# Thermoelectric Hall conductivity of the fractional quantum Hall systems on a
disk

Zi-Yi Fang  Department of Physics, Chongqing University,Chongqing 401331, P.
R. China  Dan Ye  Department of Physics, Chongqing University,Chongqing
401331, P. R. China  Yu-Yu Zhang<EMAIL_ADDRESS>Department of Physics,
Chongqing University,Chongqing 401331, P. R. China  Zi-Xiang Hu
<EMAIL_ADDRESS>Department of Physics, Chongqing University,Chongqing 401331,
P. R. China  Center for Quantum materials and device, Chongqing
University,Chongqing 401331, P. R. China

###### Abstract

For the fractional quantum Hall states on a finite disc, we study the
thermoelectric transport properties under the influence of an edge and its
reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101
(2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of
the thermoelectric conductivity $\alpha_{xy}\propto T^{\eta}$ for the gapless
composite Fermi-liquid state. The exponent $\eta\sim 0.5$ appears an
independence of the filling factors and the details of the interactions. In
the presence of an edge, we find the properties of the edge spectrum dominants
the low-temperature behaviors and breaks the universal scaling law of the
thermoelectric conductivity. In order to consider individually the effects of
the edge states, the entanglement spectrum in real space is employed and tuned
by varying the area of subsystem. In non-Abelian Moore-Read state, the
Majorana neutral edge mode is found to have more significant effect than that
of the charge mode in the low temperature.

###### pacs:

73.43.Cd, 73.43.Jn

##  I Introduction

When the strongly interacting two-dimensional electron gas (2DEG) is exposed
in a perpendicular strong magnetic field and extremely low temperature, Tsui
the formed fractional quantum Hall (FQH) droplet reveals the non-trivial
topological properties, such as the fractional charge excitations and
fractional statistics, Laughlin83  especially the FQH state NayakRMP  at some
special filling which could have Majorana zero mode excitation and non-Abelian
statistics. In FQH droplet, the bulk is a gapped insulator and the low-lying
excitation in the bulk is the neutral magneto-roton excitation Girvin  which
can be described as pairs of particle-hole excitations. boPRL12  Furthermore,
in a Hall bar sample which is used to measure the FQH state, the existence of
an edge is usually unavoidable unless using the Corbino geometry which
provides direct access to the longitudinal conductivity in the bulk. Corbino ;
Corbino1  The edge physics plays an essential role to uncover the bulk
topological properties due to the bulk-edge correspondence. On a FQH edge, the
description of Fermi-liquid theory is broken down and the interacting
electrons prefer to form a chiral Luttinger liquid (CLL). WenIJPM  It is
theoretically predicted that the current-voltage dependence in the tunneling
between a Fermi liquid and a quantum Hall edge obeys a universal power-law
$I\sim V^{\alpha}$ where $\nu=1/\alpha$ is the filling factor. Wen94  Such
universality, however, has not been conclusively observed in experiments. A
very likely reason is the occurrence of additional non-chiral edges that are
not tied to the bulk topology, ChangRMP03  namely the edge reconstruction,
which is a consequence of the competition between the positive background
confinement potential and electron-electron Coulomb repulsive interaction.
Wan02 ; Wan03 ; Jain09 ; Jain10  It breaks the chirality and universality in
the $I-V$ power law. In spite of its non-universality, one of us recently
found that in the entanglement spectrum, which represents the virtual edge
excitation of the reduced density matrix in a bipartite FQH model wave
function, still can be tuned and reconstructed by varying the area of the
subsystem. Wei

Since the edge excitation is gapless and has lower energy than that of the
neutral magneto-roton excitation in the bulk, intuitively, it dominates the
low-lying energy behavior of the system, regardless of the charge or heat
transport experiments. Moreover, while backscattering between two nearest
neighboring charged edge currents occurs or in the non-Abelian FQH edge, a
possible Majorana neutral mode appears. Because of the charge neutrality, this
type of edge mode can not be detected directly by electrical measurements.
Then the thermoelectric phenomena that provide the direct conversion between
heat and electricity are interesting and useful. In a thermoelectric
measurement, one sets up a temperature gradient $\triangledown T$, and an
electrical current $I$ is generated by the system to compensate for its
effect. They are related to each other by
$I_{i}=-\alpha_{ij}\triangledown_{j}T$ where $\alpha_{ij}$ is the
thermoelectric conductivity. Experimentally, one usually measure the
thermopower $S_{xx}$ and Nernst coefficient $S_{xy}$ and they are related to
$\alpha_{ij}$ by $S_{ik}=\alpha_{ij}\rho_{jk}$, in which $\rho$ is
resistivity. It is known that the thermoelectric response is directly related
to thermal entropy per electron. Obraztsov ; Girvin1982 ; Bergman2009 ; Kozii
Because of the huge degeneracy of the non-Abelian FQH states with
quasiparticle excitations, it was proposed to probe the non-Abelian statistics
of the quasiparticles and measure their quantum dimension. kun2009 ; Stern18
Very recently, it is shown that two-dimensional quantum Hall systems can reach
thermoelectric figure of merit on the order of unity down to low temperature,
as a consequence of the thermal entropy from the massive Landau level (LL)
degeneracy without considering the Coulomb interaction. Fu

For a Fermi-liquid, since the entropy is associated with the number of thermal
excitations within $k_{B}T$ near the Fermi energy, a linear $T$ dependence has
been conjectured. Cooper  While including the Coulomb interaction, the LL
degeneracy is lifted, and FQH states are formed at specific fillings factors.
In that case, the electron system is a non-Fermi liquid and the ground state
has only the degeneracy due to the center of mass translational symmetry in a
translational invariant system. In Ref.Sheng20, , Sheng and Fu calculated the
$\alpha_{xy}$ for FQH system in a torus geometry. A non-Fermi liquid pow-law
scaling $\alpha_{xy}\propto T^{\eta}$ with $\eta\sim 0.5$ for composite Fermi-
liquid states at $\nu=1/2$ and $\nu=1/4$ was found. $\alpha_{xy}$ vanishes
exponentially $\alpha_{xy}\sim\exp(-\Delta/k_{B}T)$ with a neutral magneto-
roton gap for $\nu=1/3$ Laughlin FQH state while $T\rightarrow 0$. However, as
we stated above, the low-lying energies of the FQH system are dominated by the
edge states in system with a boundary which was neglected in the torus
geometry. They should naturally have contributions in the low temperature
transport. In this work, we consider the thermoelectric conductivity in disk
geometry to see how does the edge state and its reconstruction affect the the
thermoelectric conductivity, especially the low temperature scaling behavior.
In order to isolate the effect of the edge excitations, the real space
entanglement spectrum is used to calculate the $\alpha_{xy}$ for virtual edge
excitations in reduced density matrix after tracing the degrees of freedom in
subsystem. We will show that the edge, and its reconstruction brings non-
universal scaling in low temperature, but still satisfies nonFermi-liquid
behavior. The Fermi-liquid behavior for dipolar fermions and importance of the
Majorana mode in non-Abelian FQH state are also found.

The rest of this paper is organized as following: In Sec. II, we compare the
thermoelectric conductivity results from energy spectrum with and without edge
states. The dipolar fermions in FQH and trivial phases are also considered. In
Sec. III, the real space entanglement spectrum is used to analyzing the effect
of the edge states. The non-Abelian FQH state and the dependence of
$\alpha_{xy}$ are also considered and Sec. IV gives the conclusions and
discussions.

##  II comparing different geometries and phases

Since the thermoelectric conductivity is directly related to the thermal
entropy per electron, we perform a fully diagonalizing the Hamiltonian for a
finite system at a given filling factor and the energy spectrum are used to
calculate the partition function $Z$ as a function of temperature $T$

$Z=\sum\limits_{i}\exp(-E_{i}/k_{B}T).$ (1)

Then the thermal entropy can be obtained as

$S=-\sum\limits_{i}\rho_{i}\ln{\rho_{i}}$ (2)

with $E_{i}$ the eigenvalue and $\rho_{i}=\exp(-E_{i}/k_{B}T)/Z$ the
probability for the $i$-th state to be occupied at temperature $T$. In the
absence of disorder, thermoelectric Hall conductivity $\alpha_{xy}$ is
proportional to the entropy density as

$\alpha_{xy}=\frac{s}{B}=\frac{S}{N_{orb}}$ (3)

where $N_{orb}$ is the number of orbitals for electrons occupying in Landau
level and $s$ is $S$ divided by the area of 2DEG which is $2\pi
l_{B}^{2}N_{orb}$. Here $l_{B}$ is the magnetic length $\l_{B}=\sqrt{\hbar
c/eB}$. Suppose the dimension of the Hilbert space is $\Omega$, in high
temperature limit such that the thermal activation overwhelms all the energy
scale of the many-body system, such as the neutral gap or quasiparticle
excitation gap in the bulk, namely $k_{B}T\gg\Delta$. Each eigenstate has
identical probability $\rho_{i}=1/\Omega$ and thus the thermal entropy is
classical $S=\ln\Omega$. In this case, $\alpha_{xy}=\ln\Omega/N_{orb}$ is a
constant. For a state with $N_{e}$ electrons at filling $\nu=N_{e}/N_{orb}$,
the dimension of the Hilbert space $\Omega=C_{N_{orb}}^{N_{e}}$ reaches
maximum at $\nu=1/2$ and thus the half filling has the largest $\alpha_{xy}$.
It has potential to enable thermoelectric cooling and power generation with
unprecedented efficiency. Fu

Figure 1:  The energy spectrum for 8 electrons in 24 orbitals with Coulomb
interaction on torus(a) and disk(b). In fig.(b), the positive confinement is
set at a distance $d=0.5\ell_{B}$ from electron layer. The gapless edge mode
dispersion is labelled by red color. (c) and (d) depict the $\alpha_{xy}$ (in
units of $k_{B}e/h$) versus the thermal energy $k_{B}T$ in units of
$e^{2}/\epsilon\ell_{B}=1$ for $6-8$ electrons at $\nu=1/3$. In the inserted
plots in (c), we fit the low temperature data with
$\alpha_{xy}\sim\exp(-\Delta/k_{B}T)$. In (d), two confinement potentials are
used. One $d=0.5$ is in the Laughlin phase and the other $d=1.5$ is after the
edge reconstruction. $\eta$ are different for two cases.

At low temperature, as shown in Fig. 1, we compare the results on a compacted
torus with that on a disc. For Coulomb interaction at $\nu=1/3$, the energy
spectrum on torus has three-fold degeneracy due to the center of mass
translational symmetry which demonstrates the Laughlin-like state. In this
case, the three-fold degenerated ground states are gapped by neutral magneto-
roton excitation in the bulk. In disk geometry, mimicking the 2DEG in
semiconductor hetero-structure, we diagonalize a Hamiltonian which contains
the 2DEG and a homogeneous positive background confinement potential at
distance $d$. Wan02 ; Wan03  As shown in Fig. 1(b), the ground state is unique
and a chiral gapless edge excitation (its dispersion is labelled in red color)
dominates the low-lying energy spectrum in thermodynamic limit. As a result,
the thermoelectric Hall conductivity $\alpha_{xy}$ at low temperature exhibits
different behaviors in two cases. In Fig. 1(c) which was also shown in Ref.
Sheng20, , the $\alpha_{xy}$ is saturated at a value $\ln 3/N_{orb}$ while
$T\rightarrow 0$ due to the three-fold ground state degeneracy. For a gapped
phase on torus, the low-temperature data before saturation could be fitted by
$\exp(-\Delta/k_{B}T)$ as shown in the inserted plot. $\Delta$ is actually the
neutral gap which is labelled by arrow in Fig. 1(a). In disk geometry, as
shown in Fig. 1(d), the $\alpha_{xy}$ for $\nu=1/3$ FQH state has a similar
behavior as that for the composite Fermi-liquid at $\nu=1/2$ and $\nu=1/4$ on
torus. It approaches to zero while $T\rightarrow 0$. After neglecting the data
below the average energy gap in the spectrum, which origins from the finite
size effects, we fit the low-temperature data using $\alpha_{xy}\propto
T^{\eta}$ with $\eta\simeq 0.895$ for $d=0.5l_{B}$ and $\eta\simeq 0.780$ for
$d=1.5l_{B}$. The reason to take these two cases is that the Laughlin-like
ground state which has total angular momentum $M_{tot}=3N_{e}(N_{e}-1)/2$,
only survives as the global ground state in window $d\in[0,1.35]l_{B}$ for $8$
electrons. When $d>1.35l_{B}$, the ground state angular momentum is changed
and the reconstruction occurs on the edge. In the presence of an edge, we find
varying confinement potential significantly changes the low temperature
scaling behavior of the $\alpha_{xy}$.

Figure 2:  For fast rotated dipolar fermions on a torus, finite size scalings
of the ground energy gap at different titled angles $\theta$ for the dipole
moment. The particle number is $8$ and the thickness in $z$ direction is set
to $q=0.01$ in unit of $l=\sqrt{\hbar/(2\mu\omega)}$ where $\mu$ is the
effective mass and $\omega$ is the frequency of the trap potential.
$\theta=54.7^{\circ}$ is the magic angle at which the dipole-dipole
interacting is purely anisotropic and the system is gapless. (b) and (c)
depict the $\alpha_{xy}$ and fittings for $6-8$ particles at
$\theta=0^{\circ}$ and $\theta=54.7^{\circ}$. The gapless phase shows the
Fermi-liquid behavior with $\eta\simeq 1$.

As another example to see the different low temperature behaviors of
$\alpha_{xy}$ in gapped and gapless phases, we consider the dipole-dipole
interaction for degenerate quantum gas in a fast rotating trap. Cooperdipole

$V_{dd}(\vec{r},\theta)=\frac{r^{2}-3(z\cos\theta+x\sin\theta)^{2}}{r^{5}},$
(4)

where $\theta$ is the angle between the dipole moment and the $z$ axis. In a
fast rotating limit at which the rotating frequency is close to that of the
harmonic trap potential, the system enters into the quantum Hall regime.
Cooperdipole  It was found qiu2011 ; hu18  that the Laughlin phase robustly
survives while the tilting angle $\theta$ is small. As the $\theta$ is
increased, the ground state gap is gradually reduced and finally closed, then
the system enters into a cluster state hu18  because of the anisotropic
attractive interaction in one direction. The interaction in Eq.(4) could be
re-arranged as

$V_{dd}(\vec{r},\theta)=\frac{3\cos^{2}{\theta}-1}{2}\frac{r^{2}-3z^{2}}{r^{5}}+\frac{3\sin^{2}{\theta}}{2}\frac{y^{2}-x^{2}}{r^{5}}.$
(5)

The first term is isotropic and the second term is anisotropic in the 2D
plane. The magic angle $\theta=54.7^{\circ}$ satisfies the condition
$3\cos^{2}{\theta}-1=0$, at which the interaction only contains the second
term. In Fig. 2(a), we plot the energy gap between the lowest two states via
diagonalizing Eq.(5) on torus for several system sizes. The extrapolation in
thermodynamic limit from finite size scaling clearly tells us the system is
gapped for $\theta=0^{\circ},30^{\circ}$ and gapless for
$\theta=54.7^{\circ},90^{\circ}$ respectively. In Fig. 2(b) and (c), we plot
the $\alpha_{xy}$ for $\theta=0^{\circ}$ and $54.7^{\circ}$ which are in two
different phases. The $\alpha_{xy}$ in FQH gapped phase is similar to that of
the Coulomb Hamiltonian in Fig. 1(a) and the low temperature part still
satisfies $\alpha_{xy}\sim\exp(-\Delta/k_{B}T)$. However, in a topological
trivial gapless phase at $1/3$ filling, as shown in In Fig. 2(c), we find the
low temperature behavior could be fitted by $\alpha_{xy}\sim T^{0.97}$ which
is very close to the Fermi-liquid exponent $\eta=1$. Therefore, from low
temperature thermoelectric Hall conductivity, we speculate the phase
transition in dipolar fermions is a topological phase to Fermi-liquid
transition.

Based on the results of above two cases, we suspect that the existence of the
gapless edge mode or gapless trivial phase could have different scaling
exponent of the thermoelectric Hall conductivity in low temperature.

##  III The effects of edge states

In previous section, we observe that the presence of an edge significantly
changes the low-temperature behaviors of the thermoelectric Hall conductivity.
In order to individually consider the edge states, we calculate $\alpha_{xy}$
for the real space entanglement spectrum Regnault12 ; Dubail12 ; Simon12  of a
model wave function. To be precise, a bipartition of a quantum system is
defined when the Hilbert space is divided into two subsystems,
$H=H_{A}\bigotimes H_{B}$. Then the ground state wave function is performed by
a Schmidt decomposition:Nielsen ; Kitaev

$|\Psi\rangle=\sum_{i}e^{-\frac{1}{2}\xi_{i}}|\psi_{A}^{i}\rangle\bigotimes|\psi_{B}^{i}\rangle)$
(6)

where the $\exp(-\xi_{i})=\lambda_{i}$ are the eigenvalues of the reduced
density matrix of subsystem $\rho_{A}=\text{Tr}_{B}\rho$. It is normalized by
$\sum_{i}e^{-\xi_{i}}=|\psi\rangle\langle\psi|=1$. If $\rho_{A}$ is finite
dimensional, the von Neumann entanglement entropy is defined as
$S_{A}=-\sum_{i}\lambda_{i}\text{log}\lambda_{i}=\sum_{i}\xi_{i}\exp(-\xi_{i})$.
It was known by Haldane HaldaneES  that the full structure of the
“entanglement spectrum” (ES) which is the logarithmic Schmidt spectrum of
level $\xi_{i}$ contains much more information about the entanglement between
two halves than that from $S_{A}$ only. It plays a key role in analyzing the
topological order of the FQH state. The structure of ES is analogous to the
low energy excitations of a many-body Hamiltonian. Especially, for the model
wave function, the counting per momentum sector of ES is identical to the
energy spectrum for the edge excitations. Beyond the counting, one of us Wei
recently found that the entanglement spectrum in real space cut appears the
signals of edge reconstruction via tuning the area of the subsystem. The edge
velocity reaches its minimum when the area of subsystem is equal to that of
the correlation hole.

Figure 3:  The real space ES for subsystem of $N_{A}=4$, $R_{A}=4.0l_{B}$(a)
and $R_{A}=4.9l_{B}$(b) in a $10$-electron Laughlin state. $N_{A}$ and $R_{A}$
is number of electrons and radius of disk for the subsystem. The typical
energy spacing is defined as the gap between the lowest two states. (c) and
(d) depict the $\alpha_{xy}$ and low temperature fittings from the spectrum
(a) and (b) correspondingly. Their fitting exponents are $\eta\simeq 0.973$
and $\eta\simeq 0.753$ respectively.

We consider the $\nu=1/3$ Laughlin wave function for $10$ electrons on a disk
which can be obtained either from diagonalizing a $V_{1}$ pseudopotential
hamiltonian HaldanePP  or from the Jacks polynomial recipe. Bernevig1 ;
Bernevig2  Because the bipartition we used conserves the rotational symmetry,
the ES for a given number of electrons in subsystem $N_{A}$ and radius of the
circular cut $R_{A}$ are shown in Fig. 3(a) and (b). Here we consider the
subsystem contains $4$ particles and the radius of the subsystem $R_{A}$ is a
parameter we tuned. It shows that the ES for the same $N_{A}$ and different
$R_{A}$ have different spectra. Consequently, their thermoelectric Hall
coefficient $\alpha_{xy}$ have different behaviors at low temperature as shown
in Fig. 3(c) and (d) respectively. To eliminating the finite size effect, we
still fit the low temperature data above the typical average energy interval.
The exponent are $\eta\simeq 0.973$ and $\eta\simeq 0.753$ for two cases, both
of which are larger than $\eta\simeq 0.5$ on torus and smaller than that of
the Fermi-liquid value. Therefore, the FQH edge states still possess non-
Fermi-liquid characteristics, but its scaling exponent could be changed by
tuning the edge velocity (slope of the dispersion) and thus non-universal.

Figure 4:  (a) ES for the $\nu=5/2$ Moore-Read state $N_{e}=12$,
$N_{A}=6$,$R_{A}=3.5$. (b) The low-lying energy spectrum of a mixed
Hamiltonian with $\lambda=0.5$ for $12$ electrons in different confinement
potentials. The red bars are the Majorana neutral edge mode which has lower
energy than that of the bosonic charge mode. (c) and (d) depicts the
$\alpha_{xy}$ versus $k_{B}T$ for (a) and (b) correspondingly. In each plots,
we take two $R_{A}$s and $d$s for the sake of observing the different fittings
in different edge spectrum.

Now let’s move to the non-Abelian FQH liquid which has more complicate edge
structures. One prominent example is the Moore-Read state Moore  which is the
model wave function for FQH state at $\nu=5/2$ filling in the first Landau
level. The model wave function could also be obtained from diagonalizing a
three-body $V_{3}$ pseudopotential Simon07  Hamiltonian or Jacks. Fig. 4(a)
shows the ES of a $12$-electron Moore-Read wave function. Unlike the ES for
Laughlin state, the counting of edge spectrum becomes to be $1,1,3,5\cdots$.
This is due to the existence of a neutral Majorana mode besides the bosonic
charge mode. Wan08  Fig. 4(b) depicts the energy spectrum of a mixed
Hamiltonian

$H=(\lambda-1)H_{2B}+\lambda H_{3B}$ (7)

with $\lambda=0.5$ for 12 electrons under different strength of the
confinement potentials. $H_{2B}$ contains the electron-electron Coulomb
interaction and confinement potential. $H_{3B}$ is the three-body
pseudopotential Hamiltonian. The reason of mixing $H_{3B}$ is to emphasize the
edge states which have zero energy in the model Hamiltonian. In this case, the
low-lying edge states are separated from the bulk states and they clearly show
the same counting rule as that in (a). The Majorana edge states are labelled
by red bars. Fig. 4(c) and (d) are their corresponding $\alpha_{xy}$ and low
temperature fittings. For the ES, we have $\eta\simeq 0.834$ and $\eta\simeq
0.774$ for $R_{A}=3.5$ and $4.5$ respectively. Again, tuning the area of the
subsystem or the edge velocities changes the scaling exponent $\eta$. For the
energy spectrum which is a mixture of edge and bulk states, we obtain
different results for different confinement potentials. In the disk geometry,
we found Wan08  the Moore-Read state as the global ground state exists in
range $d\in[0,0.5]l_{B}$. In Fig. 4(d) we plot the $\alpha_{xy}$ for
$d=0.1l_{B}$ and $d=0.3l_{B}$ the fitting exponents are $\eta\simeq 1.012$ and
$\eta\simeq 0.694$ respectively. In all cases, the $\eta$ decreases as
decreasing the strength of the confinement potential (increasing $d$ in energy
spectrum or increasing $R_{A}$ in ES before reconstruction). On the other
hand, it is known Hu09 ; Hu16  that the neutral edge mode has one order lower
velocity or excitation energy than that of charge mode. From the $\alpha_{xy}$
as shown in Fig. 4(d), the low temperature behaviors are dominated near the
scale $k_{B}T\sim 0.01$ which has the same order of the energy difference
between the ground state and the lowest edge state in neutral edge mode.
Therefore, for the non-Abelian FQH state, the Majorana neutral edge mode has
much more significant contribution in thermoelectric conductivity and could be
detected in the low temperature thermal Hall experiments. kun2009 ; Stern18

Figure 5:  The ground state quantum number $M_{tot}$ and low temperature
$\alpha_{xy}$ versus the distance between 2DEG and confinement potential
$d/l_{B}$ for $8$ electrons at $\nu=1/3$. The peak of $\alpha_{xy}$
corresponds to the ground state phase transition, or the edge reconstruction.

Besides the scaling behavior in low temperature, the edge confinement also has
effects on the value of $\alpha_{xy}$ itself. In Fig. 5, we plot the ground
state angular momentum and $\alpha_{xy}$ as a function of the confinement
parameter $d/l_{B}$. Two temperatures are considered. The $M_{tot}$ is the
total angular momentum of the ground state which is $3N_{e}(N_{e}-1)/2$ for
Laughlin state. While increasing $d$, the electrons firstly stay stably in FQH
phase and then form stripes at the edge which is accompanied with a jump of
$M_{tot}$. WenIJPM ; Wan02 . If we treat the $M_{tot}$ as the ground state
quantum number, the jumps in $M_{tot}$ correspond to ground state phase
transitions and each $M_{tot}$ could be treated as a phase. It is interesting
to see that in the Laughlin phase for $d<1.35l_{B}$, thermoelectric Hall
conductivity $\alpha_{xy}$ is smaller than that of the reconstructed phase.
This is because the edge reconstruction $d>1.35l_{B}$ induces more edge modes
and thus make more contributions to the thermal transport. Moreover, the
$\alpha_{xy}$ has maximum at the phase transition and minimum at the center of
each phases. As the temperature increasing, the structure of $\alpha_{xy}$ is
gradually erased by thermal excitations.

##  IV Discussions and Conclusions

In this work, we study the thermoelectric Hall conductivity of the FQH system
in disk geometry, especially its low temperature behaviors. Because of the
presence of gapless edge mode, the low temperature behaviors of $\alpha_{xy}$
in disk geometry are similar to that of the gapless composite Fermi-liquid
state on a torus. Because of the edge spectrum could be changed by tuning the
strength of the background confinement in energy spectrum or the area of
subsystem in the bi-partition ES, the scaling exponent $\eta$ of
$\alpha_{xy}\sim T^{\eta}$ in low temperature has strong dependence of the
system parameters. Comparing to the results on torus, we find $\eta$ decreases
as decreasing the strength of the confinement potential (increasing $d$ in
energy spectrum or increasing $R_{A}$ in ES before reconstruction). For the
dipolar fermions at the magic angle, we verify the Fermi-liquid scaling
behavior with $\eta\sim 1.0$ in the gapless phase. For non-Abelian Moore-Read
state at $\nu=5/2$, we find the Majorana neutral edge mode has more
significant contribution to the thermoelectric Hall conductivity and dominates
the low temperature scaling behavior, explaining the reason of the non-Abelian
signals appear in the low temperature thermal Hall experiments.

In conclusion, comparing to the results on torus, we find the edge excitations
of the FQH states dominate the low temperature scaling behavior of
$\alpha_{xy}\sim T^{\eta}$. Both the $\alpha_{xy}$ and its scaling exponent
$\eta$ are strongly affected by the edge confinement which makes the scaling
non-universal. More edge modes produced by edge reconstruction or the neutral
Majorana mode in non-Abelian FQH state contributes significantly to
$\alpha_{xy}$ at low temperature thermal Hall transport.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China
(Grants No.12075040 and No.11974064), the Chongqing Research Program of Basic
Research and Frontier Technology (Grants No.cstc2020jcyj-msxmX0890) and the
Fundamental Research Funds for the Central Universities under Grant No.
2020CDJQY-Z003.

## References

  * (1) D. C. Tsui, H. L. Störmer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
  * (2) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * (3) C. Nayak, S. H. Simon, A. Stern, M. Freedom, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * (4) S. M. Girvin and A. H. Macdonald, Phys. Rev. Lett. 48, 581 (1985).
  * (5) B. Yang B, Z. X. Hu, Z. Papic and F. D. M. Haldane, Phys. Rev. Lett. 108 256807 (2012).
  * (6) R. B. Laughlin, Phys. Rev. B 23 5632 (1981).
  * (7) V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S. I. Dorozhkin and K. von Klitzing K, Phys. Rev. B 46 12560 (1992).
  * (8) X. G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992).
  * (9) C. de C. Chamon and X. G. Wen, Phys. Rev. B 49, 8227 (1994).
  * (10) A. M. Chang, Rev. Mod. Phys. 75, 1449 (2003).
  * (11) X. Wan, K. Yang, and E. H. Rezayi, Phys. Rev. Lett. 88, 056802 (2002).
  * (12) X. Wan, E. H. Rezayi, and K. Yang, Phys. Rev. B 68, 125307 (2003).
  * (13) S. Jolad and J. K. Jain, Phys. Rev. Lett. 102, 116801 (2009).
  * (14) S. Jolad, D. Sen and J. K. Jain, Phys. Rev. B 82, 075315 (2010).
  * (15) L. X. Wei, N. Jiang, Q. Li, and Z. X. Hu, Phys. Rev. B 101, 075307 (2020).
  * (16) Y. N. Obraztsov, Sov. Phys. Solid State 7, 455 (1965).
  * (17) S. M. Girvin and M. Jonson, J. Phys. C 15, L1147 (1982).
  * (18) D. Bergman, and V. Oganesyan, Phys. Rev. Lett. 104, 066601 (2009).
  * (19) V. Kozii, B. Skinner, and L. Fu, Phys. Rev. B 99, 155123 (2019).
  * (20) K. Yang and B. I. Halperin, Phys. Rev. B 79, 115317 (2009).
  * (21) M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Nature 559, 205 (2018).
  * (22) L. Fu, arxiv. 1909.09506.
  * (23) N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Phys. Rev. B 55, 2344 (1997).
  * (24) D. N. Sheng, and L. Fu, Phys. Rev. B 101, 241101 (2020).
  * (25) N. R. Cooper, Advances in Physics 57, 539 (2008).
  * (26) R. Z. Qiu, S. P. Kou, Z. X. Hu, X. Wan, and S. Yi, Phys. Rev. A 83, 063633 (2011).
  * (27) Z. X. Hu, Q. Li, L. P. Yang, W. Q. Yang, N. Jiang, R. Z. Qiu and B. Yang, Phys. Rev. B 97, 035140 (2018).
  * (28) A. Sterdyniak, A. Chandran, N. Regnault, B. A. Bernevig, and P. Bonderson, Phys. Rev. B, 85, 125308 (2012).
  * (29) J. Dubail, N. Read, and E. H. Rezayi, Phys. Rev. B, 85, 115321 (2012).
  * (30) I. D. Rodriguez, S. H. Simon and J. K. Slingerland, Phys. Rev. Lett., 108, 256806 (2012).
  * (31) M. A. Nielsen and I. L. Chuang, Quantum Computation, and Quantum Information (Cambridge University Press, Cambridge, 2000). (2000).
  * (32) A. Kitaev, and J. Preskill, Phys.Rev. B 96, 110404 (2006).
  * (33) H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
  * (34) F. D. M. Haldane, Phys. Rev. Lett. 51 605 (1983).
  * (35) B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 100, 246802 (2008).
  * (36) B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 101, 246806 (2008).
  * (37) G. Moore and N. Read, Nuclear Physics B 360, 362 (1991).
  * (38) S. H. Simon, E. H. Rezayi and N. Cooper, Phys.Rev. B 75, 195306 (2007).
  * (39) X. Wan, Z. X. Hu, E. H. Rezayi, and K. Yang, Phys.Rev. B 77, 165316 (2008).
  * (40) Z. X. Hu, E. H. Rezayi, X. Wan, and K. Yang, Phys. Rev. B 80, 235330 (2009).
  * (41) N. Jiang and Z. X. Hu, Phys. Rev. B 94, 125116(2016).


# Rank deficiency of random matrices

Vishesh Jain  Department of Statistics, Stanford University, Stanford CA
94305, USA<EMAIL_ADDRESS>,  Ashwin Sah  and  Mehtaab Sawhney
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA<EMAIL_ADDRESS>
###### Abstract.

Let $M_{n}$ be a random $n\times n$ matrix with i.i.d.
$\operatorname{Bernoulli}(1/2)$ entries. We show that for fixed $k\geq 1$,

$\lim_{n\to\infty}\frac{1}{n}\log_{2}\mathbb{P}[\operatorname{corank}M_{n}\geq
k]=-k.$

##  1\. Introduction

A fundamental, and intensely studied, problem in combinatorial random matrix
theory is the determination of the probability of singularity of $n\times n$
random Bernoulli matrices (i.e. $n\times n$ matrices for which each entry is
independently $0$ or $1$ with equal probability). The study of this problem
was initiated in work of Komlós [11]. After intermediate works over a period
of over 50 years [10, 15, 16, 2], the breakthrough work of Tikhomirov [17]
showed that for any fixed $p\in(0,1/2]$,

$\mathbb{P}[M_{n}(\operatorname{Ber}(p))\text{ is
singular}]=(1-p+o_{n}(1))^{n},$

where we use the notation $M_{m\times n}(\xi)$ to denote an $m\times n$ random
matrix with i.i.d. entries distributed as $\xi$, and the lighter notation
$M_{n}(\xi)$ for $M_{n\times n}(\xi)$. Also, $\operatorname{Ber}(p)$ is the
random variable which takes on the value $1$ with probability $p$ and $0$ with
probability $1-p$.

By considering the probability of a row of the matrix being $0$, one sees that
the result of Tikhomirov is optimal up to the $o_{n}(1)$ term. Recently,
several works [7, 8, 9, 12, 1] have addressed the more refined question of
determining the probability of singularity of $M_{n}(\xi)$ up to a
$(1+o_{n}(1))$ factor; in contrast, the aforementioned result of Tikhomirov
determines this probability only up to a subexponential (in $n$) factor. While
these works have succeeded in the case of sparse Bernoulli matrices (with
sparsity allowed to depend on $n$) [7, 12, 1], as well as in the case of a
fixed $\xi$ which is not uniform on its support [8, 9], we note that for the
case of $\operatorname{Ber}(1/2)$, the estimate of Tikhomirov remains
essentially the best known.

An equivalent condition to singularity of $M_{n}(\xi)$ is that the corank of
$M_{n}(\xi)$ is at least $1$. Given this formulation, the following question
immediately suggests itself: given $k\geq 1$, what is the probability that the
corank of $M_{n}(\xi)$ is at least $k$? In the case when
$\xi=\operatorname{Ber}(p)$ for fixed $p\in(0,1/2]$, by considering the event
that first $k$ rows of the matrix are identically $0$, we see that this
probability is at least $(1-p)^{nk}$. The previously best-known upper bound
appears to be due to Kahn, Komlós, and Szemerédi [10], who showed that there
exists a function $f\colon\mathbb{N}\to\mathbb{R}^{+}$ with $f(k)\to 0$ as
$k\to\infty$ such that

$\mathbb{P}[\operatorname{corank}M_{n}(\operatorname{Ber}(1/2))\geq k]\leq
f(k)^{n}.$

The above simple lower bound shows that the decay of $f(k)$ can be at most
$2^{-k}$; it has been suggested (cf. [18, Section 4], ‘It is tempting to
conjecture…’) that this rate of decay is essentially sharp, i.e. that

$\mathbb{P}[\operatorname{corank}M_{n}(\operatorname{Ber}(1/2))\geq
k]=(1/2+o_{n}(1))^{kn}.$

The main result of this paper confirms this belief.

######  Theorem 1.1.

Fix $p\in(0,1/2]$ and let $\xi=\operatorname{Ber}(p)$. Fix $k\geq 1$ and
$\epsilon>0$. Then, for $n\geq n_{\ref{thm:main}}(p,k,\epsilon)$, we have

$\mathbb{P}[\operatorname{corank}M_{n}(\xi)\geq k]=(1-p+\epsilon)^{kn}.$

######  Remark.

A modification of our proof, with Proposition 2.7 replaced by the
corresponding versions in [8, 9], shows that for any fixed $\xi$ which is
supported on finitely many points,

$\mathbb{P}[\operatorname{corank}M_{n}(\xi)\geq
k]\leq(\max_{z\in\mathbb{R}}\mathbb{P}[\xi=z]+o_{n}(1))^{kn}.$

We conjecture that, in general, the following holds.

######  Conjecture 1.2.

Fix a random variable $\xi$ supported on finitely many points. Fix $k\geq 1$
and $\epsilon>0$. Then for $n\geq n_{\ref{conj:general}}(\xi,k,\epsilon)$, we
have

$\mathbb{P}[\operatorname{corank}M_{n}(\xi)\geq
k]\leq(\mathbb{P}[\xi=0]+\epsilon)^{kn}+(\mathbb{P}[\xi_{1}=\cdots=\xi_{k+1}]+\epsilon)^{n},$

where $\xi_{1},\ldots,\xi_{k+1}$ are independent samples of $\xi$.

A stronger conjecture is that the dominant contribution to the probability of
the corank being at least $k$ comes from the event of having $k-t+1$ rows
equal up to sign and $t$ rows zero for some $0\leq t\leq k$, or the same for
columns (i.e., this controls the probability up to a $(1+o_{n}(1))$ factor).
All such events contribute, for instance, when $\xi=\operatorname{Ber}(1/2)$.
However, in the case when $\xi=\operatorname{Ber}(p)$, $p\in(0,1/2)$, the
event of having $k$ rows which are identically zero is exponentially more
likely than any of the other events. In concurrent and independent work, Huang
[6] has proved this stronger conjecture (with accompanying singular value
bounds) for sufficiently sparse Bernoulli matrices i.e.
$\xi=\operatorname{Ber}(p_{n})$ with

$1\leq\liminf_{n\to\infty}\frac{p_{n}\cdot
n}{\log{n}}\leq\limsup_{n\to\infty}\frac{p_{n}\cdot n}{\log{n}}<\infty;$

it is plausible that, combined with the techniques in [12], the upper bound on
the $\limsup$ can be relaxed (perhaps even up to $p_{n}\leq c$ for some small
constant $c$). In the complementary dense case considered here, we leave the
resolution of this stronger conjecture as a subject for future research.

Finally, we mention that in recent years, there have been several other works
on the (co)rank of random matrices (cf. [4, 3]). However, the focus of these
works is on the asymptotic determination of the (co)rank of various models,
which is different from our focus on the determination of (sharp) non-
asymptotic rates to have corank at least $k$ for matrices which are of full
rank with high probability.

###  1.1. Notation

Given a positive integer $N\geq 1$, let $\mathbb{S}^{N-1}$ be the set of unit
vectors in $\mathbb{R}^{N}$. Let $\lVert\cdot\rVert_{2}$ be the Euclidean
norm. For a matrix $A=(A_{ij})$, let $\lVert A\rVert$ be its spectral norm
(i.e., $\ell^{2}\to\ell^{2}$ operator norm) and let $\lVert
A\rVert_{\operatorname{HS}}$ be its Hilbert-Schmidt norm, defined as

$\lVert A\rVert_{\operatorname{HS}}^{2}=\sum A_{ij}^{2}.$

We let $[N]$ denote the discrete interval $\\{1,\dots,N\\}$. Given an $m\times
n$ matrix $A$ and a subset $S\subseteq[n]$ of columns, we let $A_{S}$ be the
$m\times|S|$ submatrix of $A$ consisting only of the columns in $S$.

For an $\mathbb{R}^{N}$-valued random variable $\xi$ and a real number $r\geq
0$, we define the Lévy concentration function by

$\mathcal{L}(\xi,r):=\sup_{z\in\mathbb{R}^{N}}\mathbb{P}[\lVert\xi-z\rVert_{2}\leq
r].$

Note that the case $N=1$ coincides with the usual (scalar) Lévy concentration
function. We let $\ell_{1}(\mathbb{Z})$ denote the set of functions
$f\colon\mathbb{Z}\to\mathbb{R}$ satisfying
$\sum_{z\in\mathbb{Z}}|f(z)|<\infty$.

We also make use of asymptotic notation. Given functions $f,g$, we write
$f=O_{\alpha}(g)$ or $f\lesssim_{\alpha}g$ to mean $f\leq C_{\alpha}g$, where
$C_{\alpha}$ is some constant depending on $\alpha$. We write
$f=\Omega_{\alpha}(g)$ or $f\gtrsim_{\alpha}g$ to mean $f\geq c_{\alpha}g$,
where $c_{\alpha}>0$ is some constant depending on $\alpha$. Finally, we write
$f=\Theta_{\alpha}(g)$ to mean that both $f=O_{\alpha}(g)$ and
$f=\Omega_{\alpha}(g)$ hold. For parameters $\epsilon,\delta$, we write
$\epsilon\ll_{\alpha}\delta$ to mean that $\epsilon\leq c_{\alpha}(\delta)$
for a sufficiently decaying function $c_{\alpha}$ depending on $\alpha$.

Finally, we will omit floors and ceilings when they make no essential
difference.

##  2\. Proof of Theorem 1.1

###  2.1. Preliminaries

We collect some (by now) standard notions in the non-asymptotic theory of
random matrices. For parameters $\delta,\rho\in(0,1)$ and an integer $n\geq
1$, $\operatorname{Comp}_{n}(\delta,\rho)$ denotes the set of unit vectors in
$\mathbb{R}^{n}$ which have Euclidean distance at most $\rho$ to the set of
$\delta n$-sparse vectors.
$\operatorname{Incomp}_{n}(\delta,\rho):=\mathbb{S}^{n-1}\setminus\operatorname{Comp}_{n}(\delta,\rho)$.
When the ambient dimension is clear from context, we will drop the subscript
$n$.

We will need to consider the anticoncentration behavior of a vector with
respect to i.i.d. $\operatorname{Ber}(p)$ random variables. For this, we will
use the threshold function, which was isolated in the work of Tikhomirov [17].

######  Definition 2.1.

For $p\in(0,1/2]$, $L\geq 1$, and $x\in\mathbb{R}^{n}$, we define

$\mathcal{T}_{p}(x,L):=\sup\bigg{\\{}t\in(0,1):\mathcal{L}\bigg{(}\sum_{i=1}^{n}b_{i}x_{i},t\bigg{)}>Lt\bigg{\\}},$

where $b_{1},\dots,b_{n}$ are independent $\operatorname{Ber}(p)$ random
variables.

###  2.2. Overview of the proof

In this subsection, we present the (short) proof of Theorem 1.1, modulo the
key Propositions 2.2 and 2.7, which we will formally state and prove in the
subsequent subsections.

###### Proof of Theorem 1.1.

For $\delta,\rho\in(0,1)$, let $\mathcal{E}_{C}(\delta,\rho)$ be the event
that for every $(n-k)\times n$ sub-matrix $A$ of $M_{n}(\xi)$, and for all
$x\in\mathbb{S}^{n-1}$ such that $Ax=0$, we have that
$x\in\operatorname{Comp}(\delta,\rho)$. In words, the right-kernel unit
vectors of every $(n-k)\times n$ sub-matrix of $M_{n}(\xi)$ are in
$\operatorname{Comp}(\delta,\rho)$.

Denote the rows of $M_{n}(\xi)$ by $R_{1},\dots,R_{n}$. If
$\operatorname{rank}M_{n}(\xi)\leq n-k$, then there must be some $k$ rows of
$M_{n}(\xi)$ which are in the span of the remaining $n-k$ rows. Since the rows
are i.i.d, it follows from the union bound that

$\displaystyle\mathbb{P}[\operatorname{rank}M_{n}(\xi)\leq n-k]$
$\displaystyle\leq\mathbb{P}[\mathcal{E}_{C}(\delta,\rho)]+\mathbb{P}[\operatorname{rank}M_{n}(\xi)\leq
n-k\wedge\mathcal{E}_{C}(\delta,\rho)^{c}]$
$\displaystyle\leq\mathbb{P}[\mathcal{E}_{C}(\delta,\rho)]+\binom{n}{k}\mathbb{P}[\\{R_{1},\ldots,R_{k}\in\operatorname{span}(R_{k+1},\ldots,R_{n})\\}\wedge\mathcal{E}_{C}(\delta,\rho)^{c}].$

In Proposition 2.2, which is the key innovation of this work, we will show
that there exist $\delta,\rho\in(0,1)$ (depending on $k,p,\epsilon$) such that

$\mathbb{P}[\mathcal{E}_{C}(\delta,\rho)]\leq(1-p+\epsilon)^{kn}.$

Now, fix this choice of $\delta,\rho$ and denote the corresponding event
$\mathcal{E}_{C}(\delta,\rho)$ simply by $\mathcal{E}_{C}$. Since
$\binom{n}{k}\leq n^{k}\leq(1+\epsilon)^{n}$ for $n$ sufficiently large, it
remains to show that

$\mathbb{P}[\\{R_{1},\ldots,R_{k}\in\operatorname{span}(R_{k+1},\ldots,R_{n})\\}\wedge\mathcal{E}_{C}^{c}]\leq(1-p+\epsilon)^{nk}.$

In Proposition 2.7, we show the following dichotomy: consider the $(n-k)\times
n$ matrix $M_{(n-k)\times n}(\xi)$ formed by the rows $R_{k+1},\ldots,R_{n}$.
With probability at least $1-4^{-kn}$, either

  * •

every unit vector in the right-kernel of $M_{(n-k)\times n}(\xi)$ is in
$\operatorname{Comp}(\delta,\rho)$, or

  * •

there is a unit vector $v=v(R_{k+1},\ldots,R_{n})$ in the right-kernel of
$M_{(n-k)\times n}(\xi)$ with

$\mathcal{T}_{p}(v,L_{\ref{prop:structure}})\leq(1-p+\epsilon)^{n},$

where $L_{\ref{prop:structure}}$ is a constant depending on $k,p,\epsilon$.

Note that, on the event $\mathcal{E}_{C}^{c}$, the first case cannot occur.
Let $\mathcal{A}$ denote the set of possible realizations of
$R_{k+1},\ldots,R_{n}$ for which the second case occurs; for every such
realisation $a\in\mathcal{A}$, we have a unit vector $v=v(a)$ satisfying the
conclusion of the second case. Then,

$\displaystyle\mathbb{P}[R_{1},\ldots,R_{k}\in\operatorname{span}(R_{k+1},\ldots,R_{n})\wedge\mathcal{E}_{C}^{c}]$
$\displaystyle\leq\sup_{a\in\mathcal{A}}\mathbb{P}[\langle
R_{1},v(a)\rangle=\cdots=\langle R_{k},v(a)\rangle=0]+4^{-kn}$
$\displaystyle\leq L_{\ref{prop:structure}}^{k}(1-p+\epsilon)^{kn}+4^{-kn}.$

The last inequality uses the independence of the rows, the definition of the
threshold function, and the property
$\mathcal{T}_{p}(v(a),L_{\ref{prop:structure}})\leq(1-p+\epsilon)^{n}$. The
result now follows upon rescaling $\epsilon$. ∎

###  2.3. Compressible vectors

In this subsection, we prove Proposition 2.2. Recall the event
$\mathcal{E}_{C}(\delta,\rho)$ defined at the start of the proof of Theorem
1.1.

######  Proposition 2.2.

Fix $p\in(0,1/2]$, $k\geq 1$, and $\epsilon>0$. There exist
$\delta,\rho\in(0,1)$ (depending on $p,k,\epsilon$) such that

$\mathbb{P}[\mathcal{E}_{C}(\delta,\rho)]\leq(1-p+\epsilon)^{kn}.$

The proof of Proposition 2.2 requires two ingredients, the first of which is
the classical Kolmogorov-Lévy-Rogozin anticoncentration inequality.

######  Lemma 2.3 ([13]).

Let $\xi_{1},\dots,\xi_{n}$ be independent random variables. Then, for any
real numbers $r_{1},\dots,r_{n}>0$ and any real number
$r\geq\max_{i\in[n]}r_{i}$,

$\displaystyle\mathcal{L}\bigg{(}\sum_{i=1}^{n}\xi_{i},r\bigg{)}\leq\frac{C_{\ref{lem:LKR}}r}{\sqrt{\sum_{i=1}^{n}(1-\mathcal{L}(\xi_{i},r_{i}))r_{i}^{2}}},$

where $C_{\ref{lem:LKR}}>0$ is an absolute constant.

The second ingredient is a version of restricted invertibility which allows
one to select a subset of columns of full rank.

######  Lemma 2.4 ([5, Theorem 1]).

Let $U$ be an $n\times m$ matrix of rank $n$. Then, there exists a subset
$\mathcal{S}\subseteq[m]$ of columns of size $|\mathcal{S}|=n$ such that

$\lVert
U_{\mathcal{S}}^{-1}\rVert_{\operatorname{HS}}^{2}\leq(m-n+1)\cdot\operatorname{Tr}[(UU^{T})^{-1}].$

The following is the key lemma in the proof of Proposition 2.2.

######  Lemma 2.5.

Fix $p\in(0,1/2]$ and $k\geq 1$. There exists $\theta=\theta_{\ref{lem:levy-
estimate}}(p,k)>0$ for which the following holds. If $M$ is a $k\times n$
matrix whose rows are orthonormal vectors and $x$ is an $n$-dimensional random
vector with independent $\operatorname{Ber}(p)$ components, then

$\mathcal{L}(Mx,\theta)\leq(1-p)^{k}.$

######  Remark.

Our proof shows that $\theta(p,k)$ can be taken to be of size
$\operatorname{poly}(p)\exp(-\Omega(k))$. We suspect this is far from the
truth and conjecture that one can take $\theta(p,k)$ to be of size
$\operatorname{poly}(p,1/k)$.

###### Proof.

For each $1\leq i\leq k$, let $\mathcal{T}_{i}\subseteq[n]$ be the indices
$j\in[n]$ corresponding to the $\lfloor C_{\ref{lem:LKR}}^{2}25^{k}/p\rfloor$
(which is $\geq k$) largest values $|M_{ij}|$, where (for concreteness) we
break ties according to the natural ordering of the integers. Let
$\mathcal{T}=\cup_{i=1}^{k}\mathcal{T}_{i}$. Since
$\sum_{j=1}^{n}|M_{ij}|^{2}=1$ for every $i\in[k]$, it follows that for every
$j\in\mathcal{T}^{c}$,

$|M_{ij}|\leq 5^{-k}\sqrt{p}/C_{\ref{lem:LKR}}\quad\forall i\in[k].$

We have two cases.

Case I: There exists some $i\in[k]$ such that
$\sum_{j\in\mathcal{T}^{c}}|M_{ij}|^{2}\geq 4^{-k}$. In this case, by applying
Lemma 2.3 with $r=5^{-k}\sqrt{p}/(3C_{\ref{lem:LKR}})$ and $r_{j}=|M_{ij}|/3$,
we find that

$\mathcal{L}\bigg{(}\sum_{j\in\mathcal{T}^{c}}M_{ij}x_{j},r\bigg{)}\leq
C_{\ref{lem:LKR}}\frac{C_{\ref{lem:LKR}}^{-1}5^{-k}\sqrt{p}/3}{2^{-k}\sqrt{p/9}}<2^{-k}\leq(1-p)^{k}.$

Since

$\mathcal{L}(Mx,r)\leq\mathcal{L}((Mx)_{i},r)\leq\mathcal{L}\bigg{(}\sum_{j\in\mathcal{T}^{c}}M_{ij}x_{j},r\bigg{)},$

we have the required conclusion with
$\theta=r=5^{-k}\sqrt{p}/(3C_{\ref{lem:LKR}})$.

Case II: For every $i\in[k]$, $\sum_{j\in\mathcal{T}^{c}}|M_{ij}|^{2}\leq
4^{-k}$. Let $U=M_{\mathcal{T}}$ denote the $k\times|\mathcal{T}|$ matrix
formed by the columns of $M$ corresponding to $\mathcal{T}$. Note that
$k\leq|\mathcal{T}|\leq C_{\ref{lem:LKR}}^{2}k25^{k}/p$. Let
$C_{1},\dots,C_{k}$ denote the columns of $M$. Since $MM^{T}=I$, we have

$\displaystyle\lVert I-UU^{T}\rVert$ $\displaystyle=\lVert
MM^{T}-UU^{T}\rVert\leq\sum_{j\in\mathcal{T}^{c}}\lVert C_{j}C_{j}^{T}\rVert$
$\displaystyle=\sum_{j\in\mathcal{T}^{c}}\lVert
C_{j}\rVert_{2}^{2}=\sum_{i\in[k]}\sum_{j\in\mathcal{T}^{c}}|M_{ij}|^{2}$
$\displaystyle\leq k4^{-k}<1/2.$

In particular, the eigenvalues of $UU^{T}$ are in $(1/2,1]$ so that

$\operatorname{Tr}[(UU^{T})^{-1}]<2k.$

Therefore, by Lemma 2.4, there exists a set of columns
$\mathcal{S}\subseteq\mathcal{T}$ of size $|\mathcal{S}|=k$ such that

$\lVert M_{\mathcal{S}}^{-1}\rVert_{\operatorname{HS}}^{2}=\lVert
U_{\mathcal{S}}^{-1}\rVert_{\operatorname{HS}}^{2}\leq
2C_{\ref{lem:LKR}}^{2}k^{2}25^{k}/p.$

Hence, the smallest singular value of $M_{\mathcal{S}}$, denoted by
$\sigma_{k}(M_{\mathcal{S}})$, must satisfy

$\sigma_{k}(M_{\mathcal{S}})\geq\frac{\sqrt{p}}{2C_{\ref{lem:LKR}}k5^{k}}.$

We claim that for

$r=\frac{\sqrt{p}}{5C_{\ref{lem:LKR}}k5^{k}},$

we have that

$\mathcal{L}(Mx,r)\leq(1-p)^{k}.$

Since

$\mathcal{L}(Mx,r)\leq\mathcal{L}(M_{\mathcal{S}}x_{\mathcal{S}},r),$

it suffices to show that

$\mathcal{L}(M_{\mathcal{S}}x_{\mathcal{S}},r)\leq(1-p)^{k}.$

Since $x_{\mathcal{S}}$ is a $k$-dimensional vector with i.i.d.
$\operatorname{Ber}(p)$ entries, it follows that $x_{\mathcal{S}}$ is
supported on $\\{0,1\\}^{k}$ with maximum atom probability $(1-p)^{k}$.
Moreover, by definition of the smallest singular value, we see that for any
$x\neq y\in\\{0,1\\}^{k}$,

$\|M_{\mathcal{S}}x-M_{\mathcal{S}}y\|_{2}\geq\sigma_{k}(M_{\mathcal{S}})\|x-y\|_{2}\geq\sigma_{k}(M_{\mathcal{S}}),$

which shows that

$\mathcal{L}(M_{\mathcal{S}}x_{\mathcal{S}},r)\leq\max_{x\in\\{0,1\\}^{k}}\mathbb{P}[x_{\mathcal{S}}=x]=(1-p)^{k}.$

Thus, in either case, we can take

$\theta=\frac{\sqrt{p}}{5C_{\ref{lem:LKR}}k5^{k}}.\qed$

The previous lemma allows us to quickly deduce the following which, in the
special case $k=1$, is the usual ‘invertibility with respect to a single
vector’ (cf. [17, Lemma 3.5]).

######  Lemma 2.6.

Fix $p\in(0,1/2]$, $k\geq 1$, and $\epsilon>0$. Let
$\xi=\operatorname{Ber}(p)$. There exists $c_{\ref{lem:fixed-
vectors}}=c_{\ref{lem:fixed-vectors}}(p,k,\epsilon)>0$ for which the following
holds. Let $n\geq n_{\ref{lem:fixed-vectors}}(p,k,\epsilon)$ and let
$V=[v_{1},\ldots,v_{k}]$ be an $n\times k$ matrix with orthonormal columns.
Then,

$\mathbb{P}[\lVert M_{(n-k)\times n}(\xi)V\rVert_{\operatorname{HS}}\leq
c_{\ref{lem:fixed-vectors}}\sqrt{n}]\leq(1-p+\epsilon)^{kn}.$

###### Proof.

Denote the rows of $M_{(n-k)\times n}(\xi)$ by $R_{1},\ldots,R_{n-k}$. Let
$\theta=\theta_{\ref{lem:levy-estimate}}(p,k)$. Note that if

$\lVert M_{(n-k)\times
n}(\xi)V\rVert_{\operatorname{HS}}\leq\theta\sqrt{\epsilon^{\prime}(n-k)},$

then at most $\epsilon^{\prime}(n-k)$ rows $R_{i}$ can satisfy $\lVert
R_{i}V\rVert_{2}>\theta$. Denote the set of these rows by $\mathcal{I}$. By
Lemma 2.5 and the independence of the rows, we have for all $I\subseteq[n]$,
$|I|\leq\epsilon^{\prime}(n-k)$ that

$\displaystyle\mathbb{P}[\mathcal{I}=I]\leq\mathbb{P}[\lVert
R_{i}V\rVert_{2}\leq\theta\hskip 5.69054pt\forall i\in I^{c}]\leq\prod_{i\in
I^{c}}\mathbb{P}[\lVert
V^{T}(R_{i})^{T}\rVert_{2}\leq\theta]\leq(1-p)^{k\cdot(1-\epsilon^{\prime})(n-k)}.$

Therefore, by the union bound over the choice of $I$, we have

$\mathbb{P}[\lVert M_{(n-k)\times
n}(\xi)V\rVert_{\operatorname{HS}}\leq\theta\sqrt{\epsilon^{\prime}(n-k)}]\leq
n\cdot\binom{n-k}{\epsilon^{\prime}(n-k)}(1-p)^{k\cdot(1-\epsilon^{\prime})(n-k)}.$

Therefore taking $c_{\ref{lem:fixed-
vectors}}=\theta\sqrt{\epsilon^{\prime}}/2$ for sufficiently small
$\epsilon^{\prime}=\epsilon^{\prime}(p,k,\epsilon)$ and taking $n$
sufficiently large gives the desired conclusion. ∎

Combining this with a standard epsilon-net argument allows us to prove
Proposition 2.2.

###### Proof of Proposition 2.2.

Let $\mathcal{E}(\delta,\rho)$ denote the event that for every
$x\in\mathbb{S}^{n-1}$ such that

$(M_{(n-k)\times n}(\xi))x=0,$

we have $x\in\operatorname{Comp}(\delta,\rho)$. By the union bound,

$\mathbb{P}[\mathcal{E}_{C}(\delta,\rho)]\leq\binom{n}{k}\mathbb{P}[\mathcal{E}(\delta,\rho)],$

so that (after rescaling $\epsilon$) it suffices to show that

$\mathbb{P}[\mathcal{E}(\delta,\rho)]\leq(1-p+\epsilon)^{kn}.$

Since the right-kernel of $M_{(n-k)\times n}(\xi)$ has dimension at least $k$,
it follows that on the event $\mathcal{E}(\delta,\rho)$, we can find $k$
orthonormal vectors, $v_{1},\dots,v_{k}$, such that
$v_{i}\in\operatorname{Comp}(\delta,\rho)$ and $(M_{(n-k)\times
n}(\xi))v_{i}=0$. Let $V$ denote the $n\times k$ matrix with columns
$v_{1},\dots,v_{k}$. Then,

$M_{(n-k)\times n}(\xi)V=0.$ (2.1)

Let $H\subseteq\mathbb{R}^{n}$ denote the subspace of vectors
$x=(x_{1},\dots,x_{n})$ such that $x_{1}+\dots+x_{n}=0$. Let $\mathcal{E}_{K}$
be the event that the operator norm of $(M_{(n-k)\times n}(\xi))|_{H}$ (i.e.,
the linear operator from $H$ to $\mathbb{R}^{n-k}$ which coincides with
$M_{(n-k)\times n}(\xi)$ on its domain) is at most $K\sqrt{n}$. Since $\xi$ is
sub-Gaussian, it follows from standard estimates (cf. [17, Lemma 3.4]) that

$\mathbb{P}[\mathcal{E}_{K}]\geq 1-\exp(-c_{\xi}K^{2}n).$

In particular, by choosing $K$ to be of order $\sqrt{k}$, we can ensure that
this probability is at least $1-4^{-kn}$. Then, by the union bound, it
suffices to show that

$\mathbb{P}[\mathcal{E}(\delta,\rho)\cap\mathcal{E}_{K}]\leq(1-p+\epsilon)^{kn}.$

We will show this by combining Lemma 2.6 with a standard epsilon-net argument.
Let $\epsilon^{\prime}>0$ be a sufficiently small parameter to be chosen
later. A standard volumetric net, ‘densified’ in the all-ones direction (to
account for the fact that $M_{(n-k)\times
n}(\xi)((1,\dots,1)^{T}/\sqrt{n})=\Theta(n)$) shows (cf. [17, Proposition
3.6]) that there exists a (deterministic) net $\mathcal{N}$ of
$\operatorname{Comp}(\delta,\rho)$, of size at most
$(C/\epsilon^{\prime})^{\delta n}$ (where $C$ is allowed to depend on $K$),
such that for any $x\in\operatorname{Comp}(\delta,\rho)$, there exists
$y\in\mathcal{N}$ such that on the event $\mathcal{E}_{K}$,

$\|M_{(n-k)\times n}(\xi)(x-y)\|_{2}\leq(\epsilon^{\prime}+\rho)K\sqrt{n}.$

Let $\mathcal{V}$ denote the set of all $n\times k$ matrices whose columns are
orthonormal vectors in $\operatorname{Comp}(\delta,\rho)$. Then, by
considering the $k$-fold product of $\mathcal{N}$, we obtain a net of
$\mathcal{V}$. Using the standard trick of replacing points in this net by the
closest point in $\mathcal{V}$ (see e.g. [14, Lemma 4.2]), we can obtain a
(deterministic) net $\mathcal{M}\subseteq\mathcal{V}$ of size
$|\mathcal{M}|\leq(C/\epsilon^{\prime})^{\delta kn}$ such that for every
$V\in\mathcal{V}$, there exists $V^{\prime}\in\mathcal{M}$ such that, on the
event $\mathcal{E}_{K}$,

$\|M_{(n-k)\times n}(\xi)(V-V^{\prime})\|_{\operatorname{HS}}\leq
2(\epsilon^{\prime}+\rho)K\sqrt{kn}.$

Therefore, by 2.1 and the union bound, we have

$\displaystyle\mathbb{P}[\mathcal{E}(\delta,\rho)\cap\mathcal{E}_{K}]$
$\displaystyle\leq\mathbb{P}[\mathcal{E}_{K}\cap\\{\exists
V\in\mathcal{V}:M_{(n-k)\times n}(\xi)V=0\\}]$
$\displaystyle\leq\sum_{V^{\prime}\in\mathcal{M}}\mathbb{P}[\|M_{(n-k)\times
n}(\xi)V^{\prime}\|_{\operatorname{HS}}\leq
2(\epsilon^{\prime}+\rho)K\sqrt{kn}]$
$\displaystyle\leq\left(\frac{C}{\epsilon^{\prime}}\right)^{\delta
kn}\cdot(1-p+\epsilon/2)^{kn}$ $\displaystyle\leq(1-p+\epsilon)^{kn},$

where the penultimate line follows from Lemma 2.6 by taking
$\rho=\epsilon^{\prime}$, $n$ sufficiently large, and $\epsilon^{\prime}$
sufficiently small depending on $k,p,\epsilon$, and the last line follows by
taking $n$ sufficiently large and $\delta$ sufficiently small depending on
$\epsilon^{\prime},\epsilon$. ∎

###  2.4. Incompressible vectors

Recall the notion of the threshold of a vector (Definition 2.1). The following
is the structure theorem/dichotomy used in the proof of Theorem 1.1. The case
$k=1$ is implicit in [17], although the statement given here is closer to the
one in work of the authors [9, Proposition 3.7].

######  Proposition 2.7 (Modification of [9, Proposition 3.7]).

Let $\delta,\rho,\epsilon\in(0,1)$ and $k\geq 1$. There exist
$L_{\ref{prop:structure}}=L_{\ref{prop:structure}}(k,\delta,\rho,p,\epsilon)$
and
$n_{\ref{prop:structure}}=n_{\ref{prop:structure}}(k,\delta,\rho,p,\epsilon)$
such that for all $n\geq n_{\ref{prop:structure}}$, with probability at least
$1-4^{-kn}$, exactly one of the following holds.

  * •

Every unit vector $v$ in the right-kernel of $M_{(n-k)\times n}(\xi)$ is in
$\operatorname{Comp}(\delta,\rho)$, or

  * •

there is a unit vector $v$ in the right-kernel of $M_{(n-k)\times n}(\xi)$
with $\mathcal{T}_{p}(v,L_{\ref{prop:structure}})\leq(1-p+\epsilon)^{n}$.

######  Remark.

The proof of Proposition 2.7 follows the usual format of taking a dyadic
decomposition of possible values for the threshold function, performing
randomized rounding on potential kernel vectors at the correct scale, and then
tensorizing the resulting small ball probabilities. The difference in the
statement above compared to the versions in [17, 8, 9] is that we are missing
$k$ rows as opposed to $1$ row. Additionally, we are considering the
independent threshold model rather than the “multislice” models considered in
[9], which actually simplifies the proof. Further, since $k$ is sufficiently
small compared to $n$, the difference in row counts has essentially no effect
on the union bound computation.

## Acknowledgments

We thank Jordan Ellenberg for a Twitter question which motivated the present
work. We also thank Han Huang for sharing a preliminary version of [6] with
us.

## References

  * [1] Anirban Basak and Mark Rudelson, _Sharp transition of the invertibility of the adjacency matrices of sparse random graphs_ , arXiv:1809.08454.
  * [2] Jean Bourgain, Van H. Vu, and Philip Matchett Wood, _On the singularity probability of discrete random matrices_ , J. Funct. Anal. 258 (2010), 559–603.
  * [3] Amin Coja-Oghlan, Alperen A Ergür, Pu Gao, Samuel Hetterich, and Maurice Rolvien, _The rank of sparse random matrices_ , Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 579–591.
  * [4] Kevin P Costello and Van H Vu, _The rank of random graphs_ , Random Structures & Algorithms 33 (2008), 269–285.
  * [5] E. Gluskin and A. Olevskii, _Invertibility of sub-matrices and the octahedron width theorem_ , Israel J. Math. 186 (2011), 61–68.
  * [6] Han Huang, _Rank of sparse Bernoulli matrices_ , in preparation.
  * [7] Han Huang, _Singularity of Bernoulli matrices in the sparse regime $pn={O}(\log n)$_, arXiv:2009.13726.
  * [8] Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney, _Singularity of discrete random matrices I_ , arXiv:2010.06553.
  * [9] Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney, _Singularity of discrete random matrices II_ , arXiv:2010.06554.
  * [10] Jeff Kahn, János Komlós, and Endre Szemerédi, _On the probability that a random $\pm 1$-matrix is singular_, J. Amer. Math. Soc. 8 (1995), 223–240.
  * [11] J. Komlós, _On the determinant of $(0,\,1)$ matrices_, Studia Sci. Math. Hungar. 2 (1967), 7–21.
  * [12] Alexander E Litvak and Konstantin E Tikhomirov, _Singularity of sparse Bernoulli matrices_ , arXiv:2004.03131.
  * [13] B. A. Rogozin, _On the increase of dispersion of sums of independent random variables_ , Teor. Verojatnost. i Primenen 6 (1961), 106–108.
  * [14] Mark Rudelson, _Recent developments in non-asymptotic theory of random matrices_ , Modern aspects of random matrix theory, Proc. Sympos. Appl. Math., vol. 72, Amer. Math. Soc., Providence, RI, 2014, pp. 83–120.
  * [15] Terence Tao and Van Vu, _On random $\pm 1$ matrices: singularity and determinant_, Random Structures Algorithms 28 (2006), 1–23.
  * [16] Terence Tao and Van Vu, _On the singularity probability of random Bernoulli matrices_ , J. Amer. Math. Soc. 20 (2007), 603–628.
  * [17] Konstantin Tikhomirov, _Singularity of random Bernoulli matrices_ , Ann. of Math. (2) 191 (2020), 593–634.
  * [18] Van Vu, _Recent progress in combinatorial random matrix theory_ , arXiv:2005.02797.