Phantom relaxation in non-Hermitian systems

Bensa, Jaš (author), Žnidarič, Marko (mentor)

Oftentimes, intuition gained from simple systems is used to predict the behavior of more complex scenarios. For example, if we iterate a vector with a Hermitian matrix its norm (or inner product with another vector) is expected to evolve exponentially in time with the exponent equal to the largest eigenvalue of the matrix. Believing that this holds in non-Hermitian systems is wrong. In infinite systems and for general vectors, it is not the spectrum of the matrix that predicts how the iterated quantity will behave, but rather the spectrum of the slightly perturbed matrix, i.e., the pseudospectrum. In this thesis, we will explore quantities iterated with non-Hermitian matrices. In the majority of the examples that we will delve into, these quantities show a two-stage relaxation. In the first stage, the exponential behavior is determined by the largest pseudoeigenvalue of the iterated matrix. Interestingly, this stage persists up to times that scale with the system size, meaning that in infinite systems it is the only kind of behavior. The second stage is given by the largest eigenvalue, as one expects in Hermitian systems. Some physical examples that can be evolved using this setting are purity and out-of-time-ordered correlations in random quantum circuits and certain biased random walks. The setting we will explore is quite general and thus it might be applied in a wide variety of physical fields. Whenever one encounters iterations with a non-Hermitian transfer matrix, one can refer to the work presented here and hopefully predict the results of the iteration.

• Language: English
• Keywords: pseudospectrum, random quantum circuits, non-Hermiticity, non-normal matrix, Markov chain, transfer matrix, entanglement, operator correlations
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-162859
• COBISS.SI-ID: 209938179
• Publication date in RUL: 28.09.2024

Secondary language

• Language: Slovenian
• Title: Fantomska relaksacija v nehermitskih sistemih

Abstract

Pogosto uporabljamo intuicijo, pridobljeno iz preprostih primerov, za napovedovanje obnašanja kompleksnejših sistemov. Na primer, ko iteriramo vektor s hermitsko matriko, se bo njegova norma (ali skalarni produkt z drugim vektorjem) eksponentno razvijala v času. Eksponent ponavadi določa največja lastna vrednost matrike. Če bi isto pričakovali v nehermitskih sistemih, bi se krepko zmotili. V termodinamski limiti in za splošne vektorje se skalarni produkt še vedno razvija eksponentno v času, toda eksponent je določen z največjo vrednostjo v psevdospektru matrike, tj. spektru rahlo perturbirane matrike. V doktorski disertaciji bomo raziskovali razne fizikalne količine, ki jih propagiramo v času z iteracijami nehermitskih prehodnih matrik. V večini primerov bodo te količine relaksirale ekpsonentno v dveh fazah. V prvi fazi je eksponent enak največji vrednosti v psevdospektru prehodne matrike. Po času, ki skalira sorazmerno z velikostjo sistema, bo eksponent enak največji lastni vrednosti matrike. Čas, ko se neha prva faza, divergira z večanjem sistema, zato v termodinamski limiti spekter prehodne matrike sploh ne določa relaksacije količin. Primeri, ki si jih bomo ogledali, vključujejo prepletenost in korelacije v naključnih kvantnih vezjih in tudi nesimetrično naključno hojo. Kontekst, ki ga analiziramo, je precej splošen, zato pričakujem, da bi lahko koristil številnim fizikalnim področjem. Kadarkoli se bo kdo srečal z iteracijami nehermitske matrike, se lahko ozre na to delo in mu bo predstavljena študija pomagala pri napovedovanju obnašanja sistema.

• Keywords: psevdospekter, naključna kvantna vezja, nehermitskost, nenormalna matrika, Markovska veriga, prehodna matrika, prepletenost, korelacije operatorjev