Signatures of quantum chaos in non-integrable models of quantum field theory

Srdinšek, Miha (author), Prosen, Tomaž (mentor), Sotiriadis, Spyridon (comentor)

Classical concepts in the theory of dynamical systems, like integrability, periodicity, and chaos, were observed to have quantum analogues, which is reflected in the difference of spectral properties. Although the statement is well-grounded in evidence, we are lacking verifications to a large extent in models of quantum field theory. With the help of the Truncated Conformal Space Approach (TCSA) we investigate spectral properties of non-integrable relativistic (1+1)D models of quantum field theory defined on finite space. We study the distributions of level spacings and the distribution of ratios of consecutive level spacings, both of which are well established in literature as good signatures of quantum chaos. The models we study include the $\phi^{4}$, double sine-Gordon, massive Schwinger-Thirring model, and the integrable sine-Gordon model for comparison. We develop a conservative measure of truncation error that estimates the convergence of spectral statistics, and by increasing the integrability breaking parameters we observe a gradual onset of statistics predicted by quantum chaos theory—the statistics of random matrices. Surprisingly, we observe a strong quantum integrability breaking already at small system sizes, and by extrapolation we estimate that quantum chaos can be observed also at an infinite system size. At the same time, by studying the diagonal elements of observables in the energy eigenbasis of non-integrable models, we find preliminary evidence that the Eigenstate Thermalization Hypothesis is obeyed. The results presented in the thesis show that the TCSA can be used for analysing quantum chaos in (1+1)D models of quantum field theory, reveal that signatures of quantum chaos indeed appear in continuous quantum models as well, and unveil non-trivial properties of the parameter space of the investigated theories.

• Language: English
• Keywords: TCSA, Truncated Conformal Space Approach, sine-Gordon, double sine-Gordon, massive Sine-Gordon, massive Schwinger-Thirring model, massive Thirring model, phi4, quantum chaos, random matrix theory
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-117645
• COBISS.SI-ID: 24006915
• Publication date in RUL: 18.07.2020

SRDINŠEK, Miha, 2020, Signatures of quantum chaos in non-integrable models of quantum field theory [online]. Master’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=117645

Secondary language

• Language: Slovenian
• Title: Sledi kvantnega kaosa v neintegrabilnih modelih kvantne teorije polja

Abstract

Opazimo lahko, da imajo pojavi poznani iz klasične teorije dinamičnih sistemov, kot so integrabilnost, ergodičnost in kaos, tudi svoje kvantne analoge. Razliko med takimi sistemi lahko v kvantnem svetu opazimo v različnih spektralnih lastnostih. Čeprav je ta ugotovitev dobro podprta s teoretičnimi, numeričnimi in eksperimentalnimi opažanji, o tem v kvantni teoriji polja nimamo veliko dokazov. S pomočjo metode prirezanega konformnega prostora (TCSA) sem raziskal spektralne lastnosti neintegrabilnih relativističnih (1+1)D modelov kvantne teorije polja v končnem prostoru. Preučeval sem verjetnostne porazdelitve energijskih razmikov in razmerij med zaporednimi energijskimi razmiki. Obe porazdelitvi sta uveljavljeni kot dobri in stabilni pokazateljici kvantnega kaosa. Prečesal sem parametre sledečih modelov: $\phi^{4}$, Dvojni sine-Gordon, Masivni Schwinger-Thirring in integrabilni Sine-Gordon za primerjavo. Razvil sem zelo konzervativno mero napake zaradi prirezovanja, ki nam služi kot dobra mera konvergence spektralne statistike. Vzporedno z večanjem parametrov, ki zlomijo integrabilnost preučevanih modelov, sem opazil postopni nastop statistike, napovedane s teorijo o kvantnem kaosu — to je statistika naključnih matrik. Nepričakovano sem opazil močan zlom kvantne integrabilnosti že pri majhnih velikostih prostora in s pomočjo ekstrapolacije ocenil, da zlom opazimo tudi v limiti neskončnega prostora. Poleg tega sem z opazovanjem diagonalnih elementov opazljivk v bazi lastnih stanj neintegrabilnih modelov opazil možne dokaze o tem, da je hipotezi o termalizaciji lastnih stanj zadoščeno. Rezultati, predstavljeni v delu, kažejo na to, da se TCSA lahko uporablja pri analizi kvantnega kaosa v (1+1)D modelih kvantne teorije polja. Prav tako razkrivajo, da se sledi kvantnega kaosa lahko pojavijo tudi v zveznih kvantnih sistemih in dajejo nov vpogled v lastnosti parametričnega prostora preučevanih teorij.

• Keywords: TCSA, metoda prirezanega konformnega prostora, sine-Gordon, dvojni sine-Gordon, masivni sine-Gordon, masivni Schwinger-Thirringov model, masivni Thirringov model, phi4, kvantni kaos, teorija naključnih matrik