Entanglement Order Parameters and Critical Behavior for Topological Phase Transitions and Beyond

Autor(en)

Mohsin Iqbal, Norbert Schuch

Abstrakt

Order parameters are key to our understanding of phases of matter. Not only do they allow one to classify phases, but they also enable the study of phase transitions through their critical exponents which identify the universal long-range physics underlying the transition. Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. While probes have been developed to identify such phases, those probes are only qualitative, in that they take discrete values, and, thus, provide no means to study the scaling behavior in the vicinity of phase transitions. In this paper, we develop a unified framework based on variational tensor networks (infinite projected entangled pair states) for the quantitative study of both topological and conventional phase transitions through entanglement order parameters. To this end, we employ tensor networks with suitable physical and/or entanglement symmetries encoded and allow for order parameters detecting the behavior of any of those symmetries, both physical and entanglement ones. On the one hand, this gives rise to entanglement-based order parameters for topologically ordered phases. These topological order parameters allow one to quantitatively probe the behavior when going through topological phase transitions and, thus, to identify universal signatures of such transitions. We apply our framework to the study of the toric code model in different magnetic fields, which along some special lines maps to the (2+1)D Ising model. Our method identifies 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we, in addition, also find an unknown critical exponent β∗≈0.021 for one of our topological order parameters. We take this - together with known dualities between the toric code and the Ising model - as a motivation to also apply our framework of entanglement order parameters to conventional phase transitions. There, it enables us to construct a novel type of disorder operator (or disorder parameter), which is nonzero in the disordered phase and measures the response of the wave function to a symmetry twist in the entanglement. We numerically evaluate this disorder operator for the (2+1)D transverse field Ising model, where we again recover a critical exponent hitherto unknown in the (2+1)D Ising model, β∗≈0.024, consistent with the findings for the toric code. This shows that entanglement order parameters can provide additional means of characterizing the universal data both at topological and conventional phase transitions and altogether demonstrates the power of this framework to identify the universal data underlying the transition.

Organisation(en)

Institut für Mathematik, Quantenoptik, Quantennanophysik und Quanteninformation

Externe Organisation(en)

Munich Center for Quantum Science and Technology (MCQST), Max-Planck-Institut für Quantenoptik

Journal

Physical Review X

Band

11

Seiten

041014-1 - 041014-27

Anzahl der Seiten

27

ISSN

2160-3308

DOI

https://doi.org/10.1103/PhysRevX.11.041014

Publikationsdatum

10-2021

Peer-reviewed

Ja

ÖFOS 2012

103025 Quantenmechanik, 103015 Kondensierte Materie, 103029 Statistische Physik

Schlagwörter

ASJC Scopus Sachgebiete

Allgemeine Physik und Astronomie

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/15bd4e39-0a3f-4a25-a63e-991933f342ea