How are continued fractions related to quantum materials?

Question

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.

Answer 1

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.

Answer 2

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.

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_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].}

Answer 3

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.