Long-time divergences in the nonlinear response of gapped one-dimensional many-particle systems

TL;DR

$1$ The paper addresses long-time divergences in the nonlinear response of one-dimensional, gapped many-particle systems that host stable single-particle excitations. $2$ It develops a semiclassical wavepacket framework to predict linear-in-time divergences in momentum- and time-resolved 4-point functions, and confirms them through a TFIM form-factor analysis and arguments applicable to diagonal IQFTs. $3$ The authors show that pump-probe and 2DCS setups access the same universal $|t_{32}| olinebreak$-divergence, with subleading $ olinebreak \\sqrt{|t_{32}|}$ corrections arising from wavepacket spreading, and provide explicit expressions for NR and PP signals alongside Ising-model benchmarks. $4$ The work connects nonlinear response to quasiparticle scattering data, suggesting nonlinear spectroscopy could directly probe the two-particle S-matrix in 1D quantum materials, and outlines extensions to higher dimensions and disordered systems.

Abstract

We consider one dimensional many-particle systems that exhibit kinematically protected single-particle excitations over their ground states. We show that momentum and time-resolved 4-point functions of operators that create such excitations diverge linearly in particular time differences. This behaviour can be understood by means of a simple semiclassical analysis based on the kinematics and scattering of wave packets of quasiparticles. We verify that our wave packet analysis correctly predicts the long-time limit of the four-point function in the transverse field Ising model through a form factor expansion. We present evidence in favour of the same behaviour in integrable quantum field theories. In addition, we extend our discussion to experimental protocols where two times of the four-point function coincide, e.g. 2D coherent spectroscopy and pump-probe experiments. Finally, focusing on the Ising model, we discuss subleading corrections that grow as the square root of time differences. We show that the subleading corrections can be correctly accounted for by the same semiclassical analysis, but also taking into account wave packet spreading.

Figures (8)

• Figure 1: Two examples of scattering-connected processes (a) and fully-connected process (b). Black lines denote the trajectories of the center of QP wave packets, which, in the long-time limit can be approximated as ballistic. Grey dots denote scattering events taking place where two wave packets collide. QPs are created and annihilated by $\mathcal{A}$ operators. Scattering-connected processes are those where the removal of the scattering events would reduce the diagram to a disconnected one, i.e. a process whose amplitude can be expressed as the product of two amplitudes related to two-point functions.
• Figure 2: (a) $C_{\rm NR}$ involves amplitudes where the state is evolved forward in time from $t_1=0$ to $t_3=\tau_1+\tau_2$, and then evolves backwards to $t_4=\tau_1$. (b) Processes that give rise to the leading contribution in the limit $\tau_{1,2}\to\infty$ limit. Here $\mathcal{A}_2:=\mathcal{A}(x_2,\tau_1)$ creates a shower of $n$ QPs ($n=3$ in the figure), which spread ballistically and scatter with a QP exchanged between $\mathcal{A}_1$ and $\mathcal{A}_3$. When evolved backwards in time, the $n$ QPs refocus at the same position and are annihilated by $\mathcal{A}_4$.
• Figure 3: Left: $C_{\rm PP}$ is a correlator where the state is evolved forward in time from $t_1=0$ to $t_3=\tau_1+\tau_2$, to finally be evolved backward to time $t_4=0$ again. (Right cartoon) Types of processes giving rise to the leading contribution in the $\tau_1,\tau_2\to\infty$ limit. Here the first operator $\mathcal{A}_1:=\mathcal{A}(x_1,0)$ can create a shower of $n$ QPs ($n=3$ in the figure), which spread ballistically and can scatter against the QP exchanged between $\mathcal{A}_2$ and $\mathcal{A}_3$. When evolved backward in time, the $n$ QPs refocus at the same position and can be annihilated by $\mathcal{A}_4$.
• Figure 4: The form factor expansion of a disconnected contribution would give rise to terms where $\boldsymbol{n}$ is either of the form (i) $(m,n+m,n)$ (as in panel (a) where $n=5$, $m=3$), (ii) $(n,n+m,n)$ (as in panel (b) where $n=5$, $m=3$), or (iii) $(m,0,n)$.
• Figure 5: $C_{\rm NR}^{(1,4,3)}$ as obtained by numerically summing over momenta at finite size $L$. A black dashed line reports the WP prediction, to which the finite size data converges in the infinite volume limit.