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Long-time divergences in the nonlinear response of gapped one-dimensional many-particle systems
by M. Fava, S. Gopalakrishnan, R. Vasseur, S. A. Parameswaran, F. H. L. Essler
Submission summary
| Authors (as registered SciPost users): | Fabian Essler |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2411.06167v3 (pdf) |
| Date accepted: | Sept. 8, 2025 |
| Date submitted: | Aug. 27, 2025, 5:01 p.m. |
| Submitted by: | Fabian Essler |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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.
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Author comments upon resubmission
We thank both referees for their positive assessments of our manuscript. In the following we reply to their comments in detail and indicate the changes we have made.
Referee 1:
We thank the referee for their comments. A discussion of the class of systems we expect our theory to apply to was given in section 2.3 of our manuscript. We believe that one-dimensional models with stable, gapped, coherent quasiparticle excitations will generically exhibit the kind of nonlinear response we identify. This class includes gapped integrable models like the transverse-field Ising chain in the paramagnetic phase or the sine-Gordon QFT in the attractive regime as well as non-integrable models where stable excitations exist as a result of kinematic protection. Here examples are Haldane-gap spin chains or the spin-1/2 XXZ chain in a staggered magnetic field.
Referee 2:
We thank the referee for their comments. We have followed their advice to restructure our manuscript. We now start with a discussion of nonlinear response in integrable models, and follow this by sections presenting the wave-packet analysis (which applies more widely).
Our responses to the specific points the referee raised is as follows:
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We thank the referee for pointing out that we had not defined $\langle A(q)\rangle_0$. The latter denotes the expectation values in absence of the perturbation. We have added a sentence in our manuscript explaining this.
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The nonlinear response formalism is based on an expansion in the strength of the perturbation, and the orders in the expansion gives rise to the different nonlinear response functions. As we show, the latter exhibit divergences. However, it is clear on physical grounds that these divergences must be cut off in the actual response of the system. The way this comes about is that higher response functions have stronger divergences, and summing these to all orders gives a finite result. This is akin to summing infrared divergences in perturbation theory.
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The pump and probe processes are separated in time. For simplicity we consider an instantaneous pump process at $t=0$, which is then followed by a probe process at later times. Given this separation in time on physical grounds there is no issue with considering higher orders in the parameters $\mu$ and $\mu'$.
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We thank the referee for spotting this typo, which we have corrected.
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We thank the referee for spotting this typo, which we have corrected.
Published as SciPost Phys. 19, 086 (2025)