Time-evolution and dynamical phase transitions at a critical time in a system of one dimensional bosons after a quantum quench

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by  Aditi Mitra

Time-evolution and dynamical phase transitions at a critical time in a system of one
dimensional bosons after a quantum quench
Aditi Mitra
Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA
(Dated: October 5, 2012)
Arenormalizationgroupapproachisusedtoshowthataonedimensionalsystemofbosonssubject
toalatticequenchexhibitsafinite-timedynamicalphasetransitionwhereanorderparameterwithin
alight-coneiszerouptoacriticaltimeandbecomesnon-zeroafterthistime. Suchatransitionisalso
foundforasimultaneouslatticeandinteractionquenchwheretheeffectivescalingdimensionofthe
latticebecomestime-dependent,cruciallyaffectingthetime-evolutionofthesystem. Explicitresults
2
1 arepresentedforthetime-evolutionofthebosoninteractionparameterandtheorderparameterfor
0 thedynamical transition as well as for more general quenches.
2
PACSnumbers: 05.70.Ln,37.10.Jk,71.10.Pm,03.75.Kk
t
c
O
A fundamental and challenging topic of research is to of this order-parameterspatially averagedwithin a light

3 understand nonequilibrium strongly correlated systems cone. Our results hold relevance not only to exper-
in general, and how phase transitions occur in such sys- iments in cold-atomic gases where system parameters

] temsinparticular. Whilethetheoryofequilibriumphase can be tuned rapidly in time,12 but also to conven-
l
e transitions is well developed, and relies heavily on the tional solid state materials where time-evolution of an
-
r renormalization group, the development of an equally order-parameter may be probed with high precision us-
st powerful approach to study nonequilibrium phase tran- ing ultra-fast pump-probe13 and angle resolved photoe-
. sitions is still in its infancy. Moreover, in any progress mission spectroscopy.14
t
a on this topic, it has always appeared that nonequilib- We model the 1D Bose gas as a Luttinger liquid,9
m
rium phase transitions have one aspect in common with
d- their equilibrium counterparts, both occur by adiabati- H = u0 dx K [πΠ(x)]2+ 1 [∂ φ(x)]2 (1)
callytuningsomeparameterofthesystem,intheabsence i 2π 0 K x
n Z (cid:20) 0 (cid:21)
o orpresenceofanexternaldrive,andstrictlyspeakingoc-
c cur only in the limit of infinite time (steady-state).1–8 where ∂xφ/π represents the density of the Bose gas, Π
−
[ In contrast, here we study a completely different kind is the variable canonically conjugate to φ, K0 is the di-
2 ofanonequilibriumphasetransition,onethatoccursasa mensionlessinteractionparameter,andu0 isthevelocity
v functionoftime. Employingatime-dependentrenormal- of the sound modes. We assume that the bosons are ini-
7 ization group (RG) approach we study quench dynam- tially in the ground state of Hi. The system is driven
7 ics of interacting one-dimensional(1D) bosons in a com- outofequilibriumvia aninteractionquenchatt=0from
7 mensurate lattice. This system in equilibrium shows the K0 K, with a commensurate lattice Vsg also switched
3 Berezinskii-Kosterlitz-Thouless (BKT) transition sepa- on s→uddenly, at the same time as the quench. This trig-
.
7 rating a Mott insulating phase from a superfluid phase gersnon-trivialtime-evolutionfromt>0duetoaHamil-
0 (Fig. 1).9 For the nonequilibrium situation we explic- tonian Hf =Hf0+Vsg, where Hf0 =Hi(K0 K) and
12 itlyshowtheappearanceofadynamicalphasetransition Vsg =−αgu2 dxcos(γφ), with g >0, and Λ=→αu a short-
wherethereisafinitecriticaltimebeforewhichanorder- distance cut-off. We assume that the quench preserves
: R
v parameter is zero, and after which it is non-zero. Such a Galilean invariance so that uK=u0K0, however this is
i behavior has no analog in equilibrium systems. not critical for either the approach or the result. While
X
A dynamical transition in time was recently identified γ=2 for bosons, we keep it general so that the results
r
a inthe exactlysolvabletransversefieldIsingmodelwhere may be generalized to other 1D systems.
the Loschmidt echo was found to show non-analytic be- In the absence of the lattice, the system is ex-
havior at a critical time, whereas the behavior of the actly diagonalizable in terms of the density modes,
order-parameterwasanalytic.10Incontrasthereweiden- Hi= p=0u0|p|ηp†ηp andHf0= p=0u|p|βp†βp whereβ,η
tifyasituationwheretheorder-parameteritselfcanshow are relat6ed by a canonical transfor6mation. This fact has
P P
non-analyticities as a function of time. In addition we been used to study the dynamics of a Luttinger liquid
generalize the study of dynamical transitions to mod- exactly,andhasrevealedinterestingphysicsarisingfrom
els that are not exactly solvable, and to low-dimensions a lackof thermalizationin the system.15–18 To study the
wherestrongfluctuations negatea mean-fieldanalysis.11 system in the presence of the lattice employing RG, we
We identify the dynamical transition by studying an writethe Keldyshactionrepresentingthe time-evolution
order-parameter ∆(r,Tm) which due to the quench de- from the initial pure state |φii (hence an initial den-
qpuenendcsh.boTthheopnhapsoestitriaonnsitrionanids aassoticmiaetedTmwitahftearnothne- ssittaytemoaftrHixi,ρZK= =|φiTihrφ[iρ|)(tc)]or=resTproned−iniHgfttoφitheφigeroiHufntd
| ih |
analytic behavior as a function of time Tm on the value = D[φcl,φq]ei(S0+Ssg). S0 is the q(cid:2)uadratic part which(cid:3)
R
2
g rithms Kneqln (r τ)2. Our approach generalizes the
eff use of RG to study ±quench dynamics near classical criti-
p
cal points22 to quantum systems.
Mott−insulator
Derivation of RG equations: We split the field
(d)
φ into slow (φ< ) and fast (φ> ) compo-
0,Λ 0,Λ dΛ Λ dΛ,Λ
nents in momentum−space φ = φ< +−φ>, and inte-
superfluid
(b) grate out the fast fields pertu±rbative±ly in g±. Following
this we rescale the cut-off back to its original value and
(c) (a)
rescale position and time R,T Λ(R,T ), where
0 ε Λ = Λ dΛ. Following thismwe→wrΛit′e themaction as
′
S = S< +−S< +δS< +δS< +δS< where S< is sim-
FIG. 1. The equilibrium BKT phase diagram. Arrows con- 0 sg 0 Teff η 0
ply the quadratic action corresponding to H with the
necttheHamiltoniansbefore(H )andafter(H )thequench. 0f
i f rescaled variables, S< is the rescaled action due to the
A dynamical phase transition is found for case (d). sg
lattice, while δS< are corrections to (g2).
0,Teff,η O
describes the nonequilibrium Luttinger liquid, which at S< =g Λ 2 ∞ dR tΛdT cosγφ<(R,T )
a tSim=e t a∞fterdxthe q∞uednxch is,t1d9t tdt φ (1) φ (2) −sgcosγφ(cid:18)<+Λ(R′(cid:19),TmZ−)∞e−γ42Zh[0φ>cl(Tmm)]2(cid:2)i − m (3)
(cid:18)0G−R1Z0(−1,∞2) 1−Z−G∞−RG1G−A2K1Z(0G1,−A21)1Z(10,2)(cid:19)2(cid:0)(cid:18)φφ∗clcql((22))(cid:19)∗q (cid:1)(2) δ×S0<∂R=φg<q22γ−2dIΛΛti(ZT−m(cid:3)∞∞) d∂RTmZφ0t<cΛl/√∂2TdmTφm<q(cid:2)−IR(Tm)(cid:0)∂Rφ<c(l4(cid:1))
wrehseenreti1n(g2)fi=e(ldxs1(2t)h,(cid:2)att1(2a)r)eatnidmφe/cla(cid:3),qn=ti-φt−i√m±2φe+owrditehre−d/o+nrtehpe- δS(cid:0)T<eff =(cid:1)ig22γ2dΛΛ (cid:0)∞dR tΛ(cid:1)/(cid:0)d√T2m φ<q(cid:1)(cid:3)2ITeff(Tm) (5)
KSmsegW<l=dey1αgsdu2hweRfih−c∞nico∞ehnatdionsxuz1oreRr.r2d0ot0edirnt-T1pth[ahceroeasmlg{aaγteptφtile−ceres(s∆1pp)m}ho=ta−eshnecet,oiimasa{nγlγφdicφsln(+oxg(,nt1i)-v)ize},en]rwoibtinhy EδSqη.<(=3)−sgh2o2γw2sdΛΛthZaZ−t∞−∞td∞hReZZ0st0cΛa/ld√iTn2gmφd<q(cid:0)im(cid:2)∂enT(cid:1)msiφon<cl(cid:3)oIfη(tThme)l(a6t)-
the gapped phase, and a two-point correlation function tice depends on [φ>(T )]2 , and in particular is time-
h cl m i
uCnadb(e1r,s2ta)=ndhetiγhφea(f1r)aem−eiγwφob(r2k)iofwhtheereRaG,b=le±t .usInstuorddyerCto dependent. To leading order (g=0) γ42h[φ>cl(Tm)]2i =
ab dΛ K + Ktr . δS< shows that the quadratic
for g=0 but for the nonequilibrium Luttinger liquid Λ neq 1+(2TmΛ)2 0
(K = K). Denoting 1(2)=R+( )r,T +( )τ, C parht of the action aciquires corrections which are also
dep0en6 ds both on the time-differen−ce2τ mas we−ll 2as thaeb time-dependent.23 δS< indicates the generation of
η,Teff
mean time T after the quench, and is translationally new terms such as a time-dependent dissipation (η) and
m
invariant in space.21 C depends on three exponents a noise (ηT ) whose physical meaning is the genera-
ab eff
K =γ2K,K =γ2K 1+ K2 ,K =γ2K 1 K2 . tion of inelastic scattering processes which will eventu-
eq 4 neq 8 0 K02 tr 8 0 − K02 allyrelaxthedistributionfunction.5,6 Thesetimedepen-
For simplicity, conside(cid:16)r Cab at(cid:17)equal-time ((cid:16)τ=0) an(cid:17)d dent corrections lead to RG equations which depend on
uafnteeqruathlepoqsuietniocnhs.butThloenngfodristsahnocret-stimr e≫s Tmu/ΛΛ,≪Ca1b Itime=TIm afteIr2t3h,eaqnudendchim. eDnseifionnliensgs dΛvΛa=riaΛb−ΛlΛes′=Tdln(l),
decays in position is a power-law with the exponent u,K R ti m
± →
Kneq + Ktr=γ2K0/4 (i.e., Cab r−γ2K0/4). Hence TmΛ,η →η/Λ,Teff →Teff/Λ the RG equations are,
∼
the short time behavior is determined completely by dg K
tr
=g 2 K + (7)
the initial wave-function. In contrast, at long-times, dlnl − neq 1+4T2
T Λ 1, C decays as a power-law but with a new (cid:20) (cid:18) m(cid:19)(cid:21)
terxmapnosni≫eenntt bKenheaaqvbi(oir.e.c,onCnaebct∼ingr−thKenseeq)l.imKitstr. gFouvretrhnesr,thaet ddKln−l1 = πg82γ2IK(Tm) (8)
dT
long times after the quench T Λ 1, C also becomes m
m ab = T (9)
≫ m
translationally invariant in time (hence independent of dlnl −
T ). 1 du πg2γ2
m = I (T ) (10)
The RG in equilibrium sums the leading logarithms. Kudlnl 8 u m
We use the same philosophy to employ RG to study dη πg2γ2K
dynamics. In particular at short times (but long dis- =η+ Iη(Tm) (11)
dlnl 4
tances), the RG will resum the logarithms γ2K0 ln r d(ηT ) πg2γ2K
whereasatlongtimes(TmΛ 1),itwillresumt4helog|a-| dlneflf =2ηTeff + 8 ITeff(Tm) (12)
≫
3
Note that T not only acts as an inverse cut-off in that dependence of g (T ) andǫ(T ) will be important for
m eff m m
modes of momenta Λ > 1/T dominate the physics at the results.
m
a time Tm,24,25 but it also governs the crossover from Case (a), periodic potential always irrelevant28:
a short time behavior where the physics is determined This occurs for ǫ(T ) > 0 and g not too large (a
m eff
primarily by the initial state, and a long time behavior condition to be made more precise when discussing case
characterizedby a new nonequilibrium fixed point. This (d)). Here the periodic potential renormalizes to zero,
crossoverismosteasilyseenfromEq.(7)wherethescal- andone recoversa gaplesstheory whicheventuallylooks
ingdimensionofthelatticeǫ(Tm)= Kneq+ 1+K4tTrm2 −2 trheneromrmalalaitzeTimn≫tim1e/ηa.n5d,6iTnhpeaRrtGicuplraerdsichtoswhsowthqatuaanttliotinegs
depends on time as follows: at shorht times (Tm ≪ 1) iit timesthesteady-statestateisapproachedasapower-law
is ( 2+ γ2K0), and hence depends on the initial wave- with a non-universal exponent.
− 4
function, atlong times (Tm 1), a nonequilibriumscal- Case (b), periodic potential always relevant28:
≫
ing dimension ( 2+K ) emerges.
neq This occurs for ǫ(T ) < 0. Thus we are always in the
− m
Above,I reachsteadystatevaluesatT 1,
K,u,η,Teff m ≫ strong coupling regime. Here we integrate the RG equa-
whereas for short times, they vanish as T 0 as ex-
m → tions upto a scale l∗(Tm) where the renormalized cou-
pectedsincethe effectofthelattice potentialvanishesat
pling is O(1). Beyond this scale our RG equations are
T =0. For example, at short times I (T2).26,27
m K ∼ O m notvalid,howevertheadvantageofthebosonictheoryis
Eqns. (8) and (9) represent renormalizationof the inter- that at strong-coupling gcos(γφ) g(1 γ2φ2/2+...)
actionparameterandthe velocity. Theeffects ofthe lat- ≃ −
so that √g may be identified with a gap. The physical
ter being small will be neglected, andin what follows we
gap/order-parameteristhengivenby∆=√g/l∗ =1/l∗.
set u=1. Eqns (10), (11) show the generationof dissipa-
Since l (T ) depends on time, it tells us how the order-
∗ m
tionandnoisethatrepresentinelasticscatteringbetween parameter evolves in time.29
bosonic modes.5,6
Let us first consider short times. Here we find that
Inwhatfollows wedo ananalysisfora time T <1/η
m the order-parameter increases as ∆(T Λ 1)
m0 0
where 1/η is the time in which the distribution func- 1 ≪ ≃
tion first begins to change due to inelastic scattering. A (Λ0Tm0g0)(cid:20)2−γ24K0(cid:21). For a pure lattice quench (K0=K)
perturbativecalculation5,6 showsthatforsmallquenches at the exactly solvable Luther-Emery point,27 we find
(K0 K 0), and at steady-state, η g2(K0 K)4. ∆ (T2 )inagreementwith.30TheRGhasallowedus
S|ince−η |1→, one may easily be in the re∼gime of T−m 1 to∼obOtainmr0esults for the time evolution for more general
≪ ≫
but Tmη 1 so that inelastic scattering may be ne- quenchesthattakethesystemawayfromexactsolubility.
≪
glected. At these times, andin whatfollowswe will only
At long times after the quench, the scaling dimen-
use equations (7), (8) and (9).
sion is 2 K . Provided that ∆T 1, we find
neq m0
Thebehaviorofthesystemisverydifferentdepending − ≫ 1
upon K ,K,g. We discuss four cases (see Fig 1). Case the steady-state order-parameter, ∆ss = (geff,0)2−Kneq.
0
Compare this with the order-parameter in the ground
(a)iswhentheperiodicpotentialisirrelevantatalltimes
1
after the quench, case (b) is when the periodic potential state of Hf9 ∆eq=(geff,0)2−Keq. Since Kneq > Keq and
isalwaysrelevant,case(c)iswhentheperiodicpotential geff,0 1, the order-parameter at long times after the
≪
is relevant at short times, and irrelevant at long times, quench is always smaller than the order-parameter in
while case(d)is whenthe periodicpotentialis irrelevant equilibrium. The RG equations may also be solved at
atshorttimesandrelevantatlongtimes. Forcase(d)we intermediate times28 Λ10 ≪ Tm0 ≪ ∆1. Here we find,
showthatanorder-parameterbehavesinadiscontinuous 1
way in time. There is a critical time Tm∗ before which ∆ = (Λ0Tm0)γ24K0−Kneqgeff,0 2−γ24K0 . Thus at inter-
the order-parameter is zero, and after which the order- (cid:20) (cid:21)
mediate times the gap decreases with time if K >
neq
parameter begins to grow indicating a dynamical phase
γ2K0, or increases with time for the reverse case. Note
transition. In contrast, for case (c), no discontinuous 4
that for K =K this intermediate time dynamics is ab-
behavior of the order-parameteroccurs. 0
sent, instead the order-parameter reaches a steady-state
We use ǫ ,g ,T ,Λ to denote bare physical values.
0 0 m0 0
at times T Λ 1.
We will present analytic solutions of the RG equations m0 0
≫
Case (c), periodic potential relevant at short
in the limits of short (T Λ 1) times (but long dis-
m0 0
≪ times, and irrelevant at long times28. This oc-
tances) and at long times (T Λ 1). For the for-
m0 0
≫ curs when ǫ(T ) changes sign from negative to posi-
mer the physics is primarily determined by the initial m
tive and g is not too large. Here the short time
state. From Eq. (8) we define an effective-interaction eff
behavior is the same as Case (b), however at long
g (T )=g πγ2 I (T )γK K where g goes to
Tzeemfrof≫asm1T.mPq→hy8s0icpaanlldyKtrheamischiem2spqalieKsst0etahdayt sattatseehfofvratluteimfoesr timΛ0eT1sm,0th1e+o2r−dγAe24rK-0pa,rawmheetreerAdecr=easesgwe2fift,h0(t∞im)e−aǫs20(∆∞∼).
the particles have not had sufficient time to interact, (cid:16)Fig. 2 s(cid:17)ummarizes the behavior ofqthe order-parameter
therefore however large g may be, any renormalization for cases (b), (c) and (d), the last case to be discussed
effectsduetointeractionsisvanishinglysmall. Thetime- next. The non-monotonic dependence of the order-
4
of the pure lattice quench.
∆ Let us suppose T Λ 1. Here the RG equations
−β m0 0 ≫
Tm0 are solved in two steps, one for 1 < l < Tm0Λ0, and the
b other for T Λ < l. For the first step, since I varies
m0 0 K
∆ b,d slowly at long times, eventually reaching a steady-state
ss value, we may assume 1 dIK dlng . Thus the RG
2|dlnl| ≪ |dlnl|
equations are the conventional ones of the equilibrium
Tmβ0 b BKT transition dgeff= g ǫ, dǫ = g2 . For the sec-
dlnl − eff dlnl − eff
−γ ondstep,(Λ T <l),sinceI T2,theRGequations
Tmθ0 b,c Tm0 c d become dgeff0 =m0 g ǫ, dǫ =K ∼ge2fmf, where ¯l = l .
dln¯l − eff dln¯l − ¯l2 Λ0Tm0
1/Λ0 Tm* 1/∆ss Tm0 tThhaetsgoelfuftiiosnirsrheolewvsantthabteftohreerethisisaticmriet,icaanldtiimseaTrem∗lesvuacnht
perturbation after this time. We find,
FIG. 2. Time evolution of the order-parameter after the
qtiumeencbhehfaovriocras(eTsm0(b≪), Λ(1c0)),aanndin(tde)r.meSdoialitdetliinmeesasshyomwptsohtoicrst Λ0Tm∗=eD1 arctan(ǫD0)−D1 arctanD2,D2 =ge2ff,0−ǫ20(13)
(Λ10 ≪ Tm0 ≪ ∆1ss) and a long time behavior Tm0 ≫ ∆1ss. ThedeeperonequenchesintotheMott-phase,theshorter
At short times for cases (b) and (c), the order-parameter in- is Tm∗. Moreover, Tm∗ is longest along the critical
creasesas∆∝Tmθ0,atintermediatetimesitincreasesasTmβ0 line geff,0=ǫ0. By identifying a length-scale at which
(decreases as Tm−0β) when Kneq < γ24K0 (Kneq > γ24K0) for geff(¯l∗)∼1, we find that the order-parametergrows as
case(b)andeventuallyreachesasteady-statevalue∆ . For
caasse∆(c∝) tThe−γo.rdeFro-rpacraasmee(tder) atlhweareysisdeacrneoanse-sanfoarlyTtimc0sbs≫ehaΛv10- ∆∼θ(Tm0−Tm∗)Λ0T1m0[geff(l =Λ0Tm0)](Tm0f−2Tm∗)(14)
m0
ior at time Tm∗ after which the order-parameter increases. where f = 1 .
θ=2−γ12K0, β=θ(cid:16)|γ24K0 −Kneq|(cid:17), γ =1+Aθ. Dashed lines An im2por|tdaǫn(dlt=TmTqmu)e|sTtmio=nTm∗c|oncerns the spatial variation
4
are a guide to theeyefor thecrossover regimes. ofthe order-parameter. Quenches in gaplesssystems are
associated with light-cone dynamics where two points a
position R apart get correlated after a time T R.31
m
∼
parameter in time is due to the time-dependence of For our case any two points separated by R > T will
m
the scaling dimension which physically leads to a sit- behaveprimarilyliketheinitialstatewithpower-lawcor-
uation where quantum fluctuations are enhanced (sup- relations in position determined by K . The predictions
0
pressed) at a later time for ǫ(T = ) > ǫ(T = 0) fortheorder-parametermadeaboveisforaregionwithin
m m
∞
(ǫ(T = ) < ǫ(T = 0)), causing the order-parameter a light cone R<T . The dynamical transition at T is
m ∞ m m m∗
to decrease (increase). associatedwiththeappearanceoforderinregionsofsize
Case (d), periodic potential irrelevant at short R∗ ∼ Tm∗, after which the ordered regions will begin to
times and relevant at long times28: This occurs un- grow in size.
dertwoconditions. Eitherǫ(T )changessignfromposi- In summary, employing RG we have identified a novel
m
tivetonegativeduringthetime-evolution,orǫ(T )isal- dynamical phase transition in a strongly correlated sys-
m
wayspositive,butg (T )becomessufficientlylargeat temwhereanorder-parameteriszerountilacriticaltime
eff m
sometime T . Thelatterincludesthecaseofapurelat- and increases after this time (Eq. (14)). The order pa-
m∗
ticequench(K =K). Foreithercondition,atlongtimes, rameter shows rich dynamics both at the transition as
0
the order-parameter reaches a steady state value, while well as for more general quenches (Fig. 2). Identifying
at short times it is zero. This indicates a non-analytic similardynamicaltransitionsinhigherdimensionswhere
behavior at a critical time T . thermalfluctuations arelesseffectiveindestroyingorder
m∗
is an important direction of research.
Fig. 1 contrasts case (d) with the previous cases con-
Acknowledgements: The author gratefully acknowl-
sidered where the order-parameter behaved analytically.
edgeshelpfuldiscussionswithI.Aleiner,B.Altshuler,P.
Therenormalizedinteractionparameterg (T )isvan-
eff m
Hohenberg, A. Millis, E. Orignac and M. Tavora. This
ishingly small right after the quench. For case (b), since
work was supported by NSF-DMR (1004589).
infinitesimally small g is a relevant perturbation, an
eff
order-parameter starts growing immediately after the
quench. On the other hand for a quench corresponding
tocase(d),Fig.1showsthatg hastobelargerthana
eff
criticalvalue inordertobe inthe Mott-phase. Thus one
has to wait some finite time before which renormaliza-
tion effects become large enough for an order-parameter
to grow. We now discuss this physics in a more quanti-
tative manner, and for simplicity, consider only the case
5
Supplementary Material
is for g =0 given by
Kneq
α α
I. ELEMENTS OF THE QUADRATIC ACTION C+ (r,Tm,τ)=
− " α2+(uτ +r)2 α2+(uτ r)2#
AND OUTLINE OF THE RG PROCEDURE −
α2+ 2u(T p+τ/2) 2 α2+p2u(T τ/2) 2 Ktr
m m
The action for the non-equilibrium Luttinger liquid is { } { − }
×"p α2+(2uTm+r)2 p α2+(2uTm r)2 #
−
S = ∞ dx ∞ dx tdt tdt φ (1) φ (2) e−iKpeq[tan−1(uτα+r)+tan−1(uτα−rp)] (21)
0 1 2 1 2 ∗cl ∗q ×
Z0−∞ Z−∞ G−A1Z(01,2) Z0 (cid:0) φcl(2) ((cid:1)15)
(cid:18)G−R1(1,2) − G−R1GKG−A1 (1,2)(cid:19)(cid:18)φq(2)(cid:19) III. EXPRESSIONS FOR IR,Iti,Iη,ITeff,IK,Iu
Denoting 1=x ,t(cid:2);2=x ,t (cid:3)
1 1 2 2
G−R,1A(1,2)
=−δ(x1−x2)δ(t1−t2)πK1u ∂t21±iδ −∂x21 (16) IR(TmΛ)=− ∞ dr¯ 2TmΛdτ¯r¯2
(cid:2) (cid:3) Z−∞ Z0
while Im[B(r¯,T Λ,τ¯)] (22)
m
GK(xt1,yt2)=−iK20 (cid:18)1+ KK022(cid:19)Z0∞ dppe−αp Iti(TmΛ)=−Z−∞∞dr¯Z02TmΛdτ¯τ¯2
Im[B(r¯,T Λ,τ¯)] (23)
cos(up(x y))cos(up(t t )) m
1 2
− | | −
−iK20 (cid:18)1− KK022(cid:19)Z0∞ dppe−αp ITeff(Tm)=Z−∞∞dr¯Z−22TTmmΛΛdτ¯
cos(up(x y))cos(up(t1+t2)) (17) Re[B(r¯,TmΛ,τ¯)] (24)
− | |
∞ 2TmΛ
where Iη(TmΛ)= dr¯ dτ¯τ¯
−
u Z−∞ Z−2TmΛ
Λ= (18) Im[B(r¯,TmΛ,τ¯)] (25)
α
I (T Λ)= ∞ dr¯ 2TmΛdτ¯ r¯2+τ¯2
is a short-distance cut-off. u m
−
In doing the RG, the fields φ are split into slow fields Im[BZ(−r¯∞,T ΛZ,0τ¯)] (cid:0) (cid:1) (26)
(φ<) that have Fourier modes in momentum space be- m
tmwoedeens0i,nΛm−odmΛe,ntaunmd fsapsatcfieebldestw(φee>n) ΛthatdhΛa,vΛe.FoTurhiuers IK(TmΛ)=− ∞ dr¯ 2TmΛdτ¯ r¯2−τ¯2
φ = φ< +φ>. The fast modes are integ−rated out per- Im[BZ(−r¯∞,T ΛZ,0τ¯)] (cid:0) (cid:1) (27)
m
turbatively in the periodic potential. In doing so we use
the fact that the correlator for the field φ (G ) is re-
0,Λ
lated to the correlators for the slow (G< ) and fast where Re[B]=(B+B∗)/2, Im[B]=(B B∗)/(2i), and
0,Λ dΛ −
fields (G> ) as follows G = G< − +G> ,
Λ dΛ,Λ 0,Λ 0,Λ dΛ Λ dΛ,Λ
so that32 − − − B(r¯,TmΛ,τ¯)=C+ (r¯,TmΛ,τ¯)F(r¯,TmΛ,τ¯) (28)
−
dG
G> =dΛ 0,Λ (19) with C+ (r¯,TmΛ,τ¯) =
Λ−dΛ,Λ dΛ eiφ+(r¯,τ¯+TmΛ/2)e−iφ−−(0,τ¯−TmΛ/2) . This quantity
h i
within leading order in perturbation theory is given by
Following this we rescale the cut-off back to its origi-
Eq. 21 which we rewrite in dimensionless variables,
nal value and in the process rescale position and time to
x ,t Λ(x ,t ) where Λ =Λ dΛ.
i i → Λ′ i i ′ − Kneq
1 1
C (r¯,T Λ,τ¯)=
+ m
− " 1+(τ¯+r¯)2 1+(τ¯ r¯)2#
II. EXPRESSION FOR C+−(r,Tm,τ) −
1+ 2(T Λ+τ¯/p2) 2 1+ 2(pT Λ τ¯/2) 2 Ktr
m m
{ } { − }
The two-point correlator defined as "p 1+(2TmΛ+r¯)2 p 1+(2TmΛ r¯)2 #
−
C+ (r,Tm,τ)= eiφ+(r,Tm+τ/2)e−iφ−(0,Tm−τ/2) (20) e−piKeq[tan−1(τ¯+r¯)+tan−1(τ¯−r¯p)] (29)
− h i ×
6
while F is given by tion of the bosons changes considerably due to inelastic
scattering, we may simplify the RG equations to
1 1
F(r¯,T Λ,τ¯)=K +
m neq"1+(τ¯+r¯)2 1+(τ¯ r¯)2# dg =g 2 K + Ktr (35)
− dlnl − neq 1+4T2
1 1 (cid:20) (cid:18) m(cid:19)(cid:21)
+K + dK 1 πg2γ2
tr"1+(2TmΛ+r¯)2 1+(2TmΛ−r¯)2# dln−l = 8 IK(Tm) (36)
dT
τ¯+r¯ τ¯ r¯ m
= T (37)
iK + − (30) m
− eq"1+(τ¯+r¯)2 1+(τ¯ r¯)2# dlnl −
−
We will use the notationg ,T ,Λ as the bare physical
and 0 m0 0
values. Let us define the scaling dimension of the lattice
γ2K
as
K = (31)
eq
4
K
γ2 K2 ǫ(T )= 2+ K + tr (38)
Kneq = 8 K0 1+ K2 (32) m (cid:20)− (cid:18) neq 1+4Tm2 (cid:19)(cid:21)
(cid:18) 0(cid:19)
γ2 K2 We now consider four cases separately: Case (a) is when
K = K 1 (33)
tr 8 0 − K2 ǫ(Tm) > 0 at all times. Case (b) is when ǫ(Tm) < 0 at
(cid:18) 0(cid:19)
alltimes. Case (c) is when ǫ(T )<0 at shorttimes and
We will often use dimensionless variables T T Λ. m
m ↔ m ǫ(Tm) > 0 at long times. Case (d) corresponds to the
dynamical transition and occurs for two possible cases,
one where ǫ(T ) > 0 at short times and ǫ(T ) < 0 at
IV. SHORT TIME BEHAVIOR OF IK m m
long times. The second is when ǫ(T ) > 0 at all times,
m
but g (T ) becomes sufficiently large at long times.
eff m
I depends onthetwo-pointfunctionC whichmay
K + While one may always solve the equations 35, 36, 37
be computed explicitly within perturbation−theory. We
numerically, we can also obtain analytic solutions in two
discuss its short-time behavior (T Λ 1) here. In this
m ≪ limits, one for short times Tm0Λ0 1 and the other for
case the integrand in Eq. 27 may be Taylor expanded in ≪
long times T Λ 1. In these two limits a simpli-
m0 0
τ so that ≫
fication comes from the fact that I (T ) for T 1
K m m
≫
IK(Tm 1) changes very slowly, as a power-law with Tm, whereas
2K ≪∞ dr≃2Tmdτ r2 1 γ24K0 achtasnhgoerstmtiomreesraiptidclhyanwgietshalsthTam2n. IKFodroelosn,gsotiwmeesw,ilgl
eqZ−∞ Z0 ( (cid:18)1+r2(cid:19) × neveegrleactt sthhoertl tdimepeesn,dweencweilolfreItKainatitslonlgdetpimenedse.ncHeovwia-
1−r2 + γ2K0 2 τ + (τ3) IK(Tm ≪1)=cTm20Λ2/l2.
×(cid:20)(1+r2)2 4 (1+r2)2(cid:21) O )
=4√πK γ2K0 1 Γ(γ24K0 − 21)T2 + (T4) A. Case (a)
eq 4 − Γ(1+ γ2K0) m O m
(cid:18) (cid:19) 4
(34) Case (a) is when ǫ(Tm) > 0 at all times, and g is not
toolarge(thisstatementwillbemademoreprecisewhen
Note that I generically behaves as I T2 at short
K K ∼ m discussing Case(d)). In this case, the periodic potential
times. However at the Luther-Emery point γ2K0 = 1 so is always irrelevant.
4
thatthe termof (T2)vanishes,andtheleadingbehav- Let us first consider the short time solution. Here ǫ=
ior is I (T4O). m γ2K0 2. Further we write I (T 1) = cT2. Thus
K ∼O m 4 − K m ≪ m
Itisimportantto notethatthe Luther-Emerypointis the RG equations become
quite deepintheMottphase,andisaccessibleperturba-
dg γ2K
tively in g only at short times. At long times, the slow 0
=g 2 (39)
power-law decay in time (and in space) leads to infrared dlnl − 4
(cid:18) (cid:19)
singularitiesthatmakesperturbationtheorybreak-down. dK 1 πg2γ2 cT2 Λ2
Inthis paper we donot attempt to discussthe long-time − m0 0 (40)
dlnl ≃ 8 l2
behavior at the Luther-Emery point. (cid:18) (cid:19)
Solving this upto l (which is justified as g de-
cays rapidly with l),→the∞solution is g = 0 and 1 =
∗ K∗
V. SOLUTION OF THE RG EQUATIONS 1 + π2γ2g02cTm20Λ20 1 + (T2 Λ2). Thus the renor-
K 16 γ2K0 1 ≃ K O m0 0
(cid:16) 4 − (cid:17)
If we are interested in times that are such that T < malized interaction parameter K changes quadratically
m ∗
1/η,where1/ηisthetimeinwhichthedistributionfunc- with time at short times from its bare value of K.
7
For long times Tm0Λ0 1, the RG equa- B. Case (b)
≫
tions have to be integrated in two steps, one
fdodlrngl=Tgm(2−≫Kne1q),wdhdlKnerle=−thπeg2γ82RKG2IKe(qTumat0iΛon0/sl).becoDmee- theThstisrooncgcucrosupfolirngǫ(rTemgi)m<e. 0H.erTehwues iwneteagrreataelwthaeysRiGn
fine ǫ=Kneq−2 where dǫ= γ42KK0dK. We will now use ecqouupatliinognsisuOp(t1o).aBsecyaolendl∗t(hTisms)cawleheoruerRthGe erqeunaotrimonaslizaerde
the fact that the l dependence of I (T Λ /l) is much
K m0 0
weaker than that of g at long times i.e, (1 dlnIK not valid, however the advantage of the bosonic theory
2| dlnl | ≪ is that at strong-coupling gcos(γφ) g(1 γ2φ2/2 +
dlng ). We also define an effective-interaction, ≃ −
|dlnl| ...) so that √g may be identified with a gap or order-
parameter. The physical gap/order-parameter is then
πγ2 γK K
geff(Tm0Λ0)=g IK(Tm0Λ0) (41) givenby √g/l∗ =1/l∗. Since l∗(Tm) depends ontime, it
8 2 K
r r 0 tells us how the order-parameterevolves in time.
p
Interms ofthese new variables,in the long time limit, Let us first consider short times. Since geff(Tm0Λ0
≪
the RG equations are 1) γ2K0 2, we may approximate the RG equations
≪| 4 − |
as
dg
eff
= ǫg (42)
eff dǫ
dlnl −
0 (49)
dǫ dlnl ≃
= g2 (43)
dlnl − eff dgeff γ2K0
=( +2)g (50)
eff
dlnl − 4
The solution of the above equations are well known9,
Integrating upto a scale l such that g =1, and using
A ∗ eff
ǫ(lnl)= (44) the short-time behavior of geff,0 g0Tm0, the effective
tanh Alnl+tanh−1 ǫA0 gap is found to increase as ∼
h A (cid:16) (cid:17)i 1
g(lnl)= sinh Alnl+tanh−1 A (45) ∆(Tm0Λ0 ≪1)≃(Λ0Tm0g0)(cid:20)2−γ24K0(cid:21) (51)
ǫ0
h (cid:16) (cid:17)i An order-parameterdefined as
where
∆ =eimγφ (52)
m
A= ǫ2(T = ) g2 (T = ) (46)
0 m ∞ − eff,0 m ∞ is related to ∆= 1 as follows,
ǫ (Tq= )=K 2 (47) l∗
0 m neq
∞ −
We solve the above equations upto a scale l1=Tm0Λ0. ∆m(Tm) [∆(Tm)]m2γ2K4˜(Tm) (53)
∼
Following this, for larger values of l, the RG equations
changeduetoachangeinthescalingdimensionandalso where K˜(T ) = K T 1 and K˜(T ) =
m 0 m m
because I T2. In order to smoothly connect the so- 4 K T 1. For a∀lattice≪quench (K =K) at the
K ∼ m γ2 neq∀ m ≫ 0
lutions of the RG equations at short and at long times exactly solvable Luther-Emery point (K = γ2K0 = 1),
we will make the ansatz, IK Tm0lΛ0 = IK(∞)1+TmT2lm202Λ020Λ20 Ethqa.t3∆4 show(sTt2ha)tinIKag∼reeTmm4e.ntTwhiuths gReefffe.q30∼. O(4Tm20), so
so that in the second step, th(cid:0)e RG e(cid:1)quations changel2to At lon∼gOtimmes0after the quench, the scaling dimension
ddglenffl =− γ24K0 −2 geff,ddlnǫl=−ge2ffTm2l02Λ20. The solu- is determined by Kneq. Here the RG equations are,
tion of th(cid:16)ese equati(cid:17)ons with the initial conditions corre- dǫ
sponding to Eqns. 44, 45 evaluated at l =T Λ give, 0 (54)
1 m0 0 dlnl ≃
A dg
eff
ǫ∗(Λ0Tm0 ≫1)= tanh Aln(Λ0Tm0)+tanh−1 ǫA0 dlnl =(−Kneq+2)geff (55)
Integrating upto a scale where g 1, and provided
h (cid:16) (cid:17)i2 eff ∼
1 A that ∆Tm0 1, we find the steady-state gap/order-
≫
1
− γ22K0 −2!sinhhAln(Λ0Tm0)+tanh−1(cid:16)ǫA0(cid:17)i(48) gpaapraimnettheer,g∆rosusn=ds(tgaetfef,o0)f2H−Kf9ne∆q.eqC=o(mgepfafr,0e)t2h−iK1sewq.itShinthcee
K > K and g 1, the gap at long times after
neq eq eff,0
≪
thequenchisalwayssmallerthanthegapinequilibrium.
Thus atlongtimes a gaplessphase is recovered,where
We now turn to intermediate times 1 T 1,
the interaction parameter ǫ∗ reaches its steady-state Λ0 ≪ m0 ≪ ∆
heretheRGinvolvestwosteps,thefirstinwhichthescal-
valueofAasapower-lawwithanon-universalexponent,
ǫ∗ −Λ−0−T−m−0≫−→1 A+O Λ0T1m0 2A. sintogpdsimateln∗si=onΛi0sTdme0te,ramnidnethdebsyecKonnedqiannwdhtihcehitnhteegsrcaatliinong
(cid:16) (cid:17)
8
dimension is determined by K , and the RG is termi- ing conventional RG equations,
0
nated at g =1. Here we find (againtaking dǫ 0),
eff dlnl ≃ dg
∆ = (Λ0Tm0)γ24K0−Kneqgeff,0 2−γ124K0 . Thus at inter- dlenffl =−geffǫ (57)
dǫ
media(cid:20)te times the order-param(cid:21)eter decreases with time = g2 (58)
dlnl − eff
if K > γ2K0, or increases with time for the reverse
neq 4 (59)
case. Note that for K =K this intermediate time dy-
0
namics is absent, instead the order-parameter reaches a where
steady-state at times T Λ 1.
m0 0
≫ πγ2 γK
g =g I ( ) (60)
eff,0 0 K
∞ 8 2
r (cid:18) (cid:19)
C. Case (c) The above equations may be solved by defining a con-
stant of the flow
Case (c), periodic potential relevant at short times, D2=g2 (l) ǫ2(l)=g2 ǫ2 (61)
and irrelevant at long times. This occurs for γ2K0 < eff − eff,0− 0
2,Kneq > 2 and geff not too large. Here the short4time Integrating upto l=Λ0Tm0 we get,
behavior is the same as Case (b). At long times, the RG ǫ
0
proceeds in two steps. One where Tm0Λ0 > l > 1 where ǫ(l =Λ0Tm0)=Dtan arctan D −Dln(Λ0Tm0)(62)
thesolutionisinequations44,45. Forthesecondstepthe h (cid:16) (cid:17) i
g (l =Λ T )= D2+ǫ2(l=Λ T ) (63)
RG equations become dgeff = g (2 γ2K0), dǫ 0. eff 0 m0 0 m0
dlnl eff − 4 dlnl ≃
Atthesecondstepthe RGequationsareintegratedupto At the next step, the RpG equations change because now
paalr∗amsuecthert∆hat g1e/flf∗(li∗s)fo∼un1d. tTohbee,solution of the order- oIKf th∼e1R/Gl2.eqIunaotirodnesrattosshmoortotahnlyd caotnlnoencgtttihmeessowluetiwonilsl
∼
make the ansatz,
∆(Λ T 1)=
0 m0 ≫ T2 Λ2
1 A 2−γ124K0 IK(cid:18)Tml0Λ0(cid:19)=IK(∞)"1+mTl02m2l020Λ20# (64)
(cid:18)Λ0Tm0(cid:19)sinh Aln(Λ0Tm0)+tanh−1 ǫA0  The RG equations at the second step become,
 h (cid:16) (cid:17)i (56)
dg
eff
= g ǫ (65)
dln¯l − eff
Thus at long times, the gap/order-parameter decreases
to zero as ∆∼ Λ0T1m0 1+2−γA24K0 . ddlnǫ¯l =−ge¯2l2ff (66)
(cid:16) (cid:17) l
¯l = (67)
Λ T
0 m0
D. Case(d) with the initial conditions at¯l=1 givenby Eqns 62, 63.
Equations 65, 66 can be solved exactly. The solution is
Here we will consider the case of the dynamical tran- ǫ(¯l)= 1+√atan √a b ln¯l (68)
sition where at short times the periodic potential is ir- − −
relevant,whereasitbecomesrelevantatafinitenon-zero geff(¯l)= √asec √a(cid:2) b (cid:0)ln¯l el(cid:1)n(cid:3)¯l (69)
− −
time. Let us for simplicity consider this scenario for a (70)
pure lattice quench so that K =K, with γ2K >2 such (cid:2) (cid:0) (cid:1)(cid:3)
0 4
that ǫ > 0. This would naively imply that the periodic where
potentialis irrelevant,howeverthatis notthe casewhen a=g2 (¯l =1) ǫ(¯l=1)+1 2 (71)
geff becomeslargeenoughatsometimeTm∗. Wedemon- eff −
strate below how the dynamical transition occurs. b= 1 tan 1 (cid:0)ǫ(¯l=1)+1(cid:1) (72)
−
Let us suppose we are at a time T Λ 1. At these √a √a
m0 0 ≫ (cid:18) (cid:19)
times, the RG equations have to be solved in two steps.
The first is when 1 < l < Tm0Λ0. Here, we may ne- Thesolutionforgeff(¯l)fordifferentinitialconditionsare
glect the l-dependence of I by noting that it changes plotted in Fig. 3. There is a dynamical transition when
slowly in time at long timeKs, in particular as a power- geff(¯l =1)=geff,c where
law, eventually reaching a steady-state. So assuming
1 dlnIK(l) dlng in this regime, we get the follow- geff,c = ǫ2(¯l =1)+2ǫ(¯l=1) (73)
2| dlnl | ≪ |dlnl|
q
9
0.5 namical transition at a critical time T such that
m∗
0.4 g (l=Λ T )= ǫ2(l=Λ T )+2ǫ(l=Λ T )(74)
eff 0 m∗ 0 m∗ 0 m∗
From Eq. 63, the abopve implies
0.3
geff D2 =2ǫ(l=Λ0Tm∗) (75)
0.2 g =0.2
eff,i
geff,i=0.44 Using Eq. 62, we solve for Tm∗,
g =0.46
0.1 eff,i
Λ0Tm∗ =eD1 arctan(ǫD0)−D1 arctanD2 (76)
0
0 20 40 60 80 As before, we may identify the order-parameter/gap as
ln(l)
the length scale l at which g (l ) 1. This gives us
∗ eff ∗
∼
the following result for how the order-parameter grows
FIG. 3. RGflowof geff(¯l= Λ0Tlm0) for threedifferent initial after the critical time,
conditions g = g (¯l = 1) and hence times, and for an
eff,i eff
initialǫ =ǫ(¯l=1)=0.1. Criticalcouplingortimeislocated 1
i ∆=
at g = 2ǫ +ǫ2 =0.458. l
eff,c p i i ∗
=θ(Tm0−Tm∗)Λ T1 [geff(l =Λ0Tm0)]ǫ(l=Tm∗)−1ǫ(l=Tm0)
0 m0
∼θ(Tm0−Tm∗)Λ T1 [geff(l =Λ0Tm0)](Tm0f−2Tm∗) (77)
0 m0
The above critical coupling implies that there is a dy- where f = 1 .
2 |dǫ(dl=TmTm)|Tm=Tm∗|
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