Building on quantum Monte Carlo simulations, we study the phase diagram of a one-parameter Hamiltonian interpolating between trivial and topological Ising paramagnets in two dimensions, which are dual to the toric code and the double semion. We discover an intermediate phase with stripe order which spontaneously breaks the protecting Ising symmetry. Remarkably, we find evidence that this intervening phase is gapless due to the incommensurability of the stripe pattern and that it is dual to a U(1) gauge theory exhibiting Cantor deconfinement.
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Acknowledgments. We are grateful to F. Alet, N. Bultinck, X. Cao, S. Capponi, E. Fradkin, B. Kang, A. Paramekanti, D. Poilblanc, F. Pollmann, P. Pujol, A. W. Sandvik, R. Vasseur, C. Xu, and L. Zou for interesting discussions. M.D. was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 through the Scientific Discovery through Advanced Computing (SciDAC) program (KC23DAC Topological and Correlated Matter via Tensor Networks and Quantum Monte Carlo). S.G. acknowledges support from the Israel Science Foundation, Grant No. 1686/18. T.S. acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), in particular the Discovery Grant No. [RGPIN-2020-05842], the Accelerator Supplement [RGPAS-2020-00060], and the Discovery Launch Supplement [DGECR-2020-00222]. This research used the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory (Supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DE-AC02-05CH11231). This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. T.S. contributed to this work prior to joining Amazon.
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