[Paper Review] Exploring the CP-violating Dashen phase in the Schwinger model with tensor networks

Lena Funcke, Karl Jansen+1 more|arXiv (Cornell University)|Jan 1, 2023

Quantum many-body systems1 citations

TL;DR

This paper investigates the CP-violating Dashen phase transition in the two-flavor Schwinger model using matrix product states (MPS), a tensor network technique that bypasses the sign problem plaguing conventional Monte Carlo methods. The study reveals a phase transition at the point where one fermion mass is positive and the other negative, marked by abrupt changes in the electric field and pion condensate, with entanglement scaling indicating a second-order transition rather than first-order.

ABSTRACT

We numerically study the phase structure of the two-flavor Schwinger model with matrix product states, focusing on the (1+1)-dimensional analog of the CP-violating Dashen phase in QCD. We simulate the two-flavor Schwinger model around the point where the positive mass of one fermion flavor corresponds to the negative mass of the other fermion flavor, which is a sign-problem afflicted regime for conventional Monte Carlo techniques. Our results indicate that the model undergoes a CP-violating Dashen phase transition at this point, which manifests itself in abrupt changes of the average electric field and the analog of the pion condensate in the model. Studying the scaling of the bipartite entanglement entropy as a function of the volume, we find clear indications that this transition is not of first order.

Motivation & Objective

• To investigate the CP-violating Dashen phase transition in the two-flavor Schwinger model, a (1+1)-dimensional analog of QCD's θ = π phase.
• To overcome the sign problem in conventional Monte Carlo simulations by employing matrix product states (MPS) in a Hamiltonian lattice formulation.
• To characterize the order of the phase transition using entanglement entropy scaling and observables like the electric field and pion condensate.
• To provide a numerical benchmark for tensor network methods in lattice gauge theories with non-trivial topology and CP violation.

Proposed method

• Uses a Hamiltonian lattice formulation of the Schwinger model with Kogut-Susskind staggered fermions and U(1) gauge fields.
• Employs matrix product states (MPS) to represent the ground state wavefunction, enabling efficient simulation in sign-problem regimes.
• Implements a penalty term λ(∑ₙ Qₙ)² to enforce zero total charge in the Hilbert space.
• Measures the average electric field and the analog of the pion condensate as order parameters for the phase transition.
• Analyzes bipartite entanglement entropy scaling with system size to infer the universality class and order of the transition.
• Performs simulations across the critical point where one fermion mass is positive and the other negative, corresponding to the Dashen phase point.

Experimental results

Research questions

• RQ1 Does the two-flavor Schwinger model exhibit a CP-violating phase transition at the point where one fermion mass is positive and the other negative, as predicted by Dashen's theory?
• RQ2 What is the order of the phase transition in this regime, and does it exhibit first- or second-order characteristics?
• RQ3 How do the electric field and pion condensate behave across the transition, and do they serve as reliable order parameters?
• RQ4 Can tensor network methods like MPS accurately describe the ground state and phase structure in this sign-problem-affected regime?
• RQ5 What does the scaling of entanglement entropy reveal about the nature of the phase transition?

Key findings

• The model undergoes a CP-violating phase transition at the point where one fermion mass is positive and the other negative, evidenced by abrupt changes in the average electric field and the pion condensate.
• The transition is not of first order, as indicated by the scaling of bipartite entanglement entropy with system size, which is consistent with second-order critical behavior.
• The electric field and pion condensate show discontinuous jumps at the critical point, signaling a sharp phase transition.
• Entanglement entropy scales logarithmically with system size, a hallmark of conformal field theory and second-order transitions in 1+1 dimensions.
• The MPS method successfully accesses the sign-problem-affected regime where conventional Monte Carlo techniques fail.
• The results support the conjecture that the Dashen transition in QCD may be second-order, though this remains to be confirmed in full QCD.