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Loop-TNR analysis of CP(1) model with theta term

Hikaru Kawauchi, Shinji Takeda

Published Mar 26, 2018• 4 citations
Publisher

Abstract

The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.

DOI
10.1051/epjconf/201817511015/pdf

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The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.
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Abstract We investigate the phase structure of a 2D lattice CP(1) model with a θ\thetaθ term. In particular, we aim to identify a critical region expected to exist along a θ=π\theta =\piθ=π line. To explore the phase structure nonperturbatively and avoid the sign problem, we employ the tensor renormalization group method. We make two improvements compared to previous tensor network studies. The first improvement involves refining the initial tensor. Specifically, we construct it using a quadrature method, which achieves higher accuracy compared to the conventional approach. The second improvement consists of analyzing the phase structure using the information on the conformal field theory, namely the central charge and the scaling dimensions, which can be accessed relatively easily via the tensor renormalization group method. Thanks to these improvements, we identify both the onset of the critical region, βc=0.5952(8)\beta _{\rm c}=0.5952(8)βc​=0.5952(8), and its universality class as the SU(2)k=1{}_{k=1}k=1​ Wess–Zumino–Witten model.
Phase structure analysis of CP(1) model with θθθ term by tensor renormalization group
We analyze the phase structure of 2d lattice CP(1) model with θθθ term by using the bond-weighted tensor renormalization group method. We propose a new tensor network representation for the model using the quadrature scheme and confirm that its accuracy is better than that of the conventional character-like expansion. As a probe to study the phase structure, we adopt the central charge and the scaling dimensions. The numerical results indicate an existence of critical point at θ=πθ=πθ=π, which is consistent with the Haldane's conjecture.