Exact results for quench dynamics and defect production in a two-dimensional model

0.28 MB·English

by  K. Sengupta

Exact results for quench dynamics and defect production in a two-dimensional model
K. Sengupta1, Diptiman Sen2 and Shreyoshi Mondal1
1 TCMP division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India
2 Center for High Energy Physics, Indian Institute of Science, Bangalore, 560 012, India
(Dated: February 2, 2008)
We show that for a d-dimensional model in which a quench with a rate τ−1 takes the system
across a d−m dimensional critical surface, the defect density scales as n∼1/τmν/(zν+1), where ν
8 andz arethecorrelationlengthanddynamicalcriticalexponentscharacterizingthecriticalsurface.
0 Weexplicitly demonstrate that theKitaev model provides an example of such ascaling with d=2
0 and m = ν = z = 1. We also provide the first example of an exact calculation of some multispin
2 correlation functions for a two-dimensional model which can be used to determine the correlation
between the defects. Wesuggest possible experimentsto test ourtheory.
n
a
J PACSnumbers: 73.43.Nq,05.70.Jk,64.60.Ht,75.10.Jm

5
1
Quantum phase transitions have been studied exten- eters. Such an exact analysis of defect correlations has

] sively for several years [1]. Such a transition is accom- not been carried out so far for 2D systems.
h panied by diverging length and time scales [1] leading to The Kitaev model is a spin-1/2 model on a 2D honey-
c
the absence of adiabaticity close to the quantum critical comb lattice with the Hamiltonian [10]
e
m point. Thus the system fails to follow its instantaneous
groundstate when some parameterin its Hamiltonian is H = (J σx σx +J σy σy +J σz σz ),
- 1 j,l j+1,l 2 j 1,l j,l 3 j,l j,l+1
at varied in time at a finite rate 1/τ which takes the sys- j+Xl=even −
t tem across the critical point. Since the final state of the (1)
s
system does not conform to the ground state of its final where j and l denote the column and row indices of
.
at Hamiltonian,defectsareproduced[2,3]. Thedefectden- the lattice. This model, sketched in Fig. 1, is known
m sityndependsonthequenchtimeτ asn 1/τdν/(νz+1), to have several interesting features [12, 13, 14, 15]. It
∼
- where ν and z are the correlation length and dynamical is a rare example of a 2D model which can be exactly
d critical exponents at the critical point [4]. A theoretical solved [10, 12, 14]. It supports a gapless phase for
n
o studyofsuchaquenchdynamicsrequiresaknowledgeof |J1 − J2| ≤ J3 ≤ J1 + J2 [10] which is possibly con-
theexcitedstatesofthesystem. Suchstudieshavethere- nected to a spin liquid state and demonstrates fermion
c
[ forebeenmostlyrestrictedtophasetransitionsinexactly fractionalization at all energy scales [13]. In certain pa-

solvable models in one or infinite dimensions [5, 6, 7, 8]. rameter regimes, the ground state exhibits topological
2
v Experimentalstudiesofdefectproductionduetoquench- order and the low-energy excitations carry Abelian and
2 ingofthemagneticfieldinatwo-dimensional(2D)spin-1 non-Abelian fractional statistics; these excitations can
1 Bosecondensate havebeenundertaken[9]. However,ex- be viewed as robust qubits in a quantum computer [11].
7 actstudiesofquenchdynamicshavenotbeencarriedout There have been proposals for experimentally realizing
1
so far for 2D spin models. this model in systems of ultracold atoms and molecules
.
0 trapped in optical lattices [16]; such systems are known
1 In this Letter, we carry out such a study for the 2D
to provide easy access to the study of non-equilibrium
7 Kitaev model[10]. We show that whenthe quenchtakes
dynamics of the underlying model. However, in spite of
0
the system across a critical (gapless) line, the density of
severalstudies of the phases and low-lying excitations of
:
v defects scales as 1/√τ. The Kitaevmodel has d=2 and
the Kitaev model, its non-equilibrium dynamics has not
i ν = z = 1; hence, the above scaling is in contrast to
X been studied so far. We will study what happens in this
the n 1/τ behavior expected when the system passes
r ∼ model when J3 is varied from to at a rate 1/τ,
a through a critical point [4]. In this context we provide keeping J and J fixed. −∞ ∞
1 2
a general discussion of the scaling of the defect density
One of the main properties of the Kitaev model which
for a general d-dimensional model with arbitrary ν and
makes it theoretically attractive is that, even in 2D, it
z; we show that when the quench takes such a system
can be mapped onto a non-interacting fermionic model
through a d m dimensional gapless (critical) surface,
by a suitable Jordan-Wigner transformation [12, 14],
−
the defect density scales as n 1/τmν/(zν+1). This re-
∼
sultis ageneralizationofthe resultofRef. [4]andhence
H =i [J b a +J b a +J D b a ], (2)
constitutes a significant extension of our current under- F X 1 ~n ~n−M~1 2 ~n ~n+M~2 3 ~n ~n ~n
~n
standing of defect production due to a quench. We also
computeexactlysomemultispincorrelationfunctionsfor where a~n and b~n are Majorana fermions sitting at the
ourmodel, anduse themto study the variationofdefect top and bottom sites respectively of a bond labeled ~n,
correlations with the quench rate and the model param- ~n=√3ˆi n +(√3ˆi+ 3ˆj) n denote the midpoints of the
1 2 2 2
2
FIG. 1: Schematic representation of the Kitaev model on
a honeycomb lattice showings the bonds J , J and J .
1 2 3
Schematic pictures of the ground states, which correspond
to pairs of spins on vertical bonds locked parallel (antiparal-
lel) to each other in the limit of large negative (positive) J ,
3
are shown at one bond on the left (right) edge respectively. FIG. 2: Plot of defect density n versus Jτ and α =
M~1 and M~2 are spanning vectors of the lattice, and a and b tan−1(J2/J1). ThedensityofdefectsismaximumatJ1 =J2.
represent inequivalentsites.
values of~k leading to a gapless phase [10, 12, 14, 15].
vertical bonds shown in Fig. 1, and n1,n2 run over all We will now quench J3(t)=Jt/τ from to at a
integers. The vectors ~n form a triangular lattice. The fixedrate1/τ,keepingJ,J andJ fixedat−s∞omep∞ositive
1 2
x,y coordinates of the triangular lattice sites are given values. The ground states of H corresponding to J
F 3
by x = √3(n1 +n2/2) and y = 3n2/2. [We will refer ( ), schematically shown in Fig. 1, are gapped an→d
to the sites of the honeycomb lattice either as (j,l) as h−a∞ve∞σz σz =1( 1)foralllatticesites(j,l)oftypeb.
in Eq. (1), or as (a,~n) and (b,~n) as in Eq. (2); j +l To studj,yl tj,hl+e1time e−volution of the system, we note that
is even (odd) for a (b) sites respectively.] The vectors after an unitary transformation U = exp( iσ1π/4), we
Mf~or1t=he√2r3eˆic+ipr32oˆjcaalnldatMt~ic2e=. T√2h3eˆi−op23eˆjraatroersDpancnainngtavkeecttohres obtain HF = ~kψ~k′†H~k′ψ~k′, where H~k′ =UH−~kU† is given
~n by H = 2[J Psin(~k M~ ) J sin(~k M~ )]σ1+2[J (t)+
values 1 independently for each ~n and commutes with ~k′ 1 · 1 − 2 · 2 3
HF,so±thatthestatescanbelabeledbythevaluesofD~n J1cos(~k·M~1)+J2cos(~k·M~2)]σ3. Hence the off-diagonal
on each bond; the ground state corresponds to D~n = 1 elementsofH~k′ remaintimeindependent,andthequench
on all bonds [10, 12, 13, 14] irrespective of the sign of J dynamics reduces to a Landau-Zener problem for each
3
due toaspecialsymmetryofthe model[10]. Since D is ~k. The defect density can then be computed following a
~n
aconstantofmotion,thedynamicsofthemodelstarting standard prescription [17]: n=(1/A) ~k d2~k p~k, where
fromthegroundstatenevertakesthesystemoutsidethe R
manifold of states with D =1. p =exp[ 2πτ J sin(~k M~ ) J sin(~k M~ ) 2/J] (4)
~n ~k − { 1 · 1 − 2 · 2 }
For D = 1, Eq. (2) can be diagonalized as H =
~n F
is the probability of defect production for the state la-
~kψ~k†H~kψ~k, where ψ~k† = (a~†k, b~†k) are Fourier trans- beled by momentum~k, and A= 4π2/(3√3) denotes the
fPorms of a and b , the sum over ~k extends over half
~n ~n area of half the BZ over which the integration is carried
the Brillouin zone (BZ) of the triangular lattice formed
out. A plot of n as a function of the quench time Jτ
by the vectors ~n, and H can be expressed in terms of
~k and an angle α is shown in Fig. 2; here we have taken
the Pauli matrices σi (where σ3 is diagonal) as H =
~k J =Jcosα[sinα]. Wenotethatthedensityofdefects
2[J sin(~k M~ ) J sin(~k M~ )]σ1+2[J +J cos(~k M~ )+ 1[2]
1 1 2 2 3 1 1 producedismaximumwhenα=π/4(J =J ). Thisoc-
· − · · 1 2
J2cos(~k M~2)]σ2. The spectrum consists of two bands cursbecausethelengthofthegaplesslinethroughwhich
·
with energies E~k± =±E~k, where the system passes during the quench is maximum for
J =J . Hence the system remains in the non-adiabatic
1 2
E~k = 2[{J1sin(~k·M~1)−J2sin(~k·M~2)}2 state for the maximum time during the quench, leading
+ J +J cos(~k M~ )+J cos(~k M~ ) 2]1/2(3) to the maximum density of defects. Note that unlike
3 1 1 2 2
{ · · } some other models [5], the defects here do not corre-
For J1 J2 J3 J1+J2,thebandstoucheachother, spond to topological defects since the dynamics always
and|the−ene|rg≤y ga≤p ∆~k = E~k+−E~k− vanishes for special keeps D~n =1 on all bonds.
3
For sufficiently slow quench 2πJτ 1, p is expo-
≫ ~k
nentially small for all values of ~k except near the line
J sin(~k M~ ) = J sin(~k M~ ); the contribution to the
1 1 2 2
· ·
momentum integral in the expression for n comes from
this region. Note that the line p = 1 precisely corre-
~k
sponds to the zeros of the energy gap ∆ as J is varied
~k 3
for fixed J ,J . By expanding p about this line, we see
1 2 ~k
that for a very slow quench, the defect density scales as
n 1/√τ. This demonstrates that the scaling of n with
∼
τ crucially depends on the dimensionality of the critical
surfacesinceforaquenchwhichtakesthesystemthrough
acriticalpointinsteadofacriticalline,thedefectdensity
oftheKitaevmodel,whichhasd=2andν =z =1[18],
is expected to scale as 1/τ [4]. This observationleads to
the following general conclusion.
Consider a d-dimensional model with ν = z =
1 which is described by a Hamiltonian H = FIG. 3: Plot of hO~ri, sans the δ-function peak at the origin,
d
as a function of~r for four values of J /J, for Jτ =5.
~kψ~k†(cid:16)σ3ǫ(~k)t/τ +∆(~k)σ++∆∗(~k)σ−(cid:17)ψ~k, where σ± 2
=P(σ1 iσ2)/2. Suppose that a quenchtakes the system
±
through a critical surface of d m dimensions. The de- thedefects. Inparticular,aplotof O versus~rgivesan
~r
− h i
fect density fora sufficiently slowquenchis givenby [17] estimate of the correlations between the defects. [Since
nπthτe=Pd(mα-1d,β/iAm=1de)gnαRsiBβoZknαdakdlβkB]e−Z∼π,τ1ff/((τ~k~k)m)/≃2=,(w1∆/hAe(r~dke))A2R/dBǫZi(sd~kdt)kheveaaxnrpeis[aheo−sf OhO~r2A~rn=ift=er1,htOha~rleliqtiuhfeennmcisho,omdthednetassnydosft=eOm1~r,ifcfoanrneibsaecehvfeomnu.on]mdetnrtivuimall~ky,:
| | | | is described by a combination of ψ with probability
on the d m dimensional critical surface, α,β denote ~k
− p and ψ with probability 1 −p∞, where ψ are
one of the m directions orthogonal to that surface, and ~k ~k − ~k ~k
gαβ = [∂2f(~k)/∂kα∂kβ]~k critical surface. Note that this tchoemepiugteends∞tianteassotfraHig~khtfoforrwJ3ar→dm±a∞nn.eHr:ence hO~r±i∞can be
result depends only on th∈e property that f(~k) vanishes
on a d m dimensional surface, and not on the pre- 2
cise form− of f(~k). For general values of ν and z, we hO~ri = − δ~r,~0 + A Z d2~k p~k cos(~k·~r), (5)
notethataLandau-Zenertypeofscalingargumentyields
∆ 1/τzν/(zν+1), where ∆ is the energy gap [4]. When where the integral runs over half the BZ with area A.
one∼crossesad mdimensionalcriticalsurfaceduringthe Forlargevaluesofτ,thedominantcontributioncomes
quench,theava−ilablephasespaceΩfordefectproduction fromthe regionnear the line J1 sin(~k M~1) = J2sin(~k
· ·
sncale1s/aτsmΩν/(∼zν+km1).∼Fo∆rma/zqu∼en1c/hτtmhνr/o(uzνg+h1a);ctrhitisiclaelapdosintot Mw~h2i)chwhvaerreypa~klo=ng1.anIdntrpoedrpuceinndgictuhlearvatroiatbhliesslkinkea(nadlokn⊥g
∼
where m=d, we retrieve the results of Ref. [4]. the directions nˆ and nˆ respectively), we see that the
Next we study the correlations between the defects integrand in Eqk. (5) ta⊥kes the form exp[ a(~k0)τk2
produced during the quench. To this end, we define i(~k +k nˆ +k nˆ ) ~r], where a(~k ) is a−number⊥de±-
0 0
the operators O~r = ib~na~n+~r. In the spin language, pendingkonk~k0. ⊥Th⊥e e·valuation of the integral over k
Ouc~0t=ofσsjzp,liσnjz,ol+pe1r.aFtoorrs~rg6=oin~0g, Ofro~rmcaanbbseitweraittt~nentoasana parsoidte- sgeiveesthaatfatchteormoafgenxiptu(cid:2)d−e(~rof·nˆt⊥he)2d/(e4feacτt)(cid:3)c/o√rraelτa.tioWnes tghoue⊥ss
at~n+~r: the productbeginswith aσx orσy at(j,l)and as 1/√τ, while the spatial extent of the correlations
ends with a σx or σy at (j′,l′) with a string of σz’s in goes as √τ. This is confirmed by the following relation:
between, where the forms of the initial and final σ ma- ~r2 O = 2( 2p ) =24πτ(J2+J2+J J )/J.
trices depend on the positions of j+l and j′+l′. Note PT~roshtu~rdiy th−eco∇rr~kela~kti~ko=n~0sbetweend1efect2s, we1ev2aluate
tshpaintifsolroJck3e→dw−it∞ht(h∞a)t,owfhitesrveetrhtiecazllcyonmeparoensetnnteoigfhebaocrh, iEnqF. i(g5.)3nufomresreicvaelrlayl;vtahleue~rsdoefpJen/dJen,cewhoferheOJ~ri =is sJhoawndn
2 1 1
hψ−∞(∞)|O~r|ψ−∞(∞)i = ±δ~r,~0. For the Kitaev model, Jτ = 5. Here we have omitted the δ-function peak at
it is known that the spin correlations between sites ly- ~r = 0 in Eq. 5 so as to make the correlation at ~r = ~0
~ring=o~0n[d1i3ff]e.reTnhtebroefnodrsevaOnisha,ree.gt.h,ehσoazn,l~nyσbzn,~no+n~-rvia=nis0hifnogr claletaiornlys.vTisoibulen.dTerhsteapnldotthoef hvOar~riiatrieoflnecotfsththeseedceofrercetlactoi6rornes-
~r
6 h i
two-pointcorrelatorsofthemodel[14]. Anon-zerovalue withtheratioJ /J ,wenotethatforlargeJτ, themax-
2 1
of O for~r =~0inthefinalstateprovidesasignatureof imum contribution to O comes from around the wave
~r ~r
h i 6 h i
4
as expected from Fig. 3. We conclude that the spatial
anisotropy of the correlations between the defects O
~r
h i
depends crucially on the ratio J /J .
2 1
Toconclude,wehaveshownthatthedensityofdefects
produced during a quench of the Kitaev model through
a critical line scales with the quench time as 1/√τ, in-
steadof the 1/τ behaviorexpected for a quenchthrough
a critical point. We have provided a general result for
the defectdensitywhichreproducesthe resultofRef. [4]
as a special case. We have also discussed the variation
ofthedefectcorrelationswiththe modelparametersand
pointed out the possibility of detection of these varia-
tions in experiments. These results significantly improve
ourgeneralunderstandingofthescalingofthedensityof
defects and their correlations in 2D systems.
WethankA.DuttaandA.Polkovnikovforstimulating
discussions.
FIG.4: PlotofhO iatthepoints(1,0)onthen axis(black
~r 1
solidline),(0,2)onthen axis(bluedottedline),and(2,−2)
2
along the−45◦ linein then −n plane (reddashed line) as [1] S. Sachdev, Quantum Phase Transitions (Cambridge
1 2
a function of α=tan−1(J /J ), for J2 =1 and Jτ =5. University Press, Cambridge, England, 1999).
2 1
[2] T.W.B.Kibble,J.Phys.A9,1387(1976);W.H.Zurek,
Nature (London) 317, 505 (1985).
[3] B. Damski, Phys.Rev. Lett. 95, 035701 (2005).
vectors~k0 forwhichp(~k0)=1. ForJ2 ( )J1,thisim- [4] A. Polkovnikov, Phys. Rev. B 72, 161201(R) (2005); A.
≫ ≪
plies sin[~k M~ (M~ )]=0 whichyields~k √3ˆi ˆj. The Polkovnikov and V.Gritsev, arXiv:0706.0212.
2 1 0
maximum·contributionto O comesfrom∼cos(~k±~r)=1, [5] K. Sengupta, S. Powell, and S. Sachdev Phys. Rev. A
h ~ri 0· 69, 053616 (2004); P. Calabrese and J. Cardy, J. Stat.
i.e., ~k ~r = 0. Thus for J ( )J , O is expected
0 2 1 ~r Mech: TheoryExptP04010(2005),andPhys.Rev.Lett.
· ≫ ≪ h i
to be maximal along the lines y = (+)√3x, namely, 96, 136801 (2006); J. Dziarmaga, Phys. Rev. Lett. 95,
−
n = n (n =0)inthen n plane. Thisexpectation 245701 (2005), and Phys. Rev.B 74, 064416 (2006).
1 2 1 1 2
iscon−firmedinFig. 3where−O canbeseentobemax- [6] A. Das et al.,Phys. Rev.B 74, 144423 (2006).
~r
imal along n = n (n =0h) foir J =4(0.2)J . Such a [7] R.W.CherngandL.S.Levitov,Phys.Rev.A73,043614
1 − 2 1 2 1 (2006); V. Mukherjee et al., Phys. Rev. B 76, 174303
strong anisotropy can be understood by noting that the
(2007).
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J2 ( )J1. For intermediate values of J1/J2, a grad- (2006); F.M.Cucchietti et al.,Phys.Rev.A75, 023603
≫ ≪
ual evolution of the defect correlations can be seen in (2007); T. Caneva, R. Fazio, and G. E. Santoro, Phys.
Fig. 3. We note that if the Kitaev model can be real- Rev. B 76, 144427 (2007).
ized using ultracoldatoms in anopticallattice [16], such [9] L. E. Sadler et al.,Nature(London) 443, 312 (2006).
an evolution of the defect correlations with J /J can, [10] A. Kitaev, Ann.Phys.321, 2 (2006).
1 2 [11] A. Kitaev, Ann.Phys.303, 2 (2003).
in principle, be experimentally detected by spatial noise
[12] X.-Y.Feng,G.-M.Zhang,andT.Xiang,Phys.Rev.Lett.
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Wecanobtainadifferentviewofthespatialanisotropy [13] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev.
of the defect correlations by studying O as a function Lett. 98, 247201 (2007).
~r
of α = tan 1(J /J ). As α changesh, thie ratio J /J [14] H.-D. Chen and Z. Nussinov, arXiv:cond-mat/0703633.
− 2 1 2 1
variesfrom0to whilefixingJ2+J2 =J2 =1. Aplot [15] D.-H. Lee, G.-M. Zhang, and T. Xiang, arXiv:07053499;
∞ 1 2 K. P. Schmidt,S. Dusuel,and J. Vidal, arXiv:07093017.
of O atthreerepresentativepoints(n ,n )=(1,0)(on
h ~ri 1 2 [16] L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev.
then1axis),(0,2)(onthen2axis),and(2, 2)(alongthe Lett.91,090402 (2003); A.Micheli, G.K.Brennen,and
−
45◦ line inthe n1 n2 plane)asa functionofα, shown P. Zoller, Nature Physics 2, 341 (2006).
− −
in Fig. 4, reveals the nature of the defect correlations. [17] See for example, L. Landau and E. M. Lifshitz, Quan-
We see that as J /J =tanα is varied from 0 to , the tum Mechanics: Non-relativistic Theory, 2nd Ed. (Perg-
2 1
magnitude of the correlation at the point (1,0)∞on the amon Press, Oxford, 1965); S. Suzuki and M. Okada in
Quantum Annealing and Related Optimization Methods,
n axis increases till it reaches a maximum at J = J
1 1 2 Eds.byA.DasandB.K.Chakrabarti(Springer-Verlag,
(α=π/4),andthendecaysto0asαapproachesπ/2. For
Berlin, 2005).
the points(0,2)onthe n axisand(2, 2)alongthe line
2 [18] Notethatasweapproachthegaplessphase,∆E ∼|J −
− 3
withslope−45◦,thecorrelationbecomesmaximumwhen J3c| and ∆E ∼|k|; hence ν =z=1 for themodel [4].
J2 J1 (α = 0) and J2 J1 (α = π/2) respectively, [19] E. Altman, E. Demler, and M. D. Lukin, Phys. Rev. A
≪ ≫
5
70, 013603 (2004).