How important are continuous symmetries in tensor networks?

28.05.2024

This question is addressed in a new paper published in Phys. Rev. B which finds that in order to capture critical behavior in RVB spin liquids, continuous virtual symmetries must be preserved. At the same time, it is also demonstrated how critical behavior and continuous symmetries can emerge in a tensor network lacking such symmetries at the microscopic level.

The work, titled "Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks", and co-authored by Henrik Dreyer, Laurens Vanderstraeten, Ji-Yao Chen, Ruben Verresen, and our group leader Norbert Schuch, has just appeared in Physical Review B [Phys. Rev. B 109, 195161 (2024)].

The paper addresses the role played by continuous U(1) symmetries in critical tensor networks. Namely, known examples of tensor networks which capture critical behavior have a continuous symmetry hard-wired into the local tensor. This raises the question whether this continuous U(1) symmetry is essential for capturing critical physics.

To address this question, the paper studies the response of critical resonating valence bond (RVB) spin liquids to doping with longer-range singlets, and more generally of U⁡(1)-symmetric tensor networks to nonsymmetric perturbations. Using a field theory description, it is found that in the RVB, doping constitutes a relevant perturbation that immediately opens up a gap, contrary to previous observations. The analysis in the paper predicts a very large correlation length even at significant doping, which is verified using high-accuracy numerical simulations. This emphasizes the need for careful analysis, but it also justifies the use of such states as a variational ansatz for critical systems. 

These findings, which underline the importance of continuous symmetries in tensor networks, are contrasted by giving the very first example of an emergent symmetry in 2D tensor networks. Based on six-vertex models, an example of a 2D tensor network state is given where nonsymmetric perturbations do not open up a gap and the U⁡(1) symmetry reemerges at large length scales.

For more details, please take a look at the open-access published version in Phys. Rev. B 109, 195161 (2024), or the arXiv version.

This work has received support through the ERC grant SEQUAM and the Austrian Science Fund FWF (grant DOIs 10.55776/P36305 and 10.55776/F71). Calculations were partly performed on the Vienna Scientific Cluster (VSC).

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