/2}_{y}$, since
$(-g)^{-1/2}_{z,y}\propto\delta^{4}(z-y)$. But when $y\longrightarrow+0$
$g(y)\longrightarrow 1$, and we return to Eq. (13).

$\displaystyle S(0,y)=\left\\{i\tilde{e}^{\mu}_{a}\gamma^{a}{\cal
D}_{\mu}-m\right\\}^{-1}_{0,y}=\\{\left\\{(i\gamma^{\mu}\partial_{\mu}-m)+V\right\\}^{-1}_{0,y}$
$\displaystyle=S^{(0)}(-y)-\int\mathop{\rm
d}\nolimits^{4}xS^{(0)}(-x)V(x)S^{(0)}(x-y)+\ldots,$ $\displaystyle
S^{(0)}(x)=\int\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}e^{-ikx}\frac{\gamma^{\mu}k_{\mu}+m}{k^{2}-m^{2}+i0},$
(13)

where (in the absence of an electromagnetic field)

$\displaystyle
V=i\delta\tilde{e}^{\mu}_{a}\gamma^{a}\partial_{\mu}+i\gamma^{\mu}\omega_{\mu}+i\delta\tilde{e}^{\mu}_{a}\gamma^{a}\omega_{\mu}$
$\displaystyle=V^{(0)}+V^{(1)}+V^{(2)}.$ (14)

On the right-hand side of the last equality, $V^{(s)},s=0,1,2$, denotes the
contribution to the perturbation operator of degree $s$ relative to the
coordinates $x^{\mu}$. The contributions of the powers $s>2$ are not
interesting here, since they do not lead to divergent corrections. This
implies that the expansions (12)-(11) relative to $x^{\mu}$ are correct, since
the divergent integrals saturate at $x^{\mu}\longrightarrow 0$.

Let’s write out all $V^{(s)}$, taking into account that the degree of the
operator $\partial_{\mu}$ is equal to $(-1)$. Using (12)-(11) and (14) we
find:

$\displaystyle
V^{(0)}(x)=-\frac{i}{2}\gamma^{a}{\mathop{{\mathfrak{T}}}\nolimits}^{\mu}_{\nu
a}x^{\nu}\partial_{\mu},$ (15) $\displaystyle
V^{(1)}=i{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda
a}x^{\nu}x^{\lambda}\gamma^{a}\partial_{\mu}+\frac{i}{4}{\mathop{{\mathfrak{R}}}\nolimits}_{ab\mu\nu}x^{\mu}\gamma^{\nu}\sigma^{ab},$
(16) $\displaystyle
V^{(2)}=i{\mathop{{\mathfrak{E}}}\nolimits}^{\mu}_{\nu\lambda\rho
a}x^{\nu}x^{\lambda}x^{\rho}\gamma^{a}\partial_{\mu}+\frac{i}{2}\left(-\frac{1}{12}{\mathop{{\mathfrak{R}}}\nolimits}_{bc\nu
f}{\mathop{{\mathfrak{T}}}\nolimits}^{f}_{\lambda
a}+\frac{1}{3}{\mathop{{\mathfrak{R}}}\nolimits}_{bc\nu
a;\lambda}\right)x^{\nu}x^{\lambda}\gamma^{a}\sigma^{bc}.$ (17)

##  III Calculations

Obviously, the term of degree zero relative to $V$ on the right-hand side of
Eq. (9) is equal to zero. Indeed, we have $\mathop{\rm
tr}\nolimits\gamma^{5}\gamma^{d}S^{(0)}(-y)\equiv 0$.

###  III.1 The contribution of the electromagnetic field to the torsion tensor

Let us first calculate the contribution to the torsion from the
electromagnetic field in the first order and without taking into account
gravity. In this case $V=e\gamma^{\mu}A_{\mu}$. According to (9) and (13),
this contribution is

$\displaystyle\delta^{(1)}_{A}{\mathop{{\mathfrak{T}}}\nolimits}_{abc}=\frac{i}{4}el^{2}_{P}\varepsilon_{abcd}\mathop{\rm
tr}\nolimits\gamma^{5}\gamma^{d}\gamma^{\nu}\gamma^{\mu}\gamma^{\lambda}$
$\displaystyle\times\int\mathop{\rm
d}\nolimits^{4}x\left(\partial_{\nu}D^{(0)}(-x)\right)\left(\partial_{\lambda}D^{(0)}(x)\right)A_{\mu}(x)\equiv
0,$ (18)

since $D^{(0)}(-x)=D^{(0)}(x)$. Here $D^{(0)}(x)$ is the causal propagator of
a free Boson field.

Comparison of Eqs. (8) and (18) shows that the mean of the spin current in the
canonical representation (7) in an external electromagnetic field is equal to
zero in the first order in the field. Note an interesting fact: a similar mean
of the spin current, constructed using the representation of the spin operator
in the Pauli-Lubanski form, is not equal to zero in an external
electromagnetic field Chu and Leung (2021). Consequently, these two
representations of spin current differ fundamentally in quantum field theory.

###  III.2 Self-consistent equation for the torsion tensor

Everywhere below, we assume that the electromagnetic field is zero.

1. Let us calculate the contribution to the right-hand side of Eq. (9) in the first order in $V$.

a) Contribution from $V^{(0)}$:

$\displaystyle\delta_{V^{(0)}}{\mathop{{\mathfrak{T}}}\nolimits}_{abc}=-\frac{3}{4}l^{2}_{P}\int_{E}\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}\frac{k^{2}_{E}}{(k^{2}_{E}+m^{2})^{2}}\cdot\mathop{{\mathfrak{T}}}\nolimits_{abc}(p),$
$\displaystyle k^{2}_{E}=(k^{1})^{2}+(k^{2})^{2}+(k^{3})^{2}+(k^{4})^{2}.$
(19)

Hereinafter, the Wick rotation is used to bring the integrals to a convenient
form.

The squarely diverging integral in (19) must be cut off at a scale of the
order of the Planck. This means that the maximum possible momentum is of the
order

$\displaystyle k_{\mbox{max}}\sim\frac{2\pi}{l_{P}}.$ (20)

Therefore we have

$\displaystyle\left(C_{V^{(0)}}\right)^{2}\equiv\frac{3}{4}l^{2}_{P}\int_{E}^{k_{\mbox{max}}}\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}\frac{k^{2}_{E}}{(k^{2}_{E}+m^{2})^{2}}\sim 1.$
(21)

b) Contribution from $V^{(1)}$.

It is easy to see that this contribution is zero. Indeed, such a contribution
is proportional to integrals of the form

$\displaystyle\int_{E}\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}\frac{k^{\mu}}{(k^{2}_{E}+m^{2})^{m}}=0,$
$\displaystyle\int_{E}\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}\frac{k^{\mu}k^{\nu}k^{\lambda}}{(k^{2}_{E}+m^{2})^{m+1}}=0,$
(22)

which are equal to zero after integration over the angles. Thus

$\displaystyle\delta_{V^{(1)}}^{(1)}{\mathop{{\mathfrak{T}}}\nolimits}_{abc}=0.$
(23)

Further, we take into account only logarithmically divergent contributions,
which are obtained by taking into account the potential $V^{(2)}$ in the first
order, as well as $V^{(0)}\otimes V^{(1)}$ in the second and $V^{(0)}\otimes
V^{(0)}\otimes V^{(0)}$ in the third order in $V$. In this case, taking into
account (20), we have

$\displaystyle\int_{E}^{k_{\mbox{max}}}\frac{\mathop{\rm
d}\nolimits^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}_{E}+m^{2})^{2}}=\frac{1}{8\pi^{2}}\ln\frac{2\pi}{l_{P}m},$
$\displaystyle\ln\frac{2\pi}{l_{P}m}=\ln\frac{2\pi\hbar}{l_{P}mc}\sim
50\quad\mbox{for electron mass}.$ (24)

Note that the estimate (24) is valid in the present epoch. We adhere to the
version that the theory is regularized with the help of a lattice. This means
that in the era close to the moment of the Big Bang, the volume of momentum
space was larger in the same proportion as the volume of space decreased. For
this reason, near the Big Bang, we also have
$k_{\mbox{max}}\longrightarrow\infty$, and the estimate (24) turns out to be
much larger.

The self-consistent equation for torsion is written in terms of 4-vector
${\mathop{{\mathfrak{t}}}\nolimits}^{a}$, which in the case under
consideration is equivalent to the torsion tensor according to the equality

$\displaystyle{\mathop{{\mathfrak{T}}}\nolimits}_{abc}=\varepsilon_{abcd}{\mathop{{\mathfrak{t}}}\nolimits}^{d}.$
(25)

In the process of calculations, the Bianchi identities are used, which have
the following form in the presence of torsion:

$\displaystyle{\mathop{{\mathfrak{R}}}\nolimits}_{a[bcd]}={\mathop{{\mathfrak{T}}}\nolimits}_{a[bc;d]}+{\mathop{{\mathfrak{T}}}\nolimits}_{af[b}{\mathop{{\mathfrak{T}}}\nolimits}^{f}_{cd]},$
(26)
$\displaystyle{\mathop{{\mathfrak{R}}}\nolimits}_{ab[cd;f]}=-{\mathop{{\mathfrak{R}}}\nolimits}_{abe[f}{\mathop{{\mathfrak{T}}}\nolimits}^{e}_{cd]}.$
(27)

Everywhere we use the standard notation for any multi-index value $\varkappa$:
$\varkappa^{\ldots}_{\ldots[abc]}\equiv(\varkappa^{\ldots}_{\ldots
abc}+\varkappa^{\ldots}_{\ldots bca}+\varkappa^{\ldots}_{\ldots cab})$. It is
easy to check that, due to the representation (25), the second term on the
right-hand side of the equality (26) vanishes identically:
${\mathop{{\mathfrak{T}}}\nolimits}_{af[b}{\mathop{{\mathfrak{T}}}\nolimits}^{f}_{cd]}\equiv
0$. With (26) we get:

$\displaystyle{\mathop{{\mathfrak{R}}}\nolimits}_{abcd}-{\mathop{{\mathfrak{R}}}\nolimits}_{cdab}=\frac{1}{2}\left({\mathop{{\mathfrak{T}}}\nolimits}_{bcd;a}-{\mathop{{\mathfrak{T}}}\nolimits}_{cda;b}-{\mathop{{\mathfrak{T}}}\nolimits}_{dab;c}+{\mathop{{\mathfrak{T}}}\nolimits}_{abc;d}\right).$
(28)

The Ricci tensor by definition

$\displaystyle{\mathop{{\mathfrak{R}}}\nolimits}_{ab}\equiv\eta^{cd}{\mathop{{\mathfrak{R}}}\nolimits}_{cadb}=\eta^{cd}{\mathop{{\mathfrak{R}}}\nolimits}_{acbd},\quad{\mathop{{\mathfrak{R}}}\nolimits}_{ab}-{\mathop{{\mathfrak{R}}}\nolimits}_{ba}={\mathop{{\mathfrak{T}}}\nolimits}^{c}_{ba;c}.$
(29)

c) As a result of cumbersome calculations, we get:

$\displaystyle\delta_{V^{(2)}}{\mathop{{\mathfrak{t}}}\nolimits}^{a}=\frac{l_{P}^{2}}{96\pi^{2}}\ln\frac{2\pi}{l_{P}m}\left\\{3({\mathop{{\mathfrak{t}}}\nolimits}^{a})^{;b}_{;b}-3\left(\mathop{{\mathfrak{t}}}\nolimits^{b}_{;b}\right)^{;a}+\frac{1}{2}{\mathop{{\mathfrak{R}}}\nolimits}\cdot\mathop{{\mathfrak{t}}}\nolimits^{a}-2{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\,\,b}\mathop{{\mathfrak{t}}}\nolimits^{b}+\varepsilon^{abcd}\mathop{{\mathfrak{t}}}\nolimits_{b}\mathop{{\mathfrak{t}}}\nolimits_{c;d}\right\\}.$
(30)

2. Contribution from $V\otimes V$:

$\displaystyle\delta_{V\otimes
V}{\mathop{{\mathfrak{t}}}\nolimits}^{a}=\frac{l_{P}^{2}}{96\pi^{2}}\ln\frac{2\pi}{l_{P}m}\left\\{-\frac{1}{2}{\mathop{{\mathfrak{R}}}\nolimits}\cdot\mathop{{\mathfrak{t}}}\nolimits^{a}-{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\,\,b}\mathop{{\mathfrak{t}}}\nolimits^{b}-\frac{3}{2}\mathop{{\mathfrak{t}}}\nolimits^{2}\cdot\mathop{{\mathfrak{t}}}\nolimits^{a}+6\varepsilon^{abcd}\mathop{{\mathfrak{t}}}\nolimits_{b}\mathop{{\mathfrak{t}}}\nolimits_{c;d}\right\\}.$
(31)

3. Contribution from $V\otimes V\otimes V$:

$\displaystyle\delta_{V\otimes V\otimes
V}{\mathop{{\mathfrak{t}}}\nolimits}^{a}=\frac{l_{P}^{2}}{96\pi^{2}}\ln\frac{2\pi}{l_{P}m}\left\\{\frac{3}{2}\mathop{{\mathfrak{t}}}\nolimits^{2}\cdot\mathop{{\mathfrak{t}}}\nolimits^{a}\right\\}.$
(32)

4. Putting together the contributions (9), (19), (23), (30), (31) and (32), we arrive at a self-consistent equation for the torsion tensor:

$\displaystyle-\left[1+\left(C_{V^{(0)}}\right)^{2}\right]{\mathop{{\mathfrak{t}}}\nolimits}^{a}+\frac{l_{P}^{2}}{96\pi^{2}}\ln\frac{2\pi}{l_{P}m}\left\\{3(\mathop{{\mathfrak{t}}}\nolimits^{a})^{;b}_{;b}-3\left(\mathop{{\mathfrak{t}}}\nolimits^{b}_{;b}\right)^{;a}-3{\mathop{{\mathfrak{R}}}\nolimits}^{a}_{\,\,b}\mathop{{\mathfrak{t}}}\nolimits^{b}+7\varepsilon^{abcd}\mathop{{\mathfrak{t}}}\nolimits_{b}\mathop{{\mathfrak{t}}}\nolimits_{c;d}\right\\}=0.$
(33)

It is useful to rewrite Eq. (1) in the following form:

$\displaystyle e^{\mu}_{b}e^{\nu}_{c}\left({\cal D}_{\mu}e_{a\nu}-{\cal
D}_{\nu}e_{a\mu}\right)=\mathop{{\mathfrak{T}}}\nolimits_{abc}\equiv\varepsilon_{abcd}\mathop{{\mathfrak{t}}}\nolimits^{d}.$
(34)

Some considerations about the possibility of the existence of non-zero
solutions of Eq. (33) are given in the next Section.

##  IV Discussion

It would be interesting vacuum average the bilinear form of the Dirac fields
in the Einstein equation $\delta\mathop{{\mathfrak{A}}}\nolimits/\delta
e^{a}_{\mu}=0$ in a similar way. Thus, the Einstein equation would arise, in
which the contribution of the Dirac field to the matter energy-momentum tensor
would be expressed in terms of the curvature and torsion tensors. In deriving
this equation, there would be additional physical difficulties: it would be
necessary to make a subtraction from the cosmological constant, which diverges
as the fourth power of the cutoff momentum and is due to the Dirac sea. This
renormalization should lead to a very small cosmological constant. There was
no such difficulty in deriving of Eq. (33). Another essential difference
between the Einstein equation thus obtained and Eq. (33) is as follows. The
energy-momentum tensor of real particles and antiparticles does not vanish if
the curvature and torsion tensors are equal to zero (outside the mass shell).
Therefore, in the presence of real particles and antiparticles, the curvature
tensor cannot vanish due to the Einstein equation (on the mass shell). The
statement remains valid for zero torsion tensor. On the contrary, Eq. (33)
allows zeroing of the torsion tensor.

Let’s assume that we have explicitly described Einstein equation. In this
case, we have a closed system of equations for variables
$\mathop{{\mathfrak{t}}}\nolimits^{a}$, $e^{a}_{\mu}$ and $\omega^{ab}_{\mu}$:
this is Einstein equation and Eqs. (33), (34)). To do this, it is necessary to
express all the geometric quantities in Einstein equation in terms of the
indicated variables. However, in this paper, Einstein equation (in the
specified context) is not studied.

A step in this direction was taken in Nascimento et al. (2022): the
topological correction to the Hilbert-Einstein-Cartan action was calculated,
which is contained in the integral over the Dirac field in the one-loop
approximation. However, this problem remains unresolved in its entirety at
present.

According to the equation (33), the torsion field cannot propagate in space-
time like the field of scalar (or vector) particles. This statement is not a
new result, it can be found in the excellent review of Hehl et al Hehl et al.
(1976). To clarify this statement, consider the Klein-Gordon-Fock equation for
a scalar field in Minkowski space-time

$\displaystyle(\partial_{b}\partial^{b}\phi+\mu^{2}\phi)=0$ (35)

and an elementary consequence of this equation. In the case of a plane
monochromatic wave, $\phi\propto\mathop{\rm exp}\nolimits(-ikx)$, Eq. (35) has
a nonzero solution only if

$\displaystyle k^{0}=\pm\sqrt{\mu^{2}+{\bf k}^{2}}.$ (36)

In the case of a scalar particle, we have $\mu^{2}>0$. Therefore, scalar
particles can propagate in space-time.

However, in Eq. (33) the role of the ”squared mass” is played by the constant

$\displaystyle\mu^{2}=-32\pi^{2}l_{P}^{-2}\left(\ln\frac{2\pi}{l_{P}m}\right)^{-1}\left[1+\left(C_{V^{(0)}}\right)^{2}\right]<0.$
(37)

Indeed, consider the case $\mathop{{\mathfrak{t}}}\nolimits\longrightarrow 0$
and $\mathop{{\mathfrak{R}}}\nolimits^{a}_{\ \ bcd}\longrightarrow 0$. In this
case, one can choose a gauge such as $\omega_{\mu}\longrightarrow 0$ (see
(11), so that ${\cal D}_{\mu}\longrightarrow\partial_{\mu}$. As a result, we
can omit the last two terms in Eq. (33), and this equation breaks down into
the following two equations:

$\displaystyle\left(\partial_{b}\partial^{b}\mathop{{\mathfrak{t}}}\nolimits^{a}-32\pi^{2}l_{P}^{-2}\left(\ln\frac{2\pi}{l_{P}m}\right)^{-1}\left[1+\left(C_{V^{(0)}}\right)^{2}\right]\right)=0,$
(38)

and $\partial_{b}\mathop{{\mathfrak{t}}}\nolimits^{b}=0$. Comparison of Eqs.
(35) and (38) leads to equality (37). But then for ${\bf k}^{2}\ll|\mu^{2}|$
we have $k^{0}=-i|\mu|$, and the wave decays exponentially in time:
$\mathop{{\mathfrak{t}}}\nolimits^{a}\propto\mathop{\rm
exp}\nolimits(-|\mu|x^{0})$. Thus, only homogeneous stationary solutions or
those close to them can be nonzero.

Such a situation (with an inverted mass square) takes place in Landau’s theory
of second-order phase transitions below critical temperature.

There is also a significant difference between Eq. (33) and the corresponding
equation in Landau’s theory: in Eq. (33) there is no cubic term, but there is
a quadratic term containing the first derivative. Obviously, Eq. (33) always
has a zero solution for the torsion tensor. Finding nonzero solutions to this
equation will be the subject of further research. Here we only point out the
possible existence of a certain solution. Let
$\mathop{{\mathfrak{t}}}\nolimits^{a}\longrightarrow 0$ and gradient terms are
unimportant, and we will expand the solution in this small quantity. Then in
the equations (26) and (27) the right parts can be set equal to zero in the
leading approximation, and the curvature tensor can be taken in the form:
${\mathop{{\mathfrak{R}}}\nolimits}^{ab}_{cd}=-H^{2}\left(\delta^{a}_{c}\delta^{b}_{d}-\delta^{a}_{d}\delta^{b}_{c}\right)$
(curvature of de Sitter space). Then
${\mathop{{\mathfrak{R}}}\nolimits}^{a}_{b}=-3H^{2}\delta^{a}_{b}$, and in the
case $\mathop{{\mathfrak{t}}}\nolimits^{a}_{;b}=0$ Eq. (33) is rewritten as

$\displaystyle\left(\frac{1}{3}\mu^{2}+H^{2}\right)\mathop{{\mathfrak{t}}}\nolimits^{a}=0.$
(39)

Here $\mu^{2}$ is given in (37). An equation (39) can have a non-zero solution
if the parenthesis in that equation vanishes. This effect is achieved with a
certain and sufficiently large value of the parameter

$\displaystyle
H^{2}=\frac{32\pi^{2}}{3l_{P}^{2}}\left(\ln\frac{2\pi}{l_{P}m}\right)^{-1}\left[1+\left(C_{V^{(0)}}\right)^{2}\right]\ll\frac{4\pi^{2}}{l_{P}^{2}}\sim
k_{\mbox{max}}^{2}.$

This superficial consideration shows that if there are no conditions for the
existence of the torsion tensor at the present epoch, then such conditions
could take place near the Big Bang point.

Finally, we point out that, according to (6), we have the relation

$\displaystyle
J^{5a}=\left(4/l_{P}^{2}\right)\mathop{{\mathfrak{t}}}\nolimits^{a},\quad
J^{5a}\equiv\overline{\Psi}\gamma^{5}\gamma^{a}\Psi.$ (40)

Equation (40) shows that if the torsion tensor is not equal to zero, then the
axial current is also not equal to zero. The physical consequences following
from this are not clear to the author.

###### Acknowledgements.

I thank Prof. G.E. Volovik for drawing my attention to the problem, as well as
for useful advice in the process of work. This work was carried out as a part
of the State Program 0033-2019-0005.

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