# How are continued fractions related to quantum materials?

In my spare time, I have been studying and analysing continued fractions.

I was having a conversation with someone on Discord in a Mathematics server and he was telling me that continued fractions can be related to quantum physics. He didn't go into it too much and the concept he was describing seemed a little vague to me. I am aware that Pincherle's theorem[1] asserts there is an intimate relationship with three-term recurrence relations of the form $$x_{n+1}=b_nx_n+a_nx_{n-1}$$ and continued fractions, more accurately with their partial convergents, given that this recurrence relation has a minimum if a related cfrac converges. But I am not quite the physicist myself to compare physics with the properties of cfracs.

Although I can go on Google and do some research on this, I figured it might serve useful to have a post on this in this beta community, but I do apologise if it is too broad or open-ended and thus upsets any regulations here.

Any thoughts on this?

### References

[1] Pincherle, S. (1894). Delle funzioni ipergeometriche e di varie questioni ad esse attinenti. Giorn. Mat. Battaglini. 32:209–29

[2] Parusnikov, V. I. A Generalization of Pincherle's Theorem to k-Term Recursion Relations. Math Notes 2005, 78 (5-6), 827–840. DOI: 10.1007/s11006-005-0188-7.

• Continued fractions come up in various places in quantum physics, such as in the "continued fraction approach" for open quantum systems dynamics, but what specifically is your question? You just want an example of a continued fraction that comes up when solving a quantum mechanics problem? May 2, 2020 at 7:21
• I think this is out of our scope here. We deal with all matters modeling. More conceptual questions should be taken elsewhere. May 2, 2020 at 7:39
• @NikeDattani I was looking for an answer like yours. I was going to ask that you post it as an answer, but Cody's comment suggests, as I feared, that this question is off-topic here. May 2, 2020 at 7:41
• idk, maybe @NikeDattani knows best. we are still refining our scope so it might be constructive for now. May 2, 2020 at 7:44
• @MrPie, I'm happy to post a longer answer if the rest of the Materials Modeling community is happy with it. However I'm not sure if they will like it. We're still a bit new here. Some people might think we're expanding the scope too much. May 2, 2020 at 7:44

In the paper "A Continued-Fraction Representation of the Time-Correlation Functions", generalized susceptibilities and transport coefficients for materials are obtained using a continued-fraction expansion of the Laplace transform of the time-correlation functions.

This was the precursor to what is now called the "hierarchical equations of motion" which are used to study the dynamics of a quantum system (such as an electron) coupled to a bosonic bath (for example the vibrations of the lattice in a GaAs semi-conductor quantum dot). This area is called "dissipative quantum dynamics" or "open quantum systems" and is used to study for example, the decoherence of qubits in solid-state quantum computers.

• I don't know which answer to accept, so perhaps I'll accept the other one and give you a 50 rep bounty to somewhat equate things. Is this a good idea or nah? (I admit, it may be a bit redundant but why not :P) May 3, 2020 at 5:26
• @MrPie The bounty is completely up to you :) Yes unfortunately the smallest bounty you can give is 50 rep, but it might also attract a lot of attention over the next 7 days, so you might get the 50 rep back from upvotes that the bounty attracts (people are voting a lot right now because it's Private Beta, and we have the "Vote Early, Vote Often" featured on the side of the screen right now). You might also get better answers than mine, and if so, you can feel free to award the bounty to them! May 4, 2020 at 23:09
• @MrPie seems you earned all your bounty rep back! May 8, 2020 at 2:51

You may take a look at the Method of Continued Fractions used in quantum scattering theory—this was only formed in 19831 so is rather recent. Related is the PhD thesis by Kónya (2000)2; §3.3 onwards.

Reference

[1] Horáček, J., Sasakawa, T. (1983). Method of continued fractions with application to atomic physics. Physical Review A. 28(4):2151–2156.

[2] Kónya, B. (2000). Continued fraction representation of quantum mechanical Green's operators. PhD thesis. arXiv:0101040 [quant-ph].

• Ok I guess the question can be considered on scope! I'll add an answer too! May 2, 2020 at 7:51

The continued fraction expansion is the most common way to calculate real-frequency dynamical Green's functions using Lanczos exact diagonalization. The method was introduced in this setting in Gagliano, E. R., and Balseiro, C. A., "Dynamical Properties of Quantum Many-Body Systems at Zero Temperature," Physical Review Letters 59, 2999 (1987), but a more common reference these days is Dagotto, E., "Correlated electrons in high-temperature superconductors," Reviews of Modern Physics 66, 763 (1994).

The same expansion was also used in the first attempts to calculate dynamical correlation functions using the density-matrix renormalization group, see K. A. Hallberg, "Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems," Physical Review B 52, R9827(R) (1995). However, it was discovered that the method has some issues, especially with higher frequencie, see Kühner, T. D., and White, S. R., White"Dynamical correlation functions using the density matrix renormalization group," Physical Review B 60, 335 (1999). Today, it's mostly been displaced by other methods in the DMRG setting.

• @NikeDattani I didn't realize it was quite so venerable! I just decided to explore the quantum-mechanics tag, and ended up here. Aug 1, 2020 at 0:06