7.2 Eigenstate dynamics
7.2.2 Ramps along the free fermion line
As already noted, the ramp dynamics along the free fermion line is exactly solvable; we exploit this fact to draw a robust picture of eigenstate dynamics by comparing the analytical and numerical results. We perform linear ramps of the type (7.9), with the role of the coupling parameter λ played by the time-dependent fermion mass M :
M(t) = −2Mit/τQ, (7.29)
where Mi is the initial value of the coupling at t =−τQ/2. As discussed in Sec.
7.1.2, the critical exponents in this case are ν = 1, z = 1, so the Kibble–Zurek time (7.3) scales asτKZ∼ √τQ. To test the adiabatic-impulse-adiabatic scenario, we need to have a specified value of τKZ which we simply set as
mτKZ=√mτQ, (7.30)
wherem =|Mi|is the mass gap at the start of the ramp. Depending on the sign of Mi, the ramp is either towards the ferromagnetic phase or the paramagnetic phase; we are going to present our results in this order.
−8 −6 −4 −2 0 2 4 6 8 0
0.2 0.4 0.6 0.8 1
mt
Overlap
Impulse regime Vacuum p1−p3pairs p1−p5pairs p3−p5pairs p1pair p3pair p5pair p7pair
(a)mτQ= 16
−30 −20 −10 0 10 20 30 0
0.2 0.4 0.6 0.8 1
mt
Overlap
Impulse regime Vacuum p1−p3pairs p1−p5pairs p3−p5pairs p1pair p3pair p5pair p7pair
(b)mτQ= 64
Figure 7.4: Overlaps of the evolving wave function with instantaneous eigenstates for two different ramps from the paramagnetic to the ferromagnetic phase with mτQ = 16 and mτQ = 64 for mL = 50. The green region indicates the non-adiabatic regime. Solid lines are TCSA data forNcut= 25 while dots are obtained from the numerical solution of the exact differential equations. Analytical results are plotted only for the few low-momentum states with the most substantial overlap. Lower indices in the legends refer to the quantum numbers of the modes present in the many-body eigenstate: pn = nπ/L. The composite structure of some lines is caused by level crossings experienced by multiparticle states.
The paramagnetic-ferromagnetic (PF) direction
Ramps starting from the paramagnetic phase are defined by Mi <0. In this case the ground state is non-degenerate and lies in the Neveu–Schwarz sector, so the time evolved state is orthogonal to the Ramond sector subspace for all times.
As noted above, the solution of the dynamics amounts to finding the ap(t) and bp(t) coefficients characterising the decoupled particle pairs with momenta {p,−p}. Consequently, each excited state is specified as a set of p momentum modes. The elementary overlaps gp corresponding to a state with a single mo-mentum mode can be expressed as
| ⟨p,−p|Ψ(t)⟩ |2 ≡ |gp(t)|2 =np(t)Y
p′̸=p
(1−np′(t)), (7.31) where np = |bp|2 and the product goes over the quantised momenta, which are odd multiples of π/L in the NS sector. The extension to states with multiple particle pairs is straightforward.
The comparison of these results with the numerical data involves a subtlety.
TCSA constructs the eigenstates and the time-evolved state such that they are normalised to 1, and as a result, the overlaps |gn|2 are highly sensitive to the number of states kept after truncation. As there is no direct correspondence between the single-mode momentum cutoff applied to the infinite product of Eq.
(7.31) and the many-body TCSA energy cutoff, we set the latter such that the
−8 −6 −4 −2 0 2 4 6 8 0
0.1 0.2 0.3 0.4 0.5
mt
Overlap
Impulse regime NS Vacuum R Vacuum p1−p3pairs p2−p4pairs p1pair p2pair p3pair p4pair
(a)mτQ= 16
−30 −20 −10 0 10 20 30 0
0.1 0.2 0.3 0.4 0.5
mt
Overlap
Impulse regime NS Vacuum R Vacuum p1−p3pairs p2−p4pairs p1pair p2pair p3pair p4pair
(b)mτQ= 64
Figure 7.5: Overlaps of the evolving wave function with instantaneous eigenstates for two different ramps from the ferromagnetic to the paramagnetic phase with mτQ = 16 and mτQ = 64 for mL = 50. The green region indicates the non-adiabatic regime. Solid lines are TCSA data forNcut= 31 while dots are obtained from the numerical solution of the exact differential equations.
match between the analytical and numerical results is optimal. Note that this is a single parameter for all the states.
The time evolution of the overlaps is presented in Fig.7.4. Dots are calculated from the exact solution (7.31) and continuous lines denote TCSA data obtained by solving the many-body dynamics numerically. Fig. 7.4a depicts a curious be-haviour of the second largest overlap in TCSA: the corresponding line seemingly consists of many different segments. This is a consequence of level crossings and the errors of numerical diagonalisation near these crossings. The state in question consists of two two-particle pairs, and as the mass scaleM is ramped, its energy increases steeper than that of high-momentum states with only a single pair, hence the level crossings. At each crossing the numerical diagonalisation cannot resolve precisely levels in the degenerate subspace, so the resulting overlap is not accurate. This accounts for the most prominent difference between the numerical and analytical results. Apart from that, the agreement is quite satisfactory.
The light green background corresponds to the naive impulse regime t ∈ [−τKZ, τKZ]. Of course this is only a crude estimate for the time when adiabatic-ity breaks down, as strictly speaking Eq. (7.30) is valid only as a scaling relation.
Nevertheless, most of the change in each state population indeed happens within this coloured region. This statement is even more accentuated by Fig. 7.4b, that is, for a slower ramp. Comparing the two panels of Fig. 7.4 we observe that in-creasing the ramp time the probability of adiabaticity increases while the weight of the multiparticle states are suppressed. Note that although the two lowest available levels (the ground state and the first excited state) dominate the time-evolved state, the dynamics is far from being completely adiabatic, meaning no excitations at all. Hence, in accordance with the remarks concerning finite size effects in the previous subsection, we are within the regime of Kibble–Zurek
0 1 2 3 4 5 0.00
0.05 0.10 0.15
W/m1
P(W,tf)
A1
A2
A3
A4
A5
A6
A1A2
A1A2
A1A3
A1A3
A2A3
A2A3
A1A4
A1A4
Figure 7.6: Statistics of work after the ramp P(W, t = τQ/2) along the E8 di-rection with m1τQ = 32, m1L = 40, and Ncut = 45. States containing only zero-momentum particles are denoted by continuous lines, while dashed lines denote moving multiparticle states, different colours correspond to a different AaAb branch. The particle content is expressed as a label near the overlap of the most prominent states, with the same convention for colouring. The colour of the single-particle labels A3 and A6 reflects that they can be viewed as the bound states of A1A1 and A2A2, respectively, with a small binding energy (cf.
Table A.1).
scaling instead of being adiabatic.
The ferromagnetic-paramagnetic (FP) direction
The ferromagnetic ground state is twofold degenerate in infinite volume. For the initial state we choose the state with maximal magnetisation corresponding to the infinite volume symmetry breaking state: |Ψ0⟩ = √12 (|0⟩R+|0⟩NS). As both sectors are present in the initial state, the time-evolved state also overlaps with both sectors. This provides yet another benchmark for our numerical approach and also a somewhat richer landscape of the overlap functions.
As one can see in Fig. 7.5, the dynamics are very similar to the PF case with the main difference coming from the fact that both sectors contribute.
We also note that the number of level crossings is substantially increased, due to the gap developed between the two sectors in the paramagnetic phase. The different behaviour of the two vacua stems from the different available momentum modes in each sector: in the Ramond sector the momenta are larger in the lowest available modes and consequently they are less likely to be excited.