Yuya Tanizaki

Complex saddle points and the sign problem in complex Langevin simulation

Abstract We show that complex Langevin simulation converges to a wrong result within the semiclassical analysis, by relating it to the Lefschetz-thimble path integral, when the path-integral weight has different phases among dominant complex saddle points. Equilibrium solution of the complex Langevin equation forms local distributions around complex saddle points. Its ensemble average approximately becomes a direct sum of the average in each local distribution, where relative phases among them are dropped. We propose that by taking these phases into account through reweighting, we can solve the wrong convergence problem. However, this prescription may lead to a recurrence of the sign problem in the complex Langevin method for quantum many-body systems.

Lefschetz-thimble techniques for path integral of zero-dimensional

Zero-dimensional

O(n)

O(n)

sigma models

-symmetric sigma models are studied by using Picard--Lefschetz integration method in the presence of small symmetry-breaking perturbations. Due to approximate symmetry, downward flows turn out to show significant structures: They slowly travel along the set of pseudo classical points, and branch into other directions so as to span middle-dimensional integration cycles. We propose an efficient way to find such slow motions for computing Lefschetz thimbles. In the limit of symmetry restoration, we figure out that only special combinations of Lefschetz thimbles can survive as convergent integration cycles: Other integrations become divergent due to non-compactness of the complexified group of symmetry. We also compute downward flows of

Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral

O(2)

Vacuum structure of bifundamental gauge theories at finite topological angles

-symmetric fermionic systems, and confirm that all of these properties are true also with fermions.

Circle compactification and ’t Hooft anomaly

The fermion sign problem appearing in the mean-field approximation is considered, and the systematic computational scheme of the free energy is devised by using the Lefschetz-thimble method. We show that the Lefschetz-thimble method respects the reflection symmetry, which makes physical quantities manifestly real at any order of approximations using complex saddle points. The formula is demonstrated through the Airy integral as an example, and its application to the Polyakov-loop effective model of dense QCD is discussed in detail.

Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics

A bstractWe discuss possible vacuum structures of SU(n) × SU(n) gauge theories with bifundamental matters at finite θ angles. In order to give a precise constraint, a mixed ’t Hooft anomaly is studied in detail by gauging the center ℤn one-form symmetry of the bifundamental gauge theory. We propose phase diagrams that are consistent with the con-straints, and also give a heuristic explanation of the result based on the dual superconductor scenario of confinement.

Multi-flavor massless QED2 at finite densities via Lefschetz thimbles

A bstractAnomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an ’t Hooft anomaly involving one-form symmetries as in pure SU(N ) Yang-Mills theory at θ = π. Recent development about large-N volume independence, however, gives us a circumstantial evidence that ’t Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider ℤN