# Dynamic melting and condensation of topological dislocation modes

Sanjib Kumar Das  Department of Physics, Lehigh University, Bethlehem,
Pennsylvania, 18015, USA  Bitan Roy  Department of Physics, Lehigh University,
Bethlehem, Pennsylvania, 18015, USA

###### Abstract

Bulk dislocation lattice defects are instrumental to identify translationally
active topological insulators (TATIs), featuring band inversion at a finite
momentum (${\bf K}_{\rm inv}$). TATIs host robust gapless modes around the
dislocation core, when the associated Burgers vector (${\bf b}$) satisfies
${\bf K}_{\rm inv}\cdot{\bf b}=\pi$ (modulo $2\pi$). From the time evolution
of appropriate density matrices, we show that when a TATI via a real time ramp
enters into a trivial or topological insulating phase, devoid of any gapless
dislocation mode, signatures of the preramp defect modes survive for a long
time. More intriguingly, as the system ramps into a TATI phase from any
translationally inert insulators, signature of the dislocation mode
dynamically builds up near the defect core, which is more prominent for slow
ramps. We exemplify these generic outcomes for two-dimensional time-reversal
symmetry breaking insulators. Proposed dynamic responses at the dislocation
core can be experimentally observed on quantum crystals, optical lattices and
metamaterials with time tunable band gap.

_Introduction_. Interfaces of quantum materials serve as a powerful tool to
identify topological crystals in nature. They feature robust gapless modes at
the edges and surfaces, for example, encoding the topological invariant of the
bulk electronic wavefunctions, manifesting a bulk boundary correspondence [1,
2]. Here we solely focus on topological insulators (TIs). The hallmark band
inversion in TIs, however, can take place at the center ($\Gamma$ point) or at
finite time reversal invariant momentum points of the Brillouin zone (BZ) [3,
4, 5, 6, 7, 8, 9]. Consequently, the landscape of TIs fragments according to
the underlying band inversion momentum (${\bf K}_{\rm inv}$). But, boundary
modes cannot distinguish them as they always exist at the interfaces of
topological crystals, irrespective of ${\bf K}_{\rm inv}$.

_Background_. Rather bulk topological lattice defects, such as dislocations,
being sensitive to ${\bf K}_{\rm inv}$, are instrumental in distinguishing
them. As dislocations are characterized by the nontrivial Burgers vector
(${\bf b}$), electrons encircling the defect core picks up a hopping phase
$\Phi_{\rm dis}={\bf K}_{\rm inv}\cdot{\bf b}$ [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23]. Evidently $\Phi_{\rm dis}=0$ in the $\Gamma$
phase, as ${\bf K}_{\rm inv}=0$ therein. If, on the other hand, ${\bf b}$ and
${\bf K}_{\rm inv}$ are such that $\Phi_{\rm dis}=\pi$ (modulo $2\pi$), a
nontrivial $\pi$ hopping phase threading the defect core binds localized
gapless topological electronic modes therein (Fig. 1) [12, 15], which have
also been observed in experiments [24, 25]. As dislocations are associated
with the breaking of the local translational symmetry in the bulk of crystals,
TIs harboring such defect modes are named translationally active topological
insulators (TATIs). This general principle is applicable to two- and three-
dimensional static and Floquet TIs and superconductors [10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

Figure 1: (a) Energy spectra of the static Hamiltonian [Eq. (2)] for
$t=t_{0}=-m_{0}=1$, yielding a TATI with the band inversion at the ${\rm M}$
point (${\rm M}$ phase), in the presence of an edge dislocation-
antidislocation pair with periodic boundary condition in the $x$ and $y$
directions. The system then supports a pair of zero energy modes (inset). (b)
The local density of states (LDOS) of these two modes are highly localized
near the defect cores. (c) Phase diagram of the static Hamiltonian. Ramps out
of (into) the ${\rm M}$ phase to (from) translationally inert insulators are
shown by solid (dashed) arrows.

_Broad questions_. Although unexplored thus far, with the recent progress at
the frontier of dynamic topological phases, such as the ones realized in
periodically driven Floquet materials [26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39], for example, the role of topological lattice defects in the
dynamic realm arises as a timely issue of fundamental importance. In this
context, we provide affirmative answers to the following questions. (a) Does
the signature of topological dislocation modes survive in translationally
inert insulators, reached from a TATI via a real time ramp? Even more
intriguingly, (b) can topological dislocation modes be dynamically generated
via a ramp, taking the system into a TATI phase from translationally inert
insulators? See Figs. 2-4.

Figure 2: Time evolution of the probability $P(t)$ [Eq. (4)] of finding the
dislocation mode in the presence of a real time ramp [Eq. (3)] that takes the
system from the ${\rm M}$ phase (with $m_{i}=-1$) to a translationally inert
TI with band inversion at the $\Gamma$ point [(a), (d)] or normal insulator
close to the ${\rm M}$ phase [(b), (e)] or $\Gamma$ phase [(c), (f)]. At $t=0$
the state is _pure_ (top), composed of a single dislocation mode or _mixed_
(bottom) with $N+1$ occupied single particle state of $H$ [Eq. (2)], named HF′
state, for various choices of the ramp speed ($\alpha$). Here $N$ is the total
number of sites in a square lattice system in the presence of a dislocation-
antidislocation pair. Therefore, signatures of the dislocation modes survive
for a long time.

_Key results_. To answer these questions, we subscribe to a paradigmatic
lattice model for two-dimensional time-reversal symmetry breaking insulators.
Besides featuring the translationally active ${\rm M}$ phase with the band
inversion at the ${\rm M}=(\pi,\pi)/a$ point of the BZ, it also accommodates
translationally inert normal or trivial insulators (with no band inversion) as
well as a TI with the band inversion at the $\Gamma=(0,0)$ point [Fig. 1(c)].
Here $a$ is the lattice constant of an underlying square lattice. Then from
the time ($t$) evolution of appropriate density matrix $\rho(t)$, governed by
the Heisenberg equation of motion

$~{}\frac{d\rho(t)}{dt}=-\frac{i}{\hbar}\;\left[H(t),\rho(t)\right],$ (1)

where the Hamiltonian $H(t)$ captures the real time ramp, we make the
following key observations. When the system is initially prepared in a TATI
phase with a pair of dislocation modes and the ramp takes it into one of the
translationally inert phases, gradually decaying signatures of the defect
modes survive for a long time. This outcome is qualitatively insensitive to
the nature of the final state. See Figs. 2 and 3. By contrast, when the ramp
takes a reverse course, taking the system into the ${\rm M}$ phase from one of
the translationally inert insulators, topological dislocation modes
dynamically condense near the defect cores. Probability of such dynamic
generation of the topological defect modes increases with decreasing ramp
speed, resembling the _adiabatic theorem_. However, dynamic nucleation of the
defect modes are most prominent when the ramp begins from a normal insulator,
residing close to the TATI or ${\rm M}$ phase. These findings are showcased in
Fig. 4. Throughout we set $\hbar=1$.

_Model_. The Hamiltonian for the lattice model is [40]

$~{}H=t_{1}\sum_{j=x,y}\sin(k_{j}a)\tau_{j}+\bigg{(}t_{0}\sum_{j=x,y}\cos(k_{j}a)-m_{0}\bigg{)}\tau_{z}.$
(2)

The vector Pauli matrix ${\bm{\tau}}=(\tau_{x},\tau_{y},\tau_{z})$ operates on
the orbital indices. Translationally active ${\rm M}$ and inert $\Gamma$
topological phases are realized for $-2<m/t_{0}<0$ and $0<m/t_{0}<2$,
respectively. Otherwise the system is a normal insulator. See Supplementary
Materials [41]. Only the ${\rm M}$ phase supports a pair of near zero energy
dislocation modes that are highly localized around the defect cores, when
${\bf b}=a{\bf e}_{x}$, for example [Fig. 1]. We denote them as
$\left|\Psi^{\rm dis}_{i}\right>$ with $i=1,2$. They are related via an
antiunitary particle-hole symmetry $\left|\Psi^{\rm
dis}_{1,2}\right>=\Theta\left|\Psi^{\rm dis}_{2,1}\right>$ up to an overall
unimportant phase, where $\Theta=\tau_{x}{\mathcal{K}}$ and ${\mathcal{K}}$ is
the complex conjugation, as $\\{H,\Theta\\}=0$ [42].

Figure 3: Time evolution of site resolved LDOS [Eq. (5)], computed from the
density matrix $\rho(t)$ at various time instants for a fixed ramp speed
$\alpha=1$ [Eq. (3)]. Here $\rho(0)$ is constructed from a single isolated
dislocation mode for $t=t_{0}=-m_{i}=1$ (pure state). The final phase is a TI
with the band inversion at the $\Gamma$ point [(a)-(d)], a normal insulator
residing close to the ${\rm M}$ phase [(e)-(h)] or the $\Gamma$ phase
[(i)-(l)]. Value of $m_{f}$ is quoted in each panel. Each row demonstrates
dynamic melting of localized dislocation mode due to the ramp. Compare with
Fig. 1(b) and the color scale therein. Such time evolutions lead to deeps
(peaks) in the probability of finding the initial dislocation modes when LDOS
appears prominently away from (near) the core of the lattice defect, as shown
in the first and third (second and fourth) columns of each row. See Fig. 2
(top row) for the complete time evolution of this modes. We implement the
lattice geometry shown in Fig. 1(b). Results are identical in the mixed HF′
state once the uniform background LDOS for the half-filled system is
subtracted.

The real time ramp under which $m_{0}\to m(t)$, where

$~{}m(t)=m_{i}+\left(m_{f}-m_{i}\right)\>\left[1-\exp(-\alpha t)\right]$ (3)

is characterized by the ramp speed $\alpha$. Time is measured in units of
$\alpha^{-1}$. This ramp profile with $m(0)=m_{i}$ and $m(t\to\infty)=m_{f}$,
where the subscript $i$ ($f$) stands for initial (final), allows the system to
interpolate between any two insulators appearing in the phase diagram of $H$
through band gap closing with tunable ramp speed [43]. This ramp does not
break any discrete lattice symmetry. When $m_{0}\to m(t)$ in Eq. (2), we
denote the time dependent Hamiltonian by $H(t)$, entering Eq. (1).

Figure 4: Top: Probability $P(t)$ of dynamic condensation of dislocation
modes. (a) Variation of $P(t)$ when the initial phase is a normal insulator
residing close to the ${\rm M}$ phase for various ramp speeds $\alpha$. (b)
Variation of $P(t)$ for two different band gaps (set by $m_{i}$) of the
initial phase [same as in (a)] for two specific choices of $\alpha$. (c) Same
as (a) but for a specific $\alpha$ and when the initial state features band
minima (inverted for $m_{i}=0.10$ or noninverted for $m_{i}=2.10$) near the
$\Gamma$ point. Bottom: (d)-(g) Difference between the initial and final site
resolved LDOS for various $\alpha$, confirming prominent dynamic generation of
the dislocation modes, comparable to those in the ${\rm M}$ phase in the
static system [Fig. 1(b)], for sufficiently slow ramp. See text for details.

_Results_. We set the stage with the discussion on a simple situation.
Consider an isolated _pure state_ $\left|\Psi^{\rm dis}_{1}\right>$, for which
the density matrix $\rho(0)=\left|\Psi^{\rm dis}_{1}\right>\left<\Psi^{\rm
dis}_{1}\right|$. Under the time evolution this state always remains pure.
Once the ramp [Eq. (3)] is switched on, at any given instant of time the
probability of finding the dislocation mode is

$~{}P(t)=\left<\Psi\right|\rho(t)\left|\Psi\right>,$ (4)

where $\left|\Psi\right>=\left|\Psi^{\rm dis}_{1}\right>$, and $\rho(t)$ is
obtained by numerically solving Eq. (1). The results are shown in Fig. 2
(top). Irrespective of the nature of the final translationally inert insulator
(topological or normal), we find that $P(t)$ remains appreciably finite for a
long time over a wide range of $\alpha$. For small $\alpha$, $P(t)$ decays
slowly, as the system then takes a longer time to escape the ${\rm M}$ phase.
For faster ramp $P(t)$ initially falls rapidly. But, it then shows oscillatory
behavior for a long time. These characteristics of $P(t)$ for sufficiently
large $\alpha$ are (almost) identical to those for the survival probability of
a single dislocation mode across a sudden quench [41, 44, 45]. Time evolution
of $P(t)$ reflects on the time dependence of the site resolved LDOS, computed
from the density matrix as

$~{}D_{i}(t)=\sum_{\tau=1,2}\left<i,\tau\right|\rho(t)\left|i,\tau\right>,$
(5)

where $i$ ($\tau$) is the site (orbital) index, and $\left|i,\tau\right>$ is
the single particle state vector at site $i$ with orbital $\tau$. The results
are shown in Fig. 3. Irrespective of the nature of the final insulator, we
find that LDOS spreads over the entire system with increasing time causing the
overall decay of $P(t)$ with $t$. Nonetheless, LDOS displays periodic peaks
and deeps at the dislocation core, where the initial defect mode
$\left|\Psi^{\rm dis}_{1}\right>$ was prominently localized, resulting in
oscillatory behavior of $P(t)$ shown in Fig. 2 (top).

However, in quantum materials midgap dislocation modes can only be occupied
upon filling all the negative energy bulk states. Therefore, to experimentally
observe the dynamic melting of defect modes, one needs to consider an
appropriate many-body ground state. Due to the particle-hole symmetry, a half-
filled system displays a uniform average electronic density equal to one in
the entire system irrespective of the presence or absence of dislocation modes
at any instant of time for any $\alpha$ [41]. To capture time evolution of the
dislocation modes, we therefore need to add one fermion to the half-filled
sea, a state denoted by HF′. In HF′, the LDOS shows a sharp peak at the
dislocation core in the ${\rm M}$ phase once the average background LDOS for
the half-filled system is subtracted. The density matrix for such a _mixed_
state in a system containing total $N$ sites reads

$~{}\rho(0)=\frac{1}{N+1}\sum^{N+1}_{i=1}\left|\Psi_{i}\right>\left<\Psi_{i}\right|,$
(6)

where $\left|\Psi_{i}\right>$ is an eigenstate of $H(0)$ with energy $E_{i}$.
Time evolution of this density matrix is obtained by numerically solving Eq.
(1), from which we compute the probability of finding the dislocation mode at
$t\geq 0$ from Eq. (4) with $\left|\Psi\right>=(\left|\Psi^{\rm
dis}_{1}\right>+\left|\Psi^{\rm dis}_{2}\right>)/\sqrt{2}$. The results are
shown in Fig. 2 (bottom), which is _identical_ to the ones we obtained for the
pure state [Fig. 2 (top)], except that $P(0)=(N+1)^{-1}$ since all the $(N+1)$
filled states in HF′ are equally probable to be occupied initially. And
min.$\\{P(t)\\}=1/(2N)$, which corresponds to a maximally disordered state
(all $2N$ single-particle states are equally probable to be occupied) with
maximal von-Neumann entropy [41]. The site resolved LDOS shows identical
behavior as in Fig. 3 for the pure state once the background of uniform LDOS
for the half-filled state is subtracted.

Finally, we showcase the dynamic buildup of dislocation modes when the real
time ramp brings the system, initially prepared in one of the translationally
inert insulating phases, to the ${\rm M}$ phase. Following the discussion from
the last paragraph, we immediately recognize that the initial system must be
prepared in the HF′ state, which is now devoid of any defect mode, with the
density matrix $\rho(0)$ given in Eq. (6). From $\rho(t)$, we then compute the
probability of finding the dislocation modes [Eq. (4)] and site resolved LDOS
[Eq. (5)] at any instant of time. Results are shown in Fig. 4.

Dynamic condensation of dislocation modes is most prominent when the initial
state is a normal insulator, residing close to the ${\rm M}$ phase, with the
noninverted band minima near the ${\rm M}$ point. Recall that dislocation
modes originate from the band inversion at the ${\rm M}$ point (${\bf
K}\cdot{\bf b}$ rule), which is easier to achieve dynamically when the minima
of noninverted bands is near the ${\rm M}$ point initially, as this process
requires a small momentum transfer. Otherwise, with slower ramp speed the
probability of dynamic generation of the dislocation modes increases [Fig.
4(a)], which is insensitive to the magnitude of initial band gap [Fig. 4(b)].
These observations reconcile with the adiabatic theorem: For adiabatically
slow ramp, the system should always find itself in the instantaneous ground
state. As a consequence, for sufficiently slow ramp, the hallmark dislocation
modes of the translationally active ${\rm M}$ phase develops more prominently
as $t\to\infty$. These conclusions are further substantiated from the
difference between initial and final LDOS at each site for various $\alpha$,
shown in Fig. 4(d)-(g). Remarkably, the LDOS near the dislocation cores
recovers more than 95% of the weight of original dislocation modes in the
${\rm M}$ phase of the static system for sufficiently slow ramp
($\alpha=0.01$). Such recovery of dislocation modes, however, gets weaker with
increasing ramp speed. By contrast, when the system is initially prepared in
an insulating phase, featuring minima of inverted or noninverted bands near
the $\Gamma$ point, the probability of dynamic generation of the dislocation
modes remains negligibly small even for sufficiently slow ramp [Fig. 4(c)], as
the requisite dynamic band inversion at the ${\rm M}$ point now demands a
_large_ momentum transfer.

_Discussions & outlooks_. Defects are ubiquitous in crystals. Here we outline
a theoretical framework to capture dynamic melting and condensation of
topological dislocation modes when a real time ramp takes the system out of
and into a TATI, respectively. The proposed formalism based on the time
dynamic of appropriate density matrix governed by the Heisenberg equation of
motion is sufficiently general that can be extended to any topological system,
including topological superconductors, in arbitrary dimension and belonging to
arbitrary symmetry class to capture similar dynamic phenomena for boundary
(such as edge, surface, hinge and corner) as well as defect (line dislocation
and grain boundaries, for example) modes. Our model Hamiltonian also describes
a $p_{x}+ip_{y}$ superconductor with amplitude $t_{1}$, and the band inversion
near the $\Gamma$ and ${\rm M}$ points manifests Fermi surfaces near them.
Then a weak coupling topological BCS pairing sets in and dislocation cores
host localized Majorana modes. Tunable time dynamics of localized Majorana
modes can in principle be useful in quantum computation/information
technologies, facilitating information storage over a desired time. These
fascinating avenues will be explored in future. Meanwhile, it will be
worthwhile to investigate the impact of _dephasing_ in these open quantum
systems, which may boost the revival probability of dislocation modes when its
dynamic generation requires a large momentum transfer.

The requisite time modulated band gap in quantum crystals and cold atomic
lattices can, for example, be accomplished by time-dependent hydrostatic
pressure and on-site staggered-orbital potential, respectively, allowing a
smooth interpolation between topologically distinct insulators. And predicted
dynamic melting and more intriguing condensation of dislocation modes can be
detected from time-dependent LDOS, measured near the defect cores by scanning
tunneling spectroscopy. In metamaterials, the time-dependent band gap can be
engineered by (a) coupling electric nodes containing capacitors and/or
inductors with resistors in active topolectric circuits to minimize gain and
loss, such that the circuit decay constant provides the desired profile of
$m(t)$ [46] or (b) a time-dependent ratio of the distance between intra unit-
cell magnetomechanical resonators (optical waveguides) to the effective
lattice constant of mechanical (photonic) lattices. In these metacrystals,
such ratios have been tuned to trigger transitions between topological and
normal insulators, and lattice defects as well have been engineered, featuring
robust defect modes [47, 48, 49, 50]. The proposed dynamic responses of
defects can be measured from time-dependent impedance (in topolectric
circuits), mechanical susceptibility (in mechanical system), and two-point
pump probe (in photonic lattices), all of which mimic the LDOS. With the
recent progress on realizing Floquet topological phases in quantum [31, 32,
33] and meta [34, 35, 36, 37, 38, 39] materials, the ramp dynamics and dynamic
build-up of topological defect modes should be within the reach of current
experimental facilities.

_Acknowledgments_. This work was supported by a Startup grant of B.R. from
Lehigh University.

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$n=1000$ | $100$ | $32$ | $8.7$ | $6.0$ | $5.3$ | $5.3$ | $5.2$
$n=5000$ | $100$ | $58$ | $9.9$ | $6.0$ | $5.5$ | $4.9$ | $4.8$
% | $c=10$
$n=500$ | $99$ | $99$ | $96$ | $64$ | $29$ | $10$ | $5.5$
$n=1000$ | $100$ | $100$ | $98$ | $66$ | $26$ | $8$ | $5.2$
$n=5000$ | $100$ | $100$ | $100$ | $71$ | $23$ | $6$ | $5.4$

Table 3: We report the rejection frequency in % of statistic
$\mathscr{S}^{*}(2)$ for each combination of constants $c$ and $\kappa$ in the
parameterization of the beta variance $\sigma_{\beta,3}^{2}=cn^{-\kappa}$ of
the third factor, and cross-sectional size $n$. The time series dimension is
$T=6$.

Rejection rates of $\sqrt{n}\,\mathscr{S}(k)$, $k=2$, for DGP3

$\kappa$ | $0$ | $0.25$ | $0.4$ | $0.5$ | $0.6$ | $0.75$ | $1$
---|---|---|---|---|---|---|---
| strong | semi-strong | weak | vanishing
% | $c=0.1$
$n=500$ | $85$ | $6.8$ | $6.4$ | $5.9$ | $6.1$ | $6.0$ | $5.7$
$n=1000$ | $100$ | $8.7$ | $6.3$ | $5.8$ | $6.1$ | $5.9$ | $5.3$
$n=5000$ | $100$ | $14$ | $5.7$ | $5.3$ | $5.4$ | $5.3$ | $4.7$
% | $c=1$
$n=500$ | $100$ | $100$ | $79$ | $20$ | $9.2$ | $6.7$ | $6.0$
$n=1000$ | $100$ | $100$ | $78$ | $22$ | $8.1$ | $5.6$ | $5.2$
$n=5000$ | $100$ | $100$ | $96$ | $20$ | $6.9$ | $5.1$ | $5.3$
% | $c=10$
$n=500$ | $100$ | $100$ | $100$ | $100$ | $100$ | $85$ | $8.0$
$n=1000$ | $100$ | $100$ | $100$ | $100$ | $100$ | $68$ | $6.1$
$n=5000$ | $100$ | $100$ | $100$ | $100$ | $100$ | $30$ | $5.5$

Table 4: We report the rejection frequency in % of statistic
$\sqrt{n}\mathscr{S}(2)$ for each combination of constants $c$ and $\kappa$ in
the parameterization of the beta variance $\sigma_{\beta,3}^{2}=cn^{-\kappa}$
of the third factor, and cross-sectional size $n$. The time series dimension
is $T=6$. The errors follow individual ARCH(1) processes.

Rejection rates of $\mathscr{S}^{*}(k)$, $k=2$, for DGP3

$\kappa$ | $0$ | $0.25$ | $0.4$ | $0.5$ | $0.6$ | $0.75$ | $1$
---|---|---|---|---|---|---|---
| strong | semi-strong | weak | vanishing
% | $c=0.1$
$n=500$ | $8.8$ | $5.4$ | $5.6$ | $5.1$ | $5.7$ | $5.3$ | $6.1$
$n=1000$ | $15$ | $5.4$ | $5.5$ | $5.0$ | $4.8$ | $5.1$ | $5.1$
$n=5000$ | $53$ | $5.3$ | $5.3$ | $5.4$ | $5.0$ | $5.0$ | $5.6$
% | $c=1$
$n=500$ | $98$ | $26$ | $9.1$ | $5.6$ | $5.4$ | $5.9$ | $5.4$
$n=1000$ | $100$ | $33$ | $9.0$ | $6.1$ | $5.1$ | $5.6$ | $5.2$
$n=5000$ | $100$ | $65$ | $9.4$ | $5.8$ | $5.2$ | $5.3$ | $6.1$
% | $c=10$
$n=500$ | $98$ | $98$ | $98$ | $73$ | $30$ | $9.2$ | $5.6$
$n=1000$ | $100$ | $100$ | $99$ | $78$ | $34$ | $8.1$ | $5.4$
$n=5000$ | $100$ | $100$ | $100$ | $76$ | $24$ | $6.4$ | $5.3$

Table 5: We report the rejection frequency in % of statistic
$\mathscr{S}^{*}(2)$ for each combination of constants $c$ and $\kappa$ in the
parameterization of the beta variance $\sigma_{\beta,3}^{2}=cn^{-\kappa}$ of
the third factor, and cross-sectional size $n$. The time series dimension is
$T=12$. The errors follow individual ARCH(1) processes.

Size and power of $n\mathscr{T}(k)$ under DGP4

|  | Size ($\%$) | Power ($\%$)
---|---|---|---
| $T$ | 6 | 12 | 24 | 6 | 12 | 24
$\hat{\Sigma}_{U,1}$ | $n=500$ | 4.58 | 4.78 | 4.83 | 99 | 100 | 100
|  | (0.54) | (0.19) | (0.22) | (7.0) | (0.0) | (0.0)
| $n=1000$ | 4.83 | 4.91 | 4.94 | 100 | 100 | 100
|  | (0.23) | (0.22) | (0.19) | (4.8) | (0.0) | (0.0)
| $n=5000$ | 4.97 | 5.00 | 5.03 | 100 | 100 | 100
|  | (0.23) | (0.21) | (0.21) | (0.0) | (0.0) | (0.0)
$\hat{\Sigma}_{U,2}$ | $n=500$ | 3.94 | 3.37 | 2.30 | 99 | 100 | 100
|  | (0.42) | (0.19) | (0.16) | (7.0) | (0.0) | (0.0)
| $n=1000$ | 4.49 | 4.10 | 3.36 | 100 | 100 | 100
|  | (0.23) | (0.19) | (0.18) | (0.0) | (0.0) | (0.0)
| $n=5000$ | 4.95 | 4.80 | 4.64 | 100 | 100 | 100
|  | (0.23) | (0.21) | (0.18) | (0.0) | (0.0) | (0.0)

Table 6: For each sample size combination $(n,T)$, we provide the size and
power in $\%$ under DGP4 for the test statistic based on instruments. Nominal
size is $5\%$. Size and power refer to rejection frequencies for statistics
${n}\mathscr{T}(k)$ with $k=3$ and $k=2$, respectively. In parentheses, we
report the standard deviations for size and power across $100$ different draws
of the factor path $F$ and the matrix $\Gamma$ linking betas to instruments.
The upper panel uses the estimator
$\hat{\Sigma}_{U,1}=\hat{\sigma}^{2}[I_{T}\otimes\hat{Q}_{zz}]$ for simulating
the critical values, and the lower panel uses
$\hat{\Sigma}_{U,2}=\frac{1}{n}\sum_{i=1}^{n}[(\hat{\varepsilon}_{i}\hat{\varepsilon}^{\prime}_{i})\otimes(z_{i}z^{\prime}_{i})]$.