Adiabatic dynamics in passage across quantum critical lines and gapless phases

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by  Debanjan Chowdhury

Adiabatic dynamics in passage across quantum critical lines and gapless phases
Debanjan Chowdhury,1,∗ Uma Divakaran,1,† and Amit Dutta1,‡
1Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
(Dated: January 11, 2010)
It is well known that the dynamics of a quantum system is always non-adiabatic in passage
througha quantumcritical point and thedefect densityin thefinalstate following a quenchshows
apower-lawscalingwiththerateofquenching. However,weproposehereapossiblesituationwhere
0 the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the
1 defectdensitydecaysexponentially. Thisisachievedbyincorporatingadditionalinteractionswhich
0
lead to quantum critical behavior and gapless phases but do not participate in the time evolution
2
of the system. To illustrate the general argument, we study the defect generation in the quantum

n criticaldynamicsofaspin-1/2anisotropic quantumXYspinchainwiththreespininteractionsand
a a linearly driven staggered magnetic field.
J

1 PACSnumbers: 05.30.-d,64.60.-i,75.10.-b
1

] Understanding the dynamics of a quantum system Wigner transformation. We thus provide an example of
h passing through a quantum critical point [1, 2] has been aspecialsituationthatcontradictsthepower-lawscaling
c
a very active and fascinating area of research in recent of the defect density for a non-random quantum system
e
m years [3–14]. The dynamical evolution can be initiated [4, 5].
eitherbyasuddenchangeofaparameterintheHamilto- Letusbeginthediscussionwithad-dimensionalquan-
-
t nian,calledasuddenquench[3],orbyaslow(adiabatic) tum Hamiltonian given by
a
quenchingofaparameter[4,5]. Therelatedentanglement
t
s and fidelity properties are also being lookedat [15]. It is
. H = ψ†(~k)H(~k)ψ(~k);
t well known that when a quantum system, initially pre-
ma pared in its ground state, is swept adiabatically across a Xk
quantum criticalpoint, defects aregeneratedin the final H(~k) = αc(~k)ˆ1+ λσz +∆(~k)σ +∆∗(~k)σ (1)
- + −
d stateofthesystemduetothecriticalslowingdownwhich (cid:16) (cid:17)
n forcesthe dynamics of the systemto be non-adiabaticin where α, c(~k) and ∆(~k) are model dependent functions,
o the vicinity of the quantum critical point. In a linear σi(i=x,y,z)arePaulispinmatrices,σ =(σx+iσy)/2,
+
c passage through an isolated critical point, when a pa- σ =(σx iσy)/2 and ˆ1 denotes the 2 2 identity ma-
[ rameter (e.g., the magnetic field) is changed in time t tr−ix. Here,−λ defines the time-dependent×parameter that

3 as t/τ, the defect density (n) in the final state scales is to be quenchedadiabatically. The columnvector ψ(~k)
v with the quenching rate 1/τ following the Kibble-Zurek
definesatwocomponentfermionicoperator. Suchanex-
1 scaling relation [16, 17] given by n τ−dν/(νz+1) in the
actlysolvableHamiltonian(withα=0)isknowntorep-
6 ∼
adiabatic limit (τ )[4, 5, 12]. Here, d is the spa-
1 → ∞ resentseveralone-andtwo-dimensionalintegrablequan-
tial dimension, ν and z are the correlation length and
1 tum spin models as the Ising, the XY spin chains [19]
. the dynamical exponents, respectively, associated with and the extended Kitaev model in two dimensions [20]
6
the quantum critical point [1]. The experimental verifi-
0 when the spin operators are transformed into spinless
cation of such dynamics is now possible by studying the
9 fermions via the Jordan-Wigner transformation [19]. In
dynamics of atoms trapped in optical lattices [18].
0 thepresentcase,however,thenatureofinteractionofthe
: spin chains we study, necessitates the consideration of a
v In this work, we explore the possibility of adiabatic
i dynamics or exponentially decaying defect density even twosublattice structure,andhence the fermionoperator
X in passage through quantum critical points and gapless ψ(~k)=(a~k,b~k) where a~k (b~k) denote the Jordan-Wigner
ar phases. We showthat this occursonly ina specialsitua- Fermions for the mode ~k describing the spins on even
tioninwhichanadditionaltermoftheHamiltonianthat (odd) sublattices. The excitation energy of the Hamilto-
leadstothequantumcriticalbehaviorandgaplessphases nian is given by
doesnotparticipateinthedynamicsofthesystem. Weil-
ǫ± =αc(~k) λ2+ ∆2. (2)
lustrate the general argument using a quantum spin-1/2 ~k ± | |
p
XY chain with multispin interactions and a staggered The phase diagram of the model in the α λ plane can
magnetic field which is exactly solvable via the Jordan- be easily obtained. The presence of the ad−ditional term
αc(~k) plays a non-trivial role in determining the phase
diagram of the model by making excitation energy zero
for specific values of the parameter and the wavevector
∗Electronicaddress: [email protected] ~k. For example, if ∆(~k) = 0 for the wavevector ~k , we
†Electronicaddress: [email protected] 0
‡Electronicaddress: [email protected] obtain a critical line (ǫ~−k =0) given by λ=αc(~k0).
2
Our interest is in the defect generation when the pa- never vanish during the temporal evolution of H (t) is
~k
rameter λ is quenched in a linear fashion as t/τ from far more interesting. If the parameter ∆ attains the
t to t + and the system is swept across minimum value ∆ for some wave vector k satisfying
0 0
→ −∞ → ∞
the critical line. Let us assume that at t , the the scaling form ∆2 = ∆2 + δ~k ~k 2z2, the nona-
→ −∞ 0 | − 0|
system is prepared in its ground state |1~ki such that diabatic transition probability will show an exponen-
a†a 1 = 1 and b†b 1 = 0 for any ~k. For an adi- tial behavior and the defect density will scale as n
~k ~k| ~ki ~k ~k| ~ki exp( π∆2τ/λ )/τd/2z2. As discussed already, the sca∼l-
abatic dynamics, the expected final state is |2~ki defined ing o−f the0 defe0ct density satisfy the same scaling form
as b†b 2 = 1. In course of dynamics, a general state
~k ~k| ~ki for allvalues of α. This exponentialdecay of defect even
describing the system at an instant t can be put in the
in passage through a quantum critical point is the key
form ψ (t) = C (t)1 + C (t)2 where the time
~k 1,~k | ~ki 2,~k | ~ki feature of this communication.
dependent coefficients satisfy the Schrodinger equation QuestioniswhetheritispossibletofindaHamiltonian
which gets mapped to the Eq. (1). To show this we con-
∂ C C
i 1,~k = H (t) 1,~k sider a spin-1/2 quantum XY spin with a two sublattice
∂t(cid:18)C2,~k (cid:19) ~k (cid:18) C2,~k (cid:19) structure in the presence of a three-spin interaction and
= λ(t) ∆(~k) C1,~k (3) a staggered magnetic field h given by the Hamiltonian
(cid:18)∆(~k)∗ −λ(t)(cid:19)(cid:18)C2,~k (cid:19) H = h (σz σz ) J (σx σx +σy σy )
with ~ set equal to unity. Also, the time dependent pa- − Xi i,1− i,2 − 1Xi i,1 i,2 i,1 i,2
rameterλ(t)variesasλ0t/τ andtheinitialconditionsare J (σx σx +σy σy ) J (σx σz σx
C (t )2 = 1 and C (t )2 = 0. It is − 2 i,2 i+1,1 i,2 i+1,1 − 3 i,1 i,2 i+1,1
| 1,~k → −∞ | | 2,~k → −∞ | Xi Xi
worthnoting that althoughthe term αc(~k)ˆ1 plays a cru- + σy σz σy ) J (σx σz σx
cialroleindeterminingthe criticallineorthe gaplessre- i,1 i,2 i+1,1 − 3 i,2 i+1,1 i+1,2
Xi
gions,it does not show up in the time-dependent Hamil- + σy σz σy ), (5)
tonianH (t)whichdictatesthetemporalevolution. This i,2 i+1,1 i+1,2
~k
is because the identity operator commutes with all the where i is the site index and the additional subscript
other terms of the Hamiltonian at every instant of time. 1(2) defines the odd (even) sublattice. The parameter
The term αc(~k)ˆ1 influences the dynamics only up to a J describes the XY interaction between spins on sub-
1
phase factor and is hence truly irrelevantin deciding the lattice 1 and 2 while J describes the XY interaction
2
non-adiabatic behavior. In this sense, the time depen- between spins on sublattice 2 and 1 such that J is not
1
dent Hamiltonian H~k(t) does not capture the passage necessarily equal to J2. The parameter J3, chosen to be
throughquantumcriticallines andgaplessphases gener- positive throughout, denotes the three spin interaction.
atedby αc(~k)! The Schrodingerequations(3)describing Some variants of the Hamiltonian (5) were studied pre-
the dynamics of the system effectively boils down to the viously [23]. This spin chain is exactly solvable in terms
standard Landau-Zener problem (LZ) [21] of two time- of Jordan-Wigner fermions [23] defined on even and odd
dependentlevels λ(t)2+ ∆2 (notthe levelsgivenin sublatticesasσ+ = ( σz )( σz ) a†,andσ+ =
Eq. 2) approachin±gpeach othe|r i|n a linear fashion with a i,1 h j<i − j,1 − j,2 i i i,2
minimumgap2|∆|attimet=0. Theprobabilityofexci- h j<i(−σjz,1)(−σjz,2)(−Qσiz,1)ib†i, where σiz,1 = 2a†iai −1
tationinthefinalstateisgivenbyLandauZenerformula aQnd σz = 2b†b 1. The Fermion operators a and b
[21, 22] p~k =|C1,~k(t→∞)|2 =exp(−(π∆(~k)2τ)/λ0) . can bei,2shownitoi −satisfy Fermionic anticommutaition re-i
Let us assume that the parameter ∆(~k) vanishes at a lations.
quantum critical point for α = 0 as ∆(~k) ~k ~k z2 In terms of the Jordan-Wigner Fermions, the Hamil-
0
where ~k is the critical wave vector. Noting∼th|at−in t|he tonian(5) canbe recastin the Fourier space to the form
0
given in Hamiltonian (1) with
adiabatic limit of large τ, only modes close to ~k con-
0
tribute, the defect density in the final state is given by
1 λ (1+γe−ik)
n = (2π1)d ZBZpk ddk Hk =αcoskˆ1− 2(cid:20)−(1+γe+ik) − −λ (cid:21),
(6)
1
= ddk exp( (π~k ~k 2z2τ)/λ )
(2π)d Z − | − 0| 0
BZ whereψ† =(a†,b†)andwehavesetλ=h/J ,α=J /J
1 k k k 1 3 1
(4) and γ =J2/J1. The excitation energy is obtained as
∼ τd/2z2
1
The scaling of the density of defects hence depends only ǫ± =αcosk λ2+1+γ2+2γcosk (7)
k ± 2
on the exponent z2 as observed previously in references p
[11] in the context of quenching through a multicritical Comparing with the Hamiltonian (1), we also identify
point. The situation where the parameter ∆(~k) does c(k)=cosk and ∆(k)2 = 1+γ2+2γcosk. The phase
| |
3
6 6 −0.6 0.25
γ =1 γ =0.5 γ=1.0 γ=0.5
4 AF 4 AF 0.2
2 2 −0.9
n
λ 0o GPI GPII λ 0o GPI GPII log 10−1.1 n 0.1
−2 −2
AF AF −1.3
−4 −4
−6 0 0.5 1 α1.5 2 2.5 3 −6 0 0.5 1 1α.5 2 2.5 3 −1.5 0 0.4 0l.8og τ 1.2 1.6 1.8 0 1 3 5 τ 7 9 10
10
FIG. 1: Phase diagram of the Hamiltonian (5) in the α−λ FIG.2: Numericalintegrationresultforthedefectdensityas
plane for γ = 1 and γ = 0.5. Different phases are discussed a function of τ for γ = 1 and γ = 0.5. The parameter α is
in the main text. The vertical line shows the direction of chosen such that the system crosses both the gapless phases
quenching. in thetwocases. Forγ =1, wehaveused log-scale andslope
of the straight line is 1/2 indicating a power-law decay of
the defect density. For γ = 0.5, we have used linear scale to
accentuate theexponential fall of the defect density.
diagram of the model for both γ = 1 and γ = 1 are
showninFig.1. Forγ =1,excitationenergyǫ+ 6vanishes hence the defect density in the final state is given by
for the mode k = π at6 the phase boundary bketween an 1 π 1
n= dkexp( π(π k)2τ) . (8)
antiferromagneticphase(AF) andagaplessphase(GPI) π Z − − ∼ √τ
0
given by 2α = λ2+(1 γ)2. In the GPI phase, it
is always possiblpe to find a−wave vector k for which ǫ+k Fonorthγe6=AF1,-GonPIthpehaostehberouhnadnadr,y∆givisenmbinyim∆um2f=or1k =γπ2
vanishes. Similarly, ǫ− vanishes for the mode k = 0 at | 0| | − |
k so that the defect density
thephaseboundarygivenby2α= λ2+(1+γ)2which
marks the boundary between thepGPI and the second
gapless phase (GPII). In GPII, both ǫ+ and ǫ− vanish 1 π
k k n = dkexp( π 1+γ2+2γcosk τ)
forsomewavevector,sothattherearefourFermipoints π Z − | |
0
in contrast to two Fermi points in GPI. The transition e−π|∆0|2τ
betweenGPIandGPIIphasesisaspecialquantumphase . (9)
∼ √τ
transition that involves doubling the number of Fermi
points [24]. For γ =1, on the other hand, we arrive at a
We therefore, come across a special situation where the
simplified form ∆(k)2 = 2+2cosk = 4cos2(k/2). The
| | defect density decays exponentially with the rate τ even
phase boundaries between the antiferromagnetic phase
though the system is swept across the critical lines and
andGPIphase,andtheGPIandGPIIphasearegivenby
gaplessphasesaslong as∆ =0, i.e.,γ =1. We recover
2α=λand2α=√λ2+4,respectively. Itis noteworthy 0 6 6
the power law scaling in Eq (8) for γ =1. As mentioned
that for the case γ = 1, the parameter ∆2 vanishes
| | already, the scaling behavior shown in Eqs. (8) and (9)
at the AF-GPI phase boundary for k = π and any α
aresameforallvaluesofthescaledthreespininteraction
whereas in the anisotropic case, never does it vanish at
α. The defect density obtained by numerical integration
the quantum transitions!
of the Schrodinger equations (3) is shown in Fig 2.
It is to be noted that the time evolution governed by
In the quenching scheme we employ here, the scaled
H (t) is completely insensitive to the phase transitions
staggeredmagneticfieldλisquenchedasλ t/τ fromt k
0
→ generatedbyαc(k). Thisfactcanbeunderstoodalsous-
to+ withα=0sothatthesystemissweptacross
−∞ ∞ 6 ingthefollowingargument: thetotalnumberofFermions
the quantumcriticallines andthegaplessphases. Letus
foramodekisaconstantofmotionasfarastheLandau-
set λ = 1 for simplicity. As discussed already, so far as
0
Zener dynamics is concerned. This is because the num-
the dynamics is concerned, the term αcosk is irrelevant
andtheSchrodingerequationessentiallyreducestoatwo ber operator nk = a†kak + b†kbk commutes with Hk(t)
level LZ problem. Note that in both the limits λ for all k. In the initial (final) state λ ( + ),
, the spins should be in a perfect antiferromagnet→ic the expectation value of nk is unity wh→ich−∞as per∞the
±or∞ientationin the z-directionandwronglyorientedspins above argument is conserved throughout the dynamics.
in the final state at t (λ ) are the defects. On the other hand, the three spin term gives rise to the
→∞ →∞ gapless phases where both the energies ǫ+ and ǫ− can
k k
Using the LZ transition formula we find that the become negative or both can become positive for some
probability of excitation in the final state is given by values of k; the true ground state allows n = 2 or 0
k
p =exp( π ∆(k)2τ)=exp( 4πcos2(k/2)τ) for γ =1. in those cases. However, the instantaneous eigenstates
k
− | | −
In the adiabatic limit of τ , only the modes close ofthe time-dependent Hamiltonianwill continueto have
to k =π (for which ∆(k)2 →van∞ishes) contribute,so that n = 1 as explained above and does not reach the true
k
p takesthe simplifie|d fro|mp =exp( π(π k)2τ), and ground state.
k k
− −
4
Question remains whether the result presented here cayofthedefectdensity. Ontheotherhand,ifδω van-
2,k
shouldpersistinthecaseofageneralinteractingsystem. ishes for any mode k, then the density of defects decays
Inref.[5],itwasshownthattheproofoftheKibble-Zurek as a powerlaw with the quenching rate τ. In the present
scalingformdoesnotrequirethesystemtobe brokenup work,we have used an example of an exactly solvedsys-
into a product of two-level systems which can then be tem which satisfies the above conditions. Though pos-
analyzedby Landau-Zenertunneling formula. The argu- sible in principle , we believe that finding the example
mentonlyusestranslationalinvarianceandsomegeneral of such an interacting Hamiltonian which satisfies the
scaling arguments, namely, the momentum dependence above conditions and show an exponential decay of the
of the energies and the parameter dependence of the defectdensityisadifficultandopenproblem. Ourinter-
wave functions of those states. Extending the argument est in this work is only to point out a special situation
of ref.[5] in the present case demands that the Hamilto- where one can find exponential decay of defect density
nian should be decoupled into two parts; one time inde- evenduringpassagethroughcriticalregionsandwehave
pendent H (α) and the other time dependent H (λ(t)), illustrated the possibility using an exactly solvable spin
1 2
where λ=t/τ as defined before, and H commutes with chain with complicated interactions.
1
allthe terms of H (λ). It can be shownthat H (α) does Acknowledgement We acknowledge Diptiman Sen and
2 1
notinfluence the dynamicsexcept fora phase factor and G.E. Santoro for very interesting and critical com-
the scaling form of the density of defects is then given ments and Victor Mukherjee for carefully reading the
byscalingformoftheλ-dependentpartofthe excitation manuscript. D.C. acknowledges KVPY, Department of
energyδω (λ) whereasvanishingofthe totalexcitation Science and Technology (Government of India) for sup-
2,k
energy δω = δω (α)+δω (λ) leads to the complex port and also G. Refael for interesting discussions. AD
k 1,k 2,k
phase diagram with critical lines and gapless phases. If and UD acknowledge Abdus Salam ICTP, Trieste, Italy,
δω (λ) does notvanish, one expects anexponentialde- where some part of this work was done.
2,k
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