Dynamic melting and condensation of topological dislocation modes Sanjib Kumar Das1and Bitan Roy1

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Dynamic melting and condensation of topological dislocation modes

Sanjib Kumar Das1and Bitan Roy1

1Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA

(Dated: November 3, 2023)

Bulk dislocation lattice defects are instrumental in identifying translationally active topological

insulators (TATIs), featuring band inversion at a ﬁnite momentum (Kinv ). As such, TATIs host

robust gapless modes around the dislocation core, when the associated Burgers vector bsatisﬁes

Kinv ·b=π(modulo 2π). From the time evolution of appropriate density matrices, we show that

when a TATI via a real time ramp enters into a trivial or translationally inert topological insulating

phase, devoid of gapless dislocation modes, the 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

insulator, signature of the dislocation mode dynamically builds up near its core, which is 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 in quantum crystals, optical lattices and metamaterials with time a tunable band gap.

I. INTRODUCTION AND BACKGROUND

Interfaces of quantum materials serve as a powerful

tool to identify topological crystals in nature. They fea-

ture 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 topolog-

ical insulators (TIs). The hallmark band inversion in

TIs, however, can take place at the center (Γ point) or

at ﬁnite time reversal invariant momentum points of the

Brillouin zone (BZ) [3–9]. Consequently, the landscape

of TIs fragments according to the underlying band in-

version momentum (Kinv). However, boundary modes

cannot distinguish them as they always exist at the in-

terfaces of topological crystals, irrespective of Kinv.

Bulk topological lattice defects, such as dislocations,

being sensitive to Kinv, are instrumental in distinguish-

ing TIs. As dislocations are characterized by the non-

trivial Burgers vector (b), electrons encircling the defect

core picks up a hopping phase Φdis =Kinv ·b[10–24].

Evidently Φdis = 0 in the Γ phase, as Kinv = 0 therein.

If, on the other hand, band Kinv are such that Φdis =π

(modulo 2π), a nontrivial πhopping phase threading the

defect core binds localized gapless topological electronic

modes therein (Fig. 1) [12, 15], which have also been

observed in experiments [25, 26]. As dislocations are as-

sociated with the breaking of the local translational sym-

metry in the bulk of crystals, TIs harboring such defect

modes are named translationally active topological insu-

lators (TATIs). This general principle is applicable to

two- and three-dimensional static and Floquet TIs and

superconductors [10–26].

II. BROAD QUESTIONS AND KEY RESULTS

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 [27–

40], for example, the role of topological lattice defects in

the dynamic realm arises as a timely issue of fundamen-

tal importance. In this context, we provide aﬃrmative

answers to the following questions. (a) Does the signa-

ture of topological dislocation modes survive in transla-

tionally inert insulators, reached from a TATI via a real

time ramp? (b) Even more intriguingly, can topological

dislocation modes be dynamically generated via a ramp,

taking the system into a TATI phase from translationally

inert insulators?

FIG. 1. (a) Energy spectra of the static Hamiltonian [Eq. (2)]

for t=t0=−m0= 1, yielding a TATI with the band inver-

sion at the M point (M phase), in the presence of an edge

dislocation-antidislocation pair with the periodic boundary

condition in the xand ydirections. 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 M phase to (from) translationally in-

ert insulators are shown by solid (dashed) arrows labeled by

Roman numerals. The corresponding melting and condensa-

tion of dislocation modes are shown in Figs. 2-4. Here, TI

(NI) corresponds to topological (normal) insulator.

arXiv:2210.15661v2 [cond-mat.mes-hall] 2 Nov 2023

FIG. 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 M phase (with

mi=−1) to a translationally inert TI with band inversion

at the Γ point [(a) and (d)] or normal insulator close to the

M phase [(b) and (e)] or Γ phase [(c) and (f)]. At t= 0 the

state is pure (top), composed of a single dislocation mode or

mixed (bottom) with N+ 1 occupied single particle states

(two dislocation modes and N−1 bulk states) of H[Eq. (2)],

named the HF′state, for various choices of the ramp speed

α. Here Nis 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. The Roman numeral in each panel corresponds to

the arrow out of the M phase shown in Fig. 1.

To answer these questions, we subscribe to a paradig-

matic lattice model for two-dimensional time-reversal

symmetry breaking insulators. Besides featuring the

translationally active M phase with the band inversion at

the M = (π, π)/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 Γ = (0,0) point [Fig. 1(c)]. Here ais the lattice

constant of an underlying square lattice. Then from the

time (t) evolution of the appropriate density matrix ρ(t),

governed by the von Neumann equation

dρ(t)

dt =−i

ℏ[H(t), ρ(t)] ,(1)

where the Hamiltonian H(t) captures the real time ramp,

we make the following key observations. When the sys-

tem 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 sig-

natures 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

M phase from one of the translationally inert insulators,

topological dislocation modes dynamically condense near

the defect cores. The probability of such dynamic gener-

ation of the topological defect modes increases with de-

creasing ramp speed, resembling the adiabatic theorem.

FIG. 3. Time evolution of the site resolved LDOS [Eq. (5)],

computed from the density matrix ρ(t) at various time in-

stants for a ﬁxed ramp speed α= 1 [Eq. (3)]. Here, ρ(0)

is constructed from a single isolated dislocation mode for

t=t0=−mi= 1 (pure state). The ﬁnal phase is a TI with

the band inversion at the Γ point [(a)-(d)], a normal insulator

residing close to the M phase [(e)-(h)] or the Γ phase [(i)-(l)].

The value of mfis quoted in each panel. Each row demon-

strates dynamic melting of the localized dislocation mode due

to the ramp. Compare with Fig. 1(b) and the color scale

therein. Such time evolutions lead to valleys (peaks) in the

probability of ﬁnding the initial dislocation modes when the

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 Figs. 2(a)- 2(c) for the

complete time evolution of these 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 Roman numerals in

each panel corresponds to the arrow out of the M phase shown

in Fig. 1.

However, dynamic nucleation of the defect modes is most

prominent when the ramp begins from a normal insula-

tor, residing close to the TATI or M phase. These ﬁnd-

ings are showcased in Fig. 4. Throughout this paper we

set ℏ= 1.

III. MODEL

The Hamiltonian for the lattice model is [41]

H=t1X

j=x,y

sin(kja)τj+t0X

j=x,y

cos(kja)−m0τz.

(2)

The vector Pauli matrix τ= (τx, τy, τz) operates on the

orbital indices. Translationally active M and inert Γ

topological phases are realized for −2< m/t0<0 and

0< m/t0<2, respectively. Otherwise the system is a

normal insulator. See Supplemental Material [42]. Only

the M phase supports a pair of near zero energy dislo-

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DynamicmeltingandcondensationoftopologicaldislocationmodesSanjibKumarDas1andBitanRoy11DepartmentofPhysics,LehighUniversity,Bethlehem,Pennsylvania,18015,USA(Dated:November3,2023)Bulkdislocationlatticedefectsareinstrumentalinidentifyingtranslationallyactivetopologicalinsulators(TATIs),featuringbandinv...

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Dynamic melting and condensation of topological dislocation modes Sanjib Kumar Das1and Bitan Roy1

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