Geometric phase contribution to quantum non-equilibrium many-body dynamics

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by  Michael Tomka

Geometric phase contribution to quantum non-equilibrium many-body dynamics
Michael Tomka♣, Anatoli Polkovnikov♠ and Vladimir Gritsev♣
♣Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland
♠ Physics Department, Boston University, Commonwealth ave. 590, Boston MA, 02215, USA
(Dated: January 12, 2012)
Westudytheinfluenceofgeometryofquantumsystemsunderlyingspaceofstatesonitsquantum
many-body dynamics. We observe an interplay between dynamical and topological ingredients of
quantum non-equilibrium dynamics revealed by the geometrical structure of the quantum space of
states. AsaprimaryexampleweusetheanisotropicXYringinatransversemagneticfieldwithan
2 additionaltime-dependentflux. Inparticular,ifthefluxinsertion isslow, non-adiabatictransitions
1 in the dynamics are dominated by the dynamical phase. In the opposite limit geometric phase
0 strongly affects transition probabilities. We show that this interplay can lead to a non-equilibrium
2
phasetransitionbetweenthesetworegimes. Wealsoanalyzetheeffectofgeometricphaseondefect

n generation during crossing a quantumcritical point.
a
J PACSnumbers:

2
1
Introduction.—The profound interplay and interrela- relevanceoftopologyandgeometryofthequantumspace

] tionofgeometryandphysics wasthe focus inbothfields of the many body system for the measurable quantities
r
since creation of General Theory of Relativity, in which defining a non-equilibrium evolution of the system far
e
h quantities responsiblefor the geometryofspace-timeare from the adiabatic limit. We show that these effects are
t determinedbythephysicalpropertiesofthematterliving verysignificantintheregionsclosetothequantumphase
o
. in this space and vice versa. The relevant geometry lan- transition. We thus demonstrate a profound interplay of
t
a guageinthiscaseisaRiemanniangeometry. Gaugeprin- geometryandtopologyofthephasespaceofthequantum
m cipleofclassicalgaugetheoriesfounditsnaturaldescrip- many-body system in its out of equilibrium dynamics.
- tion and nice interpretation in terms of theory of fiber In equilibrium, a Riemannian structure is introduced
d
bundles,asubjectofdifferentialgeometry[1]. Monopoles to quantum mechanics by the Quantum Geometric Ten-
n
and instantons of the gauge theory have profound topo- sor (QGT) [4],[5]. The QGT Q is defined for an arbi-
o µν
c logical meanings which is the property of defining fiber trary eigenstate n by
| i
[ bundle. Many of these notions appeared in various con-
2 densed matter systems at equilibrium. Thus, defects in Qµν(λ,|ni):=hn|←∂−µ∂ν|ni−hn|←∂−µ|nihn|∂ν|ni, (1)
v He and liquid crystals are classified according to the ho-
for µ,ν = 1,...,p, labeling the system’s parameters λ
1 motopy theory, certain phase transitions are associated µ
which form a manifold . Its real part is a Rieman-
1 to proliferation of topological defects. Topology plays a M
6 nian metric tensor gµν on that is related to the fi-
vital role in e.g. Hall effects and topological insulators. M
4 delitysusceptibilitywhichdescribesthesystemsresponse
Another intriguing phenomena, emerging in quantum
. to a perturbation and therefore is an important quan-
8
mechanics,thatrelatesgeometryandphysicsistheBerry
0 tity, e.g. in the study of Quantum Phase Transitions
phase. When a Hamiltonian is adiabatically driven, its
1 (QPT) [5]. The imaginary part is related to the 2-form
1 eigenstatesacquirenotonlythefamiliardynamicalphase (Berry curvature) F :=∂ A ∂ A =2 Q , where
µν µ ν ν µ µν
v: factor,but additionallyaphasefactorthatdepends only Aµ(λ, n ) := i n∂µ n is the c−onnection 1ℑ-form. Geo-
on the geometry of the phase space of the Hamiltonian, | i h | | i
i metric phase [2] of the state n is given by its integral
X namely the Berry phase [2]. It can be observed in inter- | i
alongaclosedloop inparameterspaceγ = A dλµ.
r ference experiments and in the Aharonov-Bohm effect. C n C µ
a It is easy to check that after a simple gauge trRansforma-
ThedeepgeometricalsignificanceoftheBerryphasewas tion,theSchr¨odingerequationiψ˙ =Hˆ ψ writteninthe
revealed as well [3]. Therefore, it is also referred to as | i | i
instantaneousbasis n such that ψ = a n canbe
the geometric phase. We point that while the Berry | i | i n n| i
put into the following form: P
phase is usually associated with adiabatic processes, the
geometric phases describe transformations of arbitrary a˙ = M exp[iE (t) iΓ (t)] a , (2)
n nm nm nm m
eigenstates and are thus not tied to the adiabaticity. In −mX6=n −
condensed matter probably its most transparent man-
where M = n∂ m . This equation highlights the
ifestation is in the Haldane phenomena (a presence or nm t
h | | i
competition between the dynamical phase E (t) =
absenceoftheexcitationgapin1Dspinchaindepending nm
t
on the value of spins). 0 [ǫn(τ)−ǫm(τ)]dτ and the geometric phase Γnm(t) =
Until now all these manifestations of the geometric Rt[A (n ) A (m )]dτ.
0 τ | i − τ | i
phase were associated to equilibrium and adiabatic phe- R The main purpose of the present work is to demon-
nomena. Here we demonstrate for the first time a direct strate how geometric effects shows up in quantum dy-
2
namics. We do it using an example of a driven XY- ˆc are the Fourier transforms of the fermionic op-
k
model which we introduce in the next paragraph. Gen- erators resulting from the Jordan-Wigner transforma-
eralizationsofsome ofourresultsto moregenericsetups tion (see Ref. [10] for details). By applying the Bo-
are discussed in the Supplementary Information. The goliubov transformation to (5) we can map it to a
main findings of our paper is that geometric phase ef- free fermionic Hamiltonian with the known spectrum
fects on transition probabilities are small for slow nearly ǫ (g,h)= (h cosp )2+g2sin2p .
k k k
adiabatic driving protocols, i.e. that the leading non- q −
Thesetofquantumcriticalpointsofthisspinchainare
adiabatic transitions are determined by the dynami-
determinedbythevanishingoftheenergygap: 2ǫ =0,
cal phase. Contrary, in the fast limit geometric phase k0
where k is defined by minimizing the excitation energy
0
strongly affects transitions between different levels. We
∂ ǫ =0. Thisconditiondefinesquantumcriticalregions
k k
also found that the interplay of geometric and dynami-
on . For the model (4) the gap vanishes on the line
cal phases can lead to non-equilibrium phase transitions M
(g = 0, 1 h 1), marking the anisotropic transition
causing sharp singularities in density of excited quasi- and on t−he≤two≤planes (g R, h = 1), identifying the
particles and pumped energy as a function of the driv- ∈ ±
Ising transitions [8, 11]. The anisotropic transition line
ing velocity. This quantum-critical behavior can happen
belongstotheLifshitzuniversalityclasssinceitmanifests
without undergoing by the system an actual quantum
thecriticalexponentsν =1/2andz =2. Ontheother
phase transition in the instantaneous basis [6]. In the 1 1
handtheIsingtransitionplanesbelongtothed=2Ising
limit of slowly driving the system through a quantum
universality class with the critical exponents ν =1 and
critical point with an additional rotation in the parame- 2
z = 1 [11]. The points where the critical line and the
2
ter space we find that the geometric phase modifies the
critical plane cross are multicritical points. In Fig. 1 we
scaling of the observables with the driving velocity and
depict the equilibrium phase diagram of the rotated XY
enhances non-adiabatic effects.
spin chain in the parameter space (g,h,φ).
The rotated XY spin chain.—Let us consider a stan-
dard, although rich and illustrative example of XY ring
in a transverse magnetic field. The Hamiltonian of this
system [7, 8] is defined by
N
1+g 1 g
Hˆ = σˆxσˆx + − σˆyσˆy +hσˆz (3)
0 − (cid:20) 2 l l+1 2 l l+1 l(cid:21)
Xl=1
withperiodicboundaryconditions,i.e.,σˆα =σˆα. The
N+1 1
numberofspinsN isassumedtobeevenandthespin1/2
on the site l is represented by the usual Pauli matrices
σˆα, with α x,y,z . Further, the anisotropy for the
l ∈ { }
nearest neighbor spin-spin interaction along the x and y
Figure 1: The phase diagram of the rotated XY spin chain
axis is described by the parameter g and h denotes the
in atransverse magnetic field in cylindrical coordinates: The
magnetic field along the z axis. tworedplanes(h=±1)indicatetheIsingcriticalplane,(i.e.
Atg =0thisHamiltonianhasanadditionalU(1)sym- the associated QPT belongs to the d = 2 Ising universality
metryrelatedtospin-rotationsintheXY-plain. Atfinite class). Whereas the blue line (g = 0) marks the anisotropic
g this symmetry is broken. Clearly there is a continu- transition line. The black bold circle and helix describe the
ous family of ways breaking this symmetry yielding the two drivingprotocols we usein this paper.
identicalspectrum. ThecorrespondingHamiltoniansare
related by applying a unitary rotation of all the spins
Dynamics of the rotated XY spin chain.—We explore
around the z axis by angle φ:
two driving protocols. The first one is driving the spin
Hˆ(g,h,φ)=Rˆ(φ,z)Hˆ0(g,h)Rˆ†(φ,z), (4) rotation φ(t) with a constant velocity. This corresponds
with the rotation operator Rˆ(φ,z) = N exp( iφσˆz). tocircularpathsinparameterspace(seeFig.1). Itisthe
l=1 − 2 l simplestsituationinwhichanon-trivialgeometricphase
This transformation yields non-triviallQy complex instan-
emerges. The second driving protocolconsists of driving
taneous eigenstates, which is a necessary condition for
the magnetic field h(t) and the spin rotation φ(t). This
existence of the nontrivial geometric phase [9].
results in helical paths in parameter space (Fig. 1) and
The Hamiltonian (4) can be diagonalized using the
allows us to study the cross over from the well known
Jordan-Wigner and the Fourier transformations:
Landau-Zener scenario (no geometric phase) to the ro-
Hˆ(g,h,φ)= ˆc†Hˆ cˆ , (5) tating driving regime (non-trivial geometric phase).
− k k k k
X For either of the protocols we assume φ(t) = ωt in
with Hˆk =(h−cospk)σˆz+gsinpk(sin2φσˆx−cos2φσˆy), the time interval 0 < t < tf, where ω > 0 is the rate
ˆc† = (ˆc ,ˆc†), p = 2πk, k = 1, 2,..., N and of change of the spin rotation. Then the Schr¨odinger
k −k k k N ± ± ±2
3
equation for the coefficients a and a , that appear If the rotation angle is not large n 1 we see that the
1,k 2,k
∼
in the expansion of state in the instantaneous basis, transitionprobabilityinthiscaseisdirectlyproportional
ψ = a gs +a es , becomes a system of linear tothesquareoftheproductofthegeometricanddynam-
k 1,k k 2,k k
| i | i | i
differential equations with constant coefficients that can ical phase differences between the ground and excited
be solved exactly (see the Supplementary Information). states:
Fromthissolutionwecomputetheprobabilityforfinding
2
the system in the excited state p g (gs ) ∆Ek∆γφ,k , (9)
ex,k φφ k
≈ | i (cid:20) 4π (cid:21)
sin2 1Ω (ω)φ
p = a (φ )2 =g (gs ) 2 k f . (6)
ex,k | 2,k f | φφ | ik 12(cid:2)Ωk(ω) 2 (cid:3) where ∆γφ,k = ∆Aφ,kφf = 0φf Aφ,kdφ and ∆Ek =
(cid:2) (cid:3) ∆ǫkT; T = 2π/ω is the rotatRion period. In the limit
ǫHees,rke−Ωkǫ(gωs,)k:==q2(cid:2)ǫ∆kωǫkis−t∆heAφe,nke(cid:3)r2g+y4dgiφffφe(r|egnsicke),b∆etǫwke:e=n ioφtffy∆lasEragtkeur≫raottea1stitoahnteaaenxvgpalreleusaestioifinnxdefeodprefrntehdqeeuntetrnaconyfsi∆tthioEenkgp≪eroomb1aeatbrniildc-
the excited and ground states of the k-th subspace
and dynamical phases: p g (gs )/2. Interestingly
and ∆Aφ,k := Aφ(|esik) − Aφ(|gsik) designates the thisprobabilityisentirelyexde∼termφφin|edibkytheRiemannian
corresponding difference of the connection 1-forms:
metric tensor, i.e. has a purely geometric interpretation.
A (gs ) = i gs∂ gs and A (es ) = i es∂ es .
φ | ik kh | φ| ik φ | ik kh | φ| ik From the discussion above we see that if we focus on
Further, g (gs ) is the Riemannian metric tensor of
the k-thgrφoφun|disktate,whichalsodefines the fidelity sus- the limit of large φf and analyze the transition proba-
bility as a function of ω we expect a smooth crossover
ceptibility along the φ direction:
betweentwosimpleregimesbothindependentofthegeo-
gφφ(|gsik)=−khgs|∂φ|esikkhes|∂φ|gsik =|khes|∂φ|gsik|2. mp etricphgase(:gpsex,)k/∼2 agtφφω(|gsik∆)ωǫ2/.∆Aǫ2ksaimtωila≪r c∆roǫsksoavnedr
ex,k φφ k k
∼ | i ≫
With this the totaldensity ofexcitedquasi-particlesand between fast and slow regimes is expected in the many-
the energy density of excitations of the entire spin chain particle situation. Thus one can naively expect that the
in the thermodynamic limit can be calculated by influence of the geometric phase on the dynamics in the
limitoflargeφ isquitelimited. Therealityturnsoutto
π dk π dk f
n = p , ǫ = 2ǫ p . (7) be much more interesting though as we illustrate below.
ex ex,k ex k ex,k
Z−π 2π Z−π 2π In this limit we can simplify the k integrals in Eqs. (7)
using the stationary phase approximation. Then we
Before proceeding with the detailed analysis of these
find that the resulting behavior of n and ǫ exhibits
two quantities let us make some qualitative remarks on ex ex
a “cusp” at a critical driving velocity ω determined by
Eq. (6). (i) For the quench of infinitesimal amplitude c
ω = 1 h. This is illustrated in Fig. 2. Because this
φf 0 both geometric and dynamical phases are not c −
→
important and the transition probability is simply given
h=0, 1
by the product of the square of the quench amplitude g=0.01 (cid:4)φ(cid:5) g=0.1
0.012 0.12
and the fidelity susceptibility in agreement with gen-
0.010 (cid:2)ex 0.10 (cid:2)ex
efirxaeldreφsult&s [112]t.he(igi)eoImnetthriec splohwaselimisitstωill≪not∆ǫimk paonrd- n()ex(cid:1) 0.008 nex 0.08 nex
f 0.006 0.06
tant while the dynamical phase suppresses the transi- ),
tions between levels such that pex,k ∝ gφφ(|gsik)ω2/ǫ2k. (ex(cid:0)(cid:1) 00..000042 00..0042
This result is again in perfect agreement with the gen- 0.000 0.00
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
g=1.0 g=10
eral prediction for linear quenches in the absence of geo- 0.6 0.6
metric phase [12] giventhat inthis case ω is the velocity 0.5 (cid:2)ex 0.5 (cid:2)ex
) nex nex
of the quench. (iii) The most interesting and nontriv- n(ex(cid:1) 0.4 0.4
ial situation where the geometric phase strongly affects ),0.3 0.3
the dynamics occurs when both the rotation frequency (ex(cid:1) 0.2 0.2
(cid:0) 0.1 0.1
and rotation angle are not small: ω & ∆ǫ , φ & 1.
k f 0.0 0.0
In particular, in the limit ω and φ = πn we 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
→ ∞ f (cid:3) (cid:3)
recover p = 0. This trivial physical fact that in-
ex,k
finitely fast rotation can not cause transitions between
Figure2: Aplotofthetotaldensityofexcitedquasi-particles
levels actually comes from the mathematical identity: (red line) and theenergy density of excitations (blue line) as
(∆A )2 +4g (gs ) = [Tr(∂ )]2 = 4. For large but a function of the rotation frequency ω in the limit of many
φ,k φφ k φ
finite ω and φf =|πni, we find rotationsφf ≫1andforvanishingmagneticfieldh=0,with
different anisotropy g = 0.01,0.1,1.0,10, showing a cusp at
pex,k gφφ(gs k)sin2 ∆ǫk∆Aφ,kφf . (8) athseac“rdityicnaalmdirciavlinqguavneltoucmityphωacs=e t1r,anwshiticiohnc”a.n be interpreted
≈ | i (cid:20) 2ω (cid:21)
4
singularity is recoveredvia the stationary phase method for ω > 0 we describe a helical path in the parameter
weexpectittobe validforaclassofmodelswithsimilar space. InFig. 3we presentthe density ofexcitations ob-
Hamiltonians. This cusp and the associated “dynamical tained from an exact numerical integration of the time-
quantum phase transition” is directly related to the dependent Schr¨odingerequation. We recover(Fig. 3) for
effect of geometric phase. To understand this let us ω = 0 (lowest curve) the scaling n √δ, as expected
ex
apply a unitary transform Uˆ (t) = diag(e+iφ(t),e−iφ(t)), by the Kibble-Zurek scaling argument∼[13, 14]. With in-
k
to go into a rotating frame, where the geometric creasing ω the density of excitations makes a crossover
phase is removed from the Hamiltonian. The resulting to a different linear scaling regime with δ. However, in
Hamiltonian in the rotating frame reads Hˆ = accordwithour generaldiscussioninthe strictadiabatic
k,rot
[(h cosp )+∂ φ]σˆz + gsinp (sin2φσˆx cos2φσˆy), limit we always observe n √δ.
k t k ex
− − ∝
where the spectrum takes the following form
Conclusion.—In summary, we addressed how the
ǫ = (h+ω cosp )2+g2sin2p . From the geometric phase influences quantum many-body non-
k,rot k k
spectrum wqe see th−at the Hamiltonian in the rotated equilibrium dynamics. We showed that at intermediate
frame has a quantum phase transition at h + ω = 1. of fast driving regimes geometric phase strongly affects
c
This transition gives raise to the cusp in Fig. 2. We transitionprobabilitiesbetweenlevels. Weshowedthata
note though that the emergence of the cusp is non- dynamicalquantum phasetransitioncanemergeasa re-
trivial since by quenching rotation frequency we are sultofa competitionbetweenthe geometricanddynam-
pumping finite energy density to the system. In the icalphases. Thistransitionmanifestsitselfinthe“cusp”
equilibrium this model does not have any singularities inthedrivingvelocitydependence ofvariousobservables
at finite temperature. Thus this singularity is a purely (like e.g. the density of excitation and the energy den-
non-equilibrium phenomenon. Further, Fig. 2 illustrates sity) at finite energy. This allows us to probe quantum
nicely that for a small g the regime where n and ǫ criticalities “from a distance”, without actually crossing
ex ex
saturate with ω is close to ω . However for g > 1 the them. Such a possibility should be attractive from an
c
dependence ofthe saturationpointong is approximated experimentalpointofviewsincethesystemdoesn’tneed
numerically as ω (g,h=0)=51.7g0.54+21.8g1.35. to undergo a QPT. We also found that the geometric
sat
Anotherpossibilitytoanalyzetheinterplayofgeomet- phase modifies the scaling with the driving velocity as
ric and dynamical phases on excess energy and density compared to the LZ scaling. This can be related to ef-
of excitations is to consider the following helical driv- fectivetopology-inducedinteractionbetweenthedefects.
ing protocol: (h(t) = δt,φ(t) = ωδt), beginning in the Thiseffectisstrongerinthegap-lessregionsofthephase
ground state at t =0 and stopping at t = 2, i.e. cross- diagram. We also note that our results rely only on the
i f δ
ing a quantum critical point. Now δ plays the role of geometryofthephasespaceandthusrathergeneric. We
driving velocity, both in h and, for ω = 0, in circular expect that they extend to other protocols where one
directions andω determines the helicity6ofthe path. For applies a time-dependent unitary transformation to the
Hamiltonianorothertransformationwhichinvolvesnon-
trivial geometric phase. In particular, similar considera-
tionsapplytotheDickemodelrealizedinRef.[15]. This
2-2 -1g0y 1 2 and possible other generalizations of our results (e.g. for
1 open [16] or turbulent [17] systems) will be discussed in
h
0 a separate work.
-1-2 -1 g0x 1 2
[1] M.Nakahara,Geometry,Topology andPhysics,Taylor&
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5
tice Hall, 1994. areusuallynon-localandleadtoconstantenergyabsorp-
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University Press, Cambridge, 1999). the qualitative results of the Paper are valid even if we
[11] K. Damle and S. Sachdev, Phys. Rev. Lett. 76, 4412
addarbitraryinteractionstotheHamiltonian,whichpre-
(1996).
servetherotationalsymmetrybutbreakitsintegrability.
[12] C.DeGrandi,V.Gritsev,A.Polkovnikov,Phys.Rev.B
81, 012303 (2010).
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105701 (2005).
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Nature 464, 1301 (2010). tainedfor a generic two-levelsystemusing only the min-
[16] A. Tomadin, S. Diehl, and P. Zoller, Phys. Rev. A 83,
imal assumption that time dependence enters through
013611, (2011).
some U(1) rotation with constant frequency. For such
[17] B.Nowak,D.Sexty,andT. Gasenzer, Phys.Rev.B 84,
a system the Schr¨odinger equation in the instantaneous
020506(R), (2011).
basis gs , es can be written as
{| i | i}
Supplementary Information ∂ta˜gs = gs∂t es exp[iEgs,es(t) iΓgs,es(t)]a˜es, (13)
−h | | i −
∂ a˜ = es∂ gs exp[iE (t) iΓ (t)]a˜ , (14)
t es t es,gs es,gs gs
−h | | i −
Effective Hamiltonian in the rotating frame
where E (t) and Γ (t) are the dynamical and ge-
nm nm
ometric phases as defined in the main text. No-
Let us consider a more general setup, which extends
tice that we applied the gauge transformation a =
the example studied in the main text. Suppose that the n
t
systemisdescribedbysomeinteractingHamiltonianH0, a˜nexph tidτ(−iǫn(τ)+iAτ(|ni))itothecoefficientsin-
which is rotationally invariant with respect to some vec- troducedR by: ψ(t) = a (t)gs(t) + a (t)es(t) . In
gs es
tor coupling λ [1]. This coupling can represent, for ex- agreement with| thei discussion| in tihe previo|us seiction
ample, an external magnetic or electric field, anisotropic all matrix elements appearing in the Schr¨odinger equa-
interaction constant, couple to a nematic order param- tion(13),(14)aretimeindependentfortheanglelinearly
eter etc. It can also break an internal U(1) symmetry changing in time φ(t) = ωt. It is convenient to change
of like the mixing symmetry between different spin variables from time t to the angle φ in (13), (14):
0
comHponents. At nonzero value of λ the rotational sym-
E (φ)
metry is thus broken. Now let us consider a dynamical ∂ a˜ = gs∂ es exp i gs,es iΓ (φ) a˜ ,
φ gs φ gs,es es
process where this coupling uniformly rotates at a fixed −h | | i (cid:20) ω − (cid:21)
magnitude. Thenthe time dependentHamiltonianreads (15)
H(t)=U−1(t)H(λ0)U(t), (10) ∂φa˜es =−hes|∂φ|gsiexp(cid:20)iEes,ωgs(φ) −iΓes,gs(φ)(cid:21)a˜gs,
where U(t) is the unitary operatorcorresponding to this
(16)
rotation. In the rotating frame ψ˜(t) = U ψ(t) the ef-
| i | i
fectiveHamiltonianintheSchr¨odingerequationpicksup differentiating Eq. (16) one more time with respect to φ
an additional “centrifugal” term: and eliminating a˜ , ∂ a˜ gives a second order differen-
gs φ gs
i~∂ ψ˜ = ψ˜ , (11) tial equation with constant coefficients:
t eff
| i H | i
= (λ ) i~ωU−1∂ U, (12) ∆ǫ
Heff H 0 − φ ∂2a˜ i ∆A ∂ a˜ +g (gs )a˜ =0. (17)
where φ is the rotational angle and ω = φ˙ is the fre- φ es− (cid:18) ω − φ(cid:19) φ es φφ | i es
quency. The centrifugal term in the Hamiltonian has a
FromthisitistrivialtoderiveEq.(6)fromthemaintext:
number of interesting properties. (i) It is proportional
to the frequency ω and thus canbe used to continuously sin2 1Ω(ω)φ
p = a (φ )2 =g (gs ) 2 f , (18)
modify the effective Hamiltonian. (ii) At constant fre- ex | es f | φφ | i 1(cid:2)Ω(ω) 2 (cid:3)
quency the effective Hamiltonian is time independent. 2
(cid:2) (cid:3)
(iii)Thediagonalcomponentsofthecentrifugaltermare
with Ω(ω):= ∆ǫ ∆A 2+4g (gs ).
given by the connection 1-form of the corresponding en- q ω − φ φφ | i
(cid:2) (cid:3)
ergy levels. Let us point that the time independence
of the centrifugal term, which follows its rotational in-
variance, and its locality imply that the rotations in the
generic parameter space do not lead to the continuous [1] In general the Hamiltonian H0 can be invariant with re-
heatingeveninergodicnon-integrablesystems. This sit- spect to an arbitrary continuous group, not necessarily
uation is opposite to e.g. Floquet Hamiltonians. Which rotations