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Knizhnik-Zamolodchikov equations and integrable Landau-Zener models
by Suvendu Barik, Lieuwe Bakker, Vladimir Gritsev, Emil A. Yuzbashyan
Submission summary
| Authors (as registered SciPost users): | Lieuwe Bakker · Suvendu Barik |
| Submission information | |
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| Preprint Link: | scipost_202409_00030v1 (pdf) |
| Code repository: | https://doi.org/10.5281/zenodo.11620326 |
| Date submitted: | 2024-09-25 18:34 |
| Submitted by: | Bakker, Lieuwe |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the relationship between integrable Landau-Zener (LZ) models and Knizhnik-Zamolodchikov (KZ) equations. The latter are originally equations for the correlation functions of two-dimensional conformal field theories, but can also be interpreted as multi-time Schrödinger equations. The general LZ problem is to find the probabilities of tunneling from eigenstates at $t=t_\text{in}$ to the eigenstates at $t\to+\infty$ for an $N\times N$ time-dependent Hamiltonian $\hat H(t)$. A number of such problems are exactly solvable in the sense that the tunneling probabilities are elementary functions of Hamiltonian parameters and time-dependent wavefunctions are special functions. It has recently been proposed that exactly solvable LZ models map to KZ equations. Here we use this connection to identify and solve various integrable hyperbolic LZ models $\hat H(t)=\hat A+\hat B/t$ for $N=2, 3$, and $4$, where $\hat A$ and $\hat B$ are time-independent matrices. Some of these models have been considered, though not fully solved, before and others are entirely new.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- Exact solutions to various 2x2 and 3x3 hyperbolic/Coulomb Landau-Zener (LZ) problems are presented, both for the dynamics and the final transition probabilities.
2- This works explores the interesting connection between time-dependent integrability and Landau-Zener dynamics.
Weaknesses
1- The Knizhnik-Zamolodchikov (KZ) equations are not used to obtain any new solutions, but rather to identify models which are then solved in a 'brute-force' manner. The connection between the KZ equations and the exact solutions to these LZ models hence remains somewhat unclear.
2- Section 2 reads largely as a list of mathematical expressions in terms of special (Bessel, hypergeometric) functions, with limited discussion of their derivation and interpretation.
Report
In this work the authors study Landau-Zener problems and explore their connection to the Knizhnik-Zamolodchikov equations of integrability. Following previous work by one of the authors, special classes of such Landau-Zener problems satisfy a notion of Frobenius integrability, which can be shown by embedding the LZ equation in a system of multi-time Schrodinger equations. These equations here result in the Knizhnik-Zamolodchikov equations, which admit a formal solution in terms of contour integrals.
After an introduction in Section 1, in Section 2 the authors consider two classes of 2x2 and 3x3 hyperbolic LZ problems. The exact solution for the time-dependent states and transition probabilities are given in terms of Bessel functions (2x2 problem) and hypergeometric functions (3x3 problem). In Section 3 the authors show how integrable BCS models can be used to construct integrable LZ models. For the two-level LZ problem the authors show how the formal solution to the corresponding KZ equation can be explicitly evaluated to return the known solutions of the state dynamics and transition probabilities, and discuss the difficulties when trying to extend this approach to higher-level models. Using the connection with integrable BCS models, different 3x3 and 4x4 integrable LZ models are subsequently introduced in Section 4.
The connection between LZ dynamics and KZ equations is interesting and highly nontrivial, and exact solutions to the non-stationary Schrodinger equation in a relevant setting present useful contributions to the literature. While this work is hence interesting and presents exact results, I have some reservations that make me believe this manuscript would be better suited to SciPost Physics Core or would require some major revisions for acceptance for SciPost Physics. The paper clearly meets the criteria for SciPost Physics Core, addressing a significant question and providing various exact results.
My main comment is that it is unclear how useful the KZ equations are to obtain solutions to the LZ dynamics discussed in this model. All exact solutions for the dynamics seem to be found by brute-force, and are presented without any discussion of how they are obtained or can e.g. be related to known differential equations (apart from the known case of the 2x2 model). The resulting expressions [Eqs. (15), (18), (23)] are hence difficult to interpret, and it would be useful if the authors would discuss how they arrive at these solutions and discuss the behavior of these solutions. The exact solutions also seem to be decoupled from the way these models are derived, such that it is unclear if it is the integrability that is responsible for the exact solvability of these models or if they can be mapped to a solvable differential equation in some other way (which could then extend beyond the integrable models discussed in this work). Because of this, the significance of the models introduced in Section 4 is also not so clear. If the connection between the KZ equations and the exact solutions from Section 2 would be made more explicit away from the 2x2 model, I would be happy to recommend this paper for SciPost Physics, since then the paper clearly meets the journal expectation of opening a new pathway in an existing or a new research direction.
Some minor comments
- In Eq. (10) the authors mention that this model is solvable, and then focus on a specific choice of parameters. Is the model integrable/solvable for all choices of the Hamiltonian (10) or only at the specific parameters of Eq. (11)?
- In Section 2.2, the authors mention that all transition probabilities need to be obtained. However, doesn't the transition probability for the 2x2 problem follow from the transition probability from the ground state using time-reversal symmetry?
- In Eq. (40), the notation $\prod_{\alpha}^1 S(\lambda_{\alpha})$ is unnecessary, since the authors could simply write $S(\lambda)$.
- The sentence "Due to their connection to the KZ equations, all these models are integrable, yet we will obtain exact solutions to some of them." at the start of Section 4 is somewhat confusing.
- Typo: "aformentioned"
Requested changes
See report.
Recommendation
Accept in alternative Journal (see Report)