Above we presented a second order perturbative calculation for the overlaps on the post-quench basis. While the formal results are expected to hold true in any quantum field theory, in practice their applicability is more restricted:
they apply to quenches with integrable post-quench dynamics. The scope of the perturbative approach can be partially extended by covering the case of integrable pre-quench dynamics. This drives us to pursue a similar approach along the second direction introduced in Sec. 5.1, i.e. to perform an analogous expansion on the pre-quench basis.
Computationally, this amounts to solving a reversed task: in the previous calculations we had to obtain the correction to the pre-quench vacuum order by order, here it is necessary the compute the perturbative correction to each eigen-state, focusing only on the part that is proportional to the pre-quench ground state |Ω⟩. Apart from this modification, the steps of the calculations are almost identical to the preceding ones so instead of a detailed calculation we focus on the results while commenting on the differences.
5.3.1 One-particle overlaps
For the g amplitudes the major changes compared to Eq. (5.27) are in the denominators: the energy differences appearing in the perturbative expansion are now with respect to the one-particle mass instead of the vacuum. Cancellation of divergent parts is again due to disconnected pieces. Apart from the energy denominators, the only difference appears in the ordering of rapidities in the form factors. The final result is given by:
ga(0)
2 =−λFaϕ m2a −λ2
NXspec
b=1,b̸=a
FbϕFabϕ(iπ,0)
m2amb(mb−ma) +Faaϕ(iπ,0)Faϕ m4a
+X
b≤c
1 (2δbc)!
Z dϑ 2π
Fbcaϕ (iπ+ϑ, iπ+ϑbc,0) m2ap
m2c+ (mbsinhϑ)2×
× Fbcϕ(ϑ, ϑbc) mbcosh(ϑ) +p
m2c+ (mbsinhϑ)2−ma
+. . .
!
+O λ3 ,
(5.39)
where now all masses and form factors are those of the pre-quench theory.2 The ellipsis indicates contributions of intermediate states with more than two particles. Note that if ma > mb +mc then the denominator of the integrand has a zero and the integrand has a pole. This will occur for all one-particle states with mass ma >2m1 in one or more such integral terms. This pole is the consequence of a disappearing energy difference between a one-particle state and the two-particle continuum in infinite volume. We postpone the discussion of this singularity after presenting the two-particle overlaps.
5.3.2 Two-particle overlaps
Let us start with the discussion of the Kaa function. The first order contri-bution is simply
−λ Faaϕ(−ϑ, ϑ)
2m2acosh2(ϑ). (5.40)
The second order contribution is given as a sum over eigenstates [c.f. the third term of Eq. (B.5)]. Inserting the vacuum yields a divergent term which is can-celled by the disconnected piece of the diagonal matrix element, analogously to
2Note that we omitted the (0) superscript from the quantities on the right-hand side for brevity. Also note that for this expression to make sense, theAaparticle state has to be present both in the pre-quench model (where the right-hand side is evaluated) and in the post-quench model (where the overlap is defined).
the argument in Section 5.2. The connected part of the diagonal element disap-pears in the infinite volume limit due to the corresponding density factor and therefore the only remaining term resulting from inserting the vacuum is
− λ2 2
Faaϕ(iπ,0)Faaϕ(−ϑ, ϑ)
m4acosh4(ϑ) . (5.41)
Moving forward and inserting the one-particle states yields λ2
2 Xs
b=1
FbϕFbaaπ (iπ,−ϑ, ϑ)
m2ambcosh2(ϑ)(2macosh(ϑ)−mb), (5.42) where the aforementioned pole manifests itself as a divergence of the pair overlap function Kaa(ϑ) whenever there is a particle with mb >2ma.
Proceeding to the insertion of two-particle states, we can consider inserting a pair of particles Ab with b ̸= a, in which case the form factor has no pole. In finite volume, the corresponding contribution reads
λ2L2 2
X
ϑ′
Fbbϕ(−ϑ′, ϑ′)Fbbaaϕ (iπ+ϑ′, iπ−ϑ′,−ϑ, ϑ)
ρbb(ϑ′,−ϑ′)2macoshϑ(2macoshϑ−2mbcoshϑ′)maLcoshϑ. (5.43) Note that the pole is only present in infinite volume since for any finiteLthere are no exact degeneracies in the spectrum due to the Bethe-Yang equations (5.11).
Hence one might expect that the finite volume regularisation technique detailed in Section 5.2.1 is able to treat its effect properly. In the limit L→ ∞ limit the energy difference can be zero at
ϑ∗ = arccosh
macoshϑ mb
. (5.44)
Note that ϑ∗ is imaginary if ϑ <arccosh(mb/ma). However, for largerϑ the pole is on the real axis. Eq. (5.43) can be rewritten as
λ2 8
X
ϑ′
Fbbϕ(−ϑ′, ϑ′)Fbbaaϕ (iπ+ϑ′, iπ−ϑ′,−ϑ, ϑ)
˜
ρb(ϑ′)m2ambcoshϑ′cosh2ϑ(macoshϑ−mbcoshϑ′). (5.45) The sum can be represented as a sum of contour integrals3 using
X
ϑ′
f(ϑ′)
˜
ρb(ϑ′) =X
ϑ′
I
ϑ′
dϑ 2π
−f(ϑ)
1 +eiQ˜b(ϑ) , (5.46) wheref(ϑ) is assumed to be regular atϑ′, and the contours encircle theϑ′ values on the real axis that are given by the quantisation condition
Q˜b(ϑ′) =mbLsinhϑ′+δbbps(2ϑ′) = 2πJ , (5.47)
ϑ
0ϑ
∗= ϑ
0Figure 5.1: Illustration of the contour integrals involved in the finite volume regularisation of an infinite volume pole. The summation goes over the quantised ϑ′ (located at the crosses on the axes), and there is an additional pole at ϑ∗ (denoted by a dot). The contours can be joined together, but the residue at ϑ∗ has to be subtracted.
whereJ is a half-integer number. The contours can be joined to form two infinite lines below and above the real axis, see Fig. 5.1. On the first the integrand vanishes in the infinite volume limit while the second one yields
λ2 8
Z ∞+iϵ
−∞+iϵ
dϑ′ 2π
Fbbϕ(−ϑ′, ϑ′)Fbbaaϕ (iπ+ϑ′, iπ−ϑ′,−ϑ, ϑ)
m2ambcosh2ϑcoshϑ′(macoshϑ−mbcoshϑ′). (5.48) When joining the contours, it is necessary to subtract the residue of the pole at ϑ=ϑ∗,
λ2 8
iFbbϕ(−ϑ∗, ϑ∗)Fbbaaϕ (iπ+ϑ∗, iπ−ϑ∗,−ϑ, ϑ)
m2ambcosh2ϑcoshϑ′mbsinhϑ∗(1 +eiQ˜b(ϑ∗)). (5.49) Even though the result is finite, it does not have aL → ∞ limit due the factor eiQ˜b(ϑ∗) ∼ eimbLsinhϑ∗. Consequently, the sum in (5.45) still fails to have a well-defined infinite volume limit.
Therefore the singularities corresponding to vanishing energy denominators are intractable by the method of finite volume regularisation. One may try other ways to circumvent this problem and arrive at a regular expression well defined in the L→ ∞ limit, however we failed in all our attempts so far. So the proper treatment of these singularities remains an interesting open question.
Nevertheless, there exist particular quenches which are free of the complica-tions discussed above. If the matrix elements of the operatorϕ are proportional to the energy of the involved states, the divergence is cancelled and the sum can be readily transformed to an integral. For instance, this is the case for Type I quenches considered in the previous chapter, where the quenching operator is σ(x). It is proportional to the trace of the energy-momentum tensor [41], and as a consequence all of its form factors are proportional to the total energy of the appropriate states.
The above mathematical reasoning is illuminated by considering the physical picture. Note that in general the pre-quench basis is not an optimal choice to ex-press the dynamics of the post-quench Hamiltonian. For example, heavy particles whose kinematically allowed decays in the pre-quench system are only prohibited
3Analogously to the treatment of disconnected pieces for the Kaa andKab functions in the post-quench expansion. For more details, see AppendixB.2.
by integrability become unstable. These particles are expected to acquire a finite lifetime which is reflected by the divergent terms of the perturbative series. The resolution would require a resummation, which is expected to shift the singularity away from the real axis. Similarly, two-particle states acquire a finite lifetime due to inelastic processes. The situation is radically different for quenches where the pre-quench and post-quench setting corresponds to the same integrable model.
Such a protocol is simply equivalent to a rescaling of parameters describing the spectrum, which retains its structure. Accordingly, one does not expect diver-gences in perturbation theory and in fact, they are absent—apart from the ones tractable with the method of finite volume regularisation.
Consequently, for this specific class of quenches, we can express the pertur-bative overlap functions up to second order on the pre-quench basis as well. The explicit formulae for the two-particle overlap functions are once again elaborate, we report them in App. B.3.