with two different analytic descriptions. Both of them are form factor expansions but they differ in important aspects. The approach developed in Refs. [40, 121]
is a perturbative one that exploits the integrability of thepre-quenchsystem but does not require integrability of the post-quench Hamiltonian. The approach de-veloped by Refs. [33,34,122] relies on the integrability of thepost-quenchsystem and makes certain assumptions about the initial state. This approach is expected to be valid for small quenches but it is not perturbative in the same sense as the pre-quench expansion. Both methods can be applied to small quenches between integrable systems, e.g. to a parameter quench in an integrable model, which allows for a comparison of the two methods using the numerical TSA results.
In the following we present the results of the comparison in the E8 model and discuss the characteristic features of quenches in this field theory.
4.2 Modelling the time evolution in E
8field
M h
CFT Type I
Type II
Figure 4.1: Illustration of quenches considered in this chapter in the M − h parameter space. The lower (blue) dot on the axis represents the post-quench Hamiltonian for Type I quenches, and the pre-quench Hamiltonian for Type II quenches, which were performed in both (negative and positiveM) directions.
coupling hi and the post-quench parameter hf we introduce the dimensionless combination ξ through
ξ ≡ λ hf
= hi−hf hf
. (4.2)
Note that the choice ofhfsimply sets the energy scale of the post-quench system through Eq. (2.15). We choose it such that the mass gap m1 of the post-quench Hamiltonian is set to unity,m1 = 1.
The magnitude of Type II quenches is measured by the dimensionless combi-nation η of M and h, defined in Eq. (2.17). In this case the post-quench model is non-integrable, hence the choice of hi sets the energy scale of the pre-quench theory. We choose it such that the mass gapm(0)1 of the pre-quench Hamiltonian is set to unity, m(0)1 = 1, where we introduced the notation superscript (0) to differentiate between quantities on the pre- and post-quench bases.
Having set the energy scales for both types of quenches we measure each physical quantity in appropriate powers of the mass gap (m1 orm(0)1 , depending on the quench protocol considered): e.g. energy differences are measured in mass gap units, while time and distance with the inverse of the energy scale set by the mass gap.
4.2.2 Modelling methods
As alluded to in Sec. 4.1, we have three ways to describe the time evolu-tion following a global quench in the E8 field theory. The first one is a nu-merical method based on Hamiltonian truncation, the latter two are analytical approaches: a perturbative expansion in the λ parameter of Eq. (4.1) (later on referred to as ‘perturbative quench expansion’) and a linked cluster expansion on the post-quench basis (‘post-quench expansion approach’ from now on) assuming
a specific form of the post-quench initial state with a sufficiently small energy density. Before proceeding to the presentation of the post-quench dynamics, we introduce these three methods in a bit more detail. We focus on the evaluation of dynamical one-point functions, as both analytical approaches are worked out on this example.
The numerical method is the conformal Hamiltonian truncation introduced in Sec.2.2.2. The numerical treatment of the quench problem proceeds via an ini-tial determination of the ground state |0⟩ of the pre-quench Hamiltonian (with λ = 0), and then solving the Schr¨odinger equation involving the post-quench Hamiltonian with |0⟩ as an initial condition. For the TCSA computations peri-odic boundary conditions are used. As the time-evolved state is explicitly con-structed, virtually any physical quantity related to the quench can be calculated this way, the only limitation being the errors arising from the truncation of the Hamiltonian to a finite dimension. To alleviate the limitations, we perform ex-trapolation in cutoff using Eq. (2.20). The finite volume parameter induces an upper time limit, until when the numerical results provide an accurate depiction of the dynamics. We return to this question in Sec. 4.3.1.
The first analytical approach we consider is the perturbative quench expan-sion [40, 121]. Its basic assumption is that the quench starts from an integrable HamiltonianH0, which is changed during the quench by the addition of an extra local interaction to
H =H0+λ Z
dxΨ(x), (4.3)
where λ is small enough to justify the application of perturbation theory. To first order in λ and including only one-particle contributions, the perturbative prediction for the post-quench time evolution of a local operator Φ is [40]
⟨Φ(t)⟩=⟨0|Φ|0⟩+λ X8
j=1
2
m(0)j 2Fj(0)Ψ∗Fj(0)Φcos m(0)j t
+· · ·+CΦ, (4.4)
where |0⟩ is the pre-quench vacuum, m(0)j are pre-quench one-particle masses and CΦ is included to satisfy the initial condition that the ⟨Φ(t)⟩ function is continuous at t = 0. The amplitudes Fj(0)Φ are the one-particle form factors of the Φ operator defined by Eq. (3.8), with an additional particle species index j.
Note that all quantities are taken at their pre-quench values reflecting that this approach only assumes integrability of the pre-quench Hamiltonian.
The ellipsis denotes the contribution of higher particle states; their omis-sion corresponds to a low-energy approximation valid for long enough times t ≳1/m(0)1 . Validity of perturbation theory for the time evolution operator also places a theoretical upper time limit for the validity of this expression in terms of the quench amplitude
t∗ ∼λ−1/(2−∆Ψ), (4.5)
where ∆Ψ is the scaling dimension of the quenching Ψ operator.
The second method, introduced for quenches in the free massive Majorana field theory [33] and developed further in Refs. [34, 122, 123], builds upon the premise that the post-quench system is integrable. It can be considered as a systematic expansion in the post-quench particle density as a small parameter, which means that it is limited to small enough quenches, which are, however, not necessarily perturbative in the Hamiltonian sense used above. Consequently, it has no upper time limit, while due to being a low-energy expansion it has a lower time limit in terms of the post-quench mass m1. Adapting the results of [122] to the case of the E8 model, the following time evolution is obtained for operator Φ to leading order:
⟨Φ(t)⟩=⟨Ω|Φ|Ω⟩+ X8
j=1
|gj|2
4 Re[FjjΦ(iπ,0)] + X8
j=1
Re[gjFjΦe−imjt]
+X
k̸=j
Re gk∗gj
2 FkjΦ(iπ,0)e−i(mj−mk)t
+. . . , (4.6) where|Ω⟩ is the post-quench vacuum state,
gj
2 =⟨Ψ(0)|Aj(0)⟩ (4.7)
is the overlap of the initial state|Ψ(0)⟩with a zero-momentum post-quench one-particle state of species j, and the FΦ form factors are matrix elements on the post-quench basis.
Once again, the ellipsis indicates the contribution involving higher many-particle states. In Refs. [33, 34, 122, 123] the first few of them were evaluated, and they were found to contain secular terms proportional to powers oft. Their resummation may lead to the appearance of frequency shifts and decay factors through functions ∼ eiat and ∼ e−bt respectively. Both effects are consequences of the finite post-quench particle densities. We neglected these terms in our con-siderations for two reasons. First, the TCSA data for the relevant quenches show no signs of damping or substantial frequency shifts on the time-scales of our simulations (see below). Second, their computation requires the knowledge of two-particle overlap functions, and their numerical determination requires con-siderable effort, see Chapter 5.
We note that unlike the perturbative approach, the second method does not construct the initial state but needs the one-particle overlaps gj as inputs. We determined them by explicitly constructing the corresponding states in TCSA and computing their scalar products with the initial state. Apart from this nu-merical input, the evaluation of Eqs. (4.4,4.6) is a mere substitution exploiting the exact relations for the particle masses and the form factors. For the latter we use the ones available at Ref. [92].1
1The only caveat is that we normalise our operators according to their short-distance
op-With the three approaches at hand, we turn to the presentation of the re-sults. We treat the two types of quenches separately. For Type I quenches that connect two points on the integrable E8 axis, all three approaches introduced in the previous section can be used to describe the post-quench time evolution. In contrast, Type II quenches do not allow for the application of the post-quench expansion as the resulting model is not integrable.
For each type, we consider three dynamical quantities: the Loschmidt echo, and the one-point functions of the σ and ϵ fields. The Loschmidt echo can be used to test the validity of TCSA data, outlining the limitations to the appli-cability of the approach in light of truncation errors and finite volume effects.
The dynamical one-point functions then serve as a benchmark for the analytical approaches in comparison with our numerical method. The time evolution is also analysed through the lens of Fourier transformation in the spirit of the “quench spectroscopy” idea put forward in Ref. [40].