Lefschetz thimbles and quantum phases in zero-dimensional bosonic models
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Regular Article - Theoretical Physics
Lefschetz thimbles and quantum phases in zero-dimensional bosonic models R. Bharathkumara , Anosh Josephb Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab 140306, India
Received: 2 July 2020 / Accepted: 24 September 2020 © The Author(s) 2020
Abstract In this paper, by analyzing the underlying Lefschetz-thimble structure, we investigate quantum phases (or quantum critical points) in zero-dimensional scalar field theories with complex actions. Using first principles, we derive the thimble equations of these models for various values of the coupling parameters. In the thimble decomposition of complex path integrals, determination of the so-called intersection numbers appears as an important ingredient. In this paper, we obtain the analytic expressions for the combined intersection number of thimbles and anti-thimbles of these zero-dimensional theories. We also derive the conditional expressions involving relations among the coupling parameters of the model, that would help us predict quantum phase transitions in these systems. We see that the underlying thimble structure undergoes a drastic change when the system passes through such a phase transition.
Contents 1 2 3 4
Introduction . . . . . . . . . . . . . . . . . . . A primer on Lefschetz thimbles . . . . . . . . . Quartic model with a source term . . . . . . . . Thimble equations and observables . . . . . . . 4.1 Thimble equations . . . . . . . . . . . . . 4.2 Partition function and observables . . . . . 5 Determining the intersection numbers . . . . . . 5.1 A simple demonstration using Airy integral 5.2 Quartic model without source term . . . . . 5.2.1 Real coupling . . . . . . . . . . . . . 5.2.2 Complex coupling . . . . . . . . . . 5.3 Quartic model with source term . . . . . . 5.3.1 Real source parameter . . . . . . . . 5.3.2 Imaginary source parameter . . . . .
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5.4 Theory with PT symmetry . . . . . . . . . . . . 6 Quantum phase transition and change in thimble structure 7 Summary of results . . . . . . . . . . . . . . . . . . . 8 Conclusions and future directions . . . . . . . . . . . Appendix A: Expressions for boundaries of phase transitions References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction We encounter path integrals with complex actions in many branches of physics. The prominent examples are the Minkowski path integral, Yang–Mills theory in the theta vacuum, Chern–Simons gauge theories, chiral gauge theories, and QCD with chemical potential. There are also quantum theories with complex actions that are invariant under PT symmetry [1–3]. In the context of string theory, the IKKT matrix model, a zero-dimensional supersymmetric quantum field theory that serves as a promising candidate for a nonperturbative formulation of superstring theory, is shown to have a complex fermion operator [4–6]. Investigating the n
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