5.4 Testing the results in the E 8 Ising field theory
5.4.1 Integrable post-quench dynamics
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.
5.4 Testing the results in the E
8Ising field
i.e. they correspond to a sudden change of h from the initial hi to the final hf
at t = 0, expressed by the Heaviside function Θ(t). For t ≤ 0 the system is in the ground state of the pre-quench Hamiltonian, which is the initial state of the out-of-equilibrium time evolution which happens for t > 0. The quench can be characterised by the dimensionless quench magnitude
ξ ≡ hf−hi
hf . (5.51)
These quenches have integrable post-quench dynamics, hence the post-quench expansion provides predictions for the overlap functions in this case. They also belong to the special class of quenches where the pre-quench expansion yields sensible results. As a consequence, the numerical data can be compared with both approaches at the same time. All the TCSA calculations are performed in finite volume, so in order to compare the numerical results to the perturbative predictions it is convenient use the finite volume normalisation of Eqs. (5.17) and (5.30). In the numerical calculations we measure everything in appropriate powers of the post-quench mass gap m1, so the perturbative parameter λ is obtained by multiplying ξ with the post-quench parameter hf.
Let us remark that the overlap functions are defined up to a phase factor, since we can freely choose the phase of any quantum state. The TCSA uses a basis in which all vectors are real, consequently the overlaps obtained from this approach are also real. Thus the comparison is performed such that we take the absolute value of the perturbative overlap functions.
Before turning to the discussion of the comparison with TCSA calculations, let us comment on the numerical evaluation of the perturbative formulae. The second order contributions involve a sum over all possible intermediate states, which means an infinite summation over all possible number N of inserted par-ticles. In the above calculations we truncated the infinite sum at pair states with N = 2. We observe that the contribution from the terms involving an integral over the momentum of a pair state is very small in most of the cases. Conse-quently, we argue that the error we make by truncating the form factor expansion at two-particle intermediate states is orders of magnitude smaller than the main contributions in second order. This argument is supported by the numerical eval-uation of the various terms presented in AppendixB.4.
Coincidentally, there is another source of truncation in the analytic expan-sions, related to the large number of particle species in theE8 model. Currently, the set of available form factors is incomplete already at the three-particle level, hence the full second order contribution cannot be evaluated even for N ≤ 2 inserted particles. Our calculation presented in Chapter3significantly increased the number of available three- and four-particle form factors, reducing the error in the perturbative expansions coming from the truncation of the form factor series.
0 2·10−2 4·10−2 (a)
g1
3.95·10−3
0 5·10−3 1·10−2
(b)
g2
1.65·10−3
−0.2 0 0.2 0.4
0 1·10−3 2·10−3 3·10−3 (c)
ξ g3
7.7·10−4
−0.2 0 0.2 0.4
0 5·10−4 1·10−3 1.5·10−3
(d)
ξ
g4
5.28·10−4
TCSA Post-1st Post-2nd
Pre-1st Pre-2nd
Figure 5.2: Comparison between TCSA overlaps of the lightest four one-particle states and two different perturbative expansions as functions of the quench mag-nitude ξ for quenches along the E8 axis in volume m1R = 40. Dashed lines indicate the first-order predictions of the post-quench expansion and continuous lines depict the sum of the first two orders. The pre-quench result up to the first and second order is shown in dotted and dot-dashed lines, respectively. TCSA data is shown by orange dots. Inset: the absolute deviation of second order re-sults from TCSA. The numbers above the inset indicate the maximal deviation in the plotted ξ interval.
One-particle overlaps
We begin the discussion of the comparison for the case of one-particle over-laps. The ga functions for the four lowest-lying states are presented in Fig. 5.2.
The observation of these functions reveals that the perturbative expressions de-scribe the overlaps very well for a quite wide range of quench magnitudes.
The agreement is most precise for the A1 and A2 particles in the top row, while the analytical data is slightly less accurate for the A3 and A4 states. This can partially be attributed to the limited knowledge of three-particle form factors remarked above. The largest number of the form factors are accessible for the case of the lightest particle A1, therefore the agreement is the best for this case and the domain of validity almost covers the whole region of the plot. For heavier
0 1 2 0
0.5 1
·10−3
(a)
p/m
1| K
11(p) | , ξ = 0.05
0.25 0.35 8
8.5
·10−4
0 1 2
0 1 2
·10−3
(b)
p/m
1| K
11(p) | , ξ = 0.1
0.25 0.35 1.6
1.7 1.8
·10−3
TCSA Post-1
stPost-2
ndPre-2
ndFigure 5.3: Perturbative predictions against the TCSA data for the two-particle overlap K11 as a function of the dimensionless momentum p1/m after quenches of size (a)ξ = 0.05,and (b)ξ = 0.1.The overlap was determined by TCSA using volumesm1R= 30. . .65.. Black lines correspond to the first-order predictions of the post-quench expansion, and blue continuous lines depict the sum of first two orders. The result of the pre-quench expansion is denoted by black dashdotted lines. The inset shows a magnified section around the local maxima of the curves.
particles it is expected that extending the set of available form factors would result in a better agreement with TCSA data, although the domain of validity presumably remains smaller than for A1 (see TableB.1).
We also note that including the second order leads to a major improvement of the agreement between the perturbative and TCSA results in almost the whole parameter region presented here. This is valid for both the post- and pre-quench perturbative expansions, but the former gives more accurate predictions, in line with the expectation that the post-quench basis is better suited to describe the quench problem. This is reflected by the figures in the inset which show the absolute deviation of the two perturbative expansions from the overlaps obtained from TCSA: the post-quench curve remains consistently below the pre-quench mismatch in all but one case.
Two-particle overlap functions
The multiple particle species present in the Ising Field Theory provide an opportunity to observe both the Kaa and Kab functions. In this case we used the data of quenches at a few different values of ξ and plotted the overlaps as
0 0.5 1 1.5 2 2.5 0.5
1 1.5 2
·10−4
(a)
p/m
1| K
12(p) |
0 0.5 1 1.5 2 2.5
0.4 0.6 0.8
·10−4 1
(b)
p/m
1| K
22(p) |
TCSA Post Pre
Figure 5.4: Overlap functions of pairs of heavier particles (a) K12(p) and (b) K22(p) after a quench of size ξ = 0.05 along the E8 axis. The first-order results are omitted, all other conventions are as in Fig. 5.3.
functions of the momentum parameterising the particle pair.
The comparison for the overlap function of a pair of the lightest particle is presented in Fig.5.3, which shows that the perturbative expansion performs well also for pair overlaps in matching the numerical results of TCSA for quenches with ξ = 0.05 and ξ = 0.1. The change from first to second order is less spec-tacular as it was in the one-particle case, but it still significantly improves the agreement, more noticeably as the quench magnitude increases, see the inset in Fig. 5.3b. In addition, the second order correction dramatically alters the quali-tative behaviour of the overlap since it introduces a pole for zero momentum (cf.
the discussion at the end of Sec. 5.2).
It would be worthwhile to extend the numerical data to investigate the pres-ence of the pole. Unfortunately, this is hindered by two reasons: first, to obtain lower momenta, the volume parameter has to be increased to a regime where TCSA becomes less accurate. Second, due to the quantisation condition ofA1−A1
pair states, their energy gets within touching distance of each other as the volume is increased. Since numerical diagonalisation techniques do not always properly discern between near degenerate states, the resulting overlap will be imprecise.
Nevertheless, the agreement shown in Fig. 5.3 is quite convincing.
It is also possible to compare the predictions regarding overlap functions of pair states of heavier particles, as well as that of pairs composed of different species. Fig.5.4illustrates that, similar to the case ofK11, the analytic prediction still agrees very well with the TCSA numerics, with the difference that there is a
slightly bigger difference between the two approaches compared to the case shown in Fig.5.3. Similarly to the single-particle overlaps, the post-quench approach is closer to the numerical data. As we observe a remarkable agreement between the perturbative expressions and the numerics in Fig.5.4a, we conclude that Type I quenches, despite the equilibrium models both being integrable, do not meet the criteria (5.3) of integrable quenches. This observation reinforces the similar but purely perturbative argumentation of Ref. [40].