To use the framework of adiabatic perturbation theory in theE8 model we assume that the time-evolved state can be expressed as
|Ψ(t)i=X
n
αn(t) exp{−ıΘn(t)} |n(t)i , (A.1) with the dynamical phase factor Θn(t) = Rt
tiEn(t0) dt0. We also assume that there is no Berry phase and thus to leading order in the small parameter λ˙ the αn coefficients take the form Higher derivatives as well as higher order terms inλ˙ are neglected from now on.
Theαncoefficients can be used to formally express quantities that have known matrix elements on the instantaneous basis of the Hamiltonian:
hO(t)i=X
m,n
α∗m(λ(t))αn(λ(t))Omn. (A.3) In what follows, we present the evaluation of this sum - approximately, under conditions of low energy density discussed in the main text - for the case of O(t) = H(t)−E0(t) in theE8 model. To generalise this calculation to the defect density or to higher moments of the statistics of work function is straightforward. The work density (or excess heat density) after the ramp reads The energy and momentum eigenstates are the asymptotic states of the model labelled by a set of relativistic rapidities{ϑ1, ϑ2, . . . ϑN}and particle species indices {a1, a2, . . . aN}: sum-mation in Eq. (A.4) in principle goes over the infinite set of asymptotic states. As discussed in the main text, for low enough density we can approximate the sum in Eq. (A.4) with the contribution of one- and two-particle states, analogously to the calculation in the sine–
Gordon model in Ref. [17].
A.1 One-particle states
Contribution of the one-particle states can be expressed as w1p = lim
where ma is the mass of the particle species a and the summation runs over the eight species. We can write the coefficientαa as
αa(λf) =
whereh{0}a(λ)| denotes the asymptotic state with a single zero-momentum particle. The matrix elements and masses depend onλthrough the Hamiltonian that defines the spec-trum. The matrix element can be evaluated as
h{0}a(λ)|∂λ|0(λ)i=−h{0}a(λ)|V |0(λ)i
ma(λ) . (A.8)
For an E8 ramp that conserves momentum, V is the integral of the local magnetisation operatorσ(x):V =RL
0 σ(x)dx. Utilising this we further expand h{0}a(λ)|∂λ|0(λ)i=− LFaσ∗(λ)
ma(λ)p
ma(λ)L, (A.9)
where the square root in the denominator emerges from the finite volume matrix element [109] andFaσ is the (infinite volume) one-particle form factor of the magnetisation operator.
It only depends on the couplingλ through its proportionality to the vacuum expectation value of σ. The particle masses scale as the gap: ma(λ) = Ca|λ|zν, where Ca are some constants. This allows us to write
|αa(λf)|2 =L
We can perform the integral in the exponent that leads to a τQ|λ|1+zν dependence there.
To get rid of the largeτQ factor in the denominator, we introduce the rescaled coupling ζ with
where C˜a and Ca0 are constants that depend on Ca, the one-particle form factors and the critical exponents. We note the integral is convergent for large ζ due to the strongly oscillating phase factor and also forζ →0since2ν−1−3/2zν =−11/15 in theE8 model.
Substitutingz= 1 in the exponent ofτQ leads to the correct KZ exponent of a relativistic model,ν/(1 +ν).
A.2 Two-particle states
The contribution of a two-particle state with speciesaandbis going to be denotedwab
and reads whereϑab is a function of ϑdetermined by the constraint that the state has zero overall momentum. The summation goes over the rapidities that are quantised in finite volumeL by the Bethe–Yang equations: whereIi are integers numbers and
δab=−ılogSab (A.15)
is the scattering phase shift of particles of typeaandb. For a two-particle state Eq. (A.14) amounts to two equations of which only one is independent due to the zero-momentum constraint. It reads
Q(ϑ) =˜ maLsinhϑ+δab(ϑ−ϑab) = 2πI , I ∈Z. (A.16) In the thermodynamic limitL → ∞ the summation is converted to an integral with the integral measure dϑ2πρ(ϑ), where˜ ρ(ϑ)˜ is the density of zero-momentum states defined by
˜ whereΦ(ϑ) is the derivative of the phase shift function. The resulting integral is
1 Analogously to the one-particle case we can evaluate the matrix element in the E8 field theory as the density factor is the Jacobian of the two-particle Bethe–Yang equations (A.14) arising from the normalisation of the finite-volume matrix element [109]. It can be expressed as
ρab(ϑ1, ϑ2) =maLcoshϑ1mbLcoshϑ2+(maLcoshϑ1+mbLcoshϑ2)Φab(ϑ1−ϑ2). (A.21) Observing Eqs. (A.17) and (A.21) one finds that the details of the interaction enter via the derivative of the phase shift function but crucially, they are of order1/L compared to the free field theory part. So leading order inL we find that
wab(λf) =
A change of variables in the outer integral to the one-particle momentum p = masinhϑ we obtain
Now we can introduce the momentum p in the inner integral as well by noting that the energy can be expressed as a function of momentum via the relativistic dispersion and that
the relativistic rapidity alsoϑ=arcsinh(p/m). Sincem∝ |λ|zν withz= 1 any expression that is a function ofϑcan be expressed as a function of p/|λ|ν. Having this in mind, the result is analogous to the free case so all the machinery developed there can be used. The key assumptions from this point regard the scaling properties of the energy gap and the matrix elementG(ϑ) in this brief notation:
Ep(λ) =|λ|zνF(p/|λ|ν) (A.24) G(ϑ) =λ−1G(p/|λ|ν). (A.25) These equations are trivially satisfied with the proper asymptotics forF(x)∝xz. ForG(x) one can verify using that in theE8 model we have
L→∞lim h{ϑ, ϑab}(λ)|∂λ|0(λ)iL= hσiFabσ∗(ϑ, ϑab)
√macoshϑmbcoshϑab(macoshϑ+mbcoshϑab)
=λ1/15−8/15−8/15G(ϑ) =λ−1G(ϑ), (A.26) where we neglected theO(1/L)term from the finite volume normalisation and usedhσi ∝ λ1/15,m∝λ8/15.Fab(ϑ, ϑab) is the two-particle form factor of theE8 theory that does not depend on the coupling. They satisfy the asymptotic bound [99]:
|ϑlimi|→∞Fσ(ϑ1, ϑ2 . . . , ϑn)≤exp(∆σ|ϑi|/2). (A.27) Since the matrix elements considered here are of zero-momentum states, ϑ → ∞ means ϑab → −∞ and Fabσ(ϑ, ϑab) ≤ exp(∆σϑ) as the form factors depend on the rapidity dif-ference. Dividing by the factorexp(2ϑ) in the denominator yields the correct asymptotics G(x) ∝x∆−2 =x−1/ν as an upper bound due to Eq. (A.27). We remark that the scaling forms (A.24) hold true for any value of the coupling λin the field theory, in contrast to the lattice where they are valid only in the vicinity of the critical point. From this per-spective Eq. (A.24) follows from the definition of the field theory as a low-energy effective description of the lattice model near its critical point.
As a consequence, one can introduce new variables in place of λ and p such that the explicit τQ dependence disappears from the integrand. This is achieved by the following rescaling:
The result for the energy density is wab=τ− in front of the integral. If it converges, Eq. (A.29) completely accounts for the KZ scaling.
Second, if |λf|= 0, the energy gap isEp ∝pz and an additional factor of τ−
ν 1+zν
Q appears
in front of the integral. Note that this is the scaling ofκ1 on Fig. 5.2. The high-energy tail of the integrand is modified due to the extra term ofηz from the energy gap. This leads to a convergence criterion such that once again the crossover to quadratic scaling happens when the exponent of τQ in front of the integral is less then −2. It is easy to generalise this argument to thenth moment of the statistics of work which amounts to substituting Epninstead ofEp to Eq. (A.29). As argued in the main text, this is the leading term in the nth cumulant of the distribution as well, that concludes the perturbative reasoning behind the results of Sec. 5.