Tensor Network States: Algorithms and Applications @ NCCU, 4-6 Dec. 2019 HOTRG study on partition function zeros in the p-state clock model Dong-Hee Kim Dept. Physics and Photon Science Gwangju Institute of Science and Technology, Korea
Outline 1. Fisher zero characterization of a phase transition — Numerical methods of computing the partition function — BKT transitions in the p-state clock model? 2. Monte Carlo: Numerical issues due to the stochastic nature [D.-H. Kim, PRE 96, 052130 (2017)] — How large systems can we consider for Fisher zeros? It depends on the type of phase transition: BKT has an issue. 3. Higher-Order Tensor Renormalization Group [S. Hong and D.-H. Kim, arXiv:1906.09036] — Characterization of the two BKT transitions in the p-state clock model — Finite-size scaling analysis: logarithmic corrections — Fisher-zero determination of the BKT transition temperature Department of Physics & Photon Science
Emergent symmetry in the 2D p-state clock model Z(p) symmetry p-state clock model in square lattices X H p = − J cos( θ i − θ j ) θ = 2 π n : “clock” spin p h i,j i n = 0 , 1 , 2 , .., p − 1 p → ∞ B erezinskii- K osterlitz- T houless transition XY model: U(1) symmetry Emergent U(1) symmetry: BKT transition occurs even at finite p! Z(p) broken disordered 2 ≤ p ≤ 4 2nd order Z(p) broken disordered critical region x x 5 ≤ p < ∞ BKT BKT critical disordered x p → ∞ BKT Review: “40 years of BKT theory”, ed. by J. V. Jose Department of Physics & Photon Science
Debates and remaining issues There are approximations involved! Z(p) broken disordered 2 ≤ p ≤ 4 2nd order Z(p) broken disordered critical region (Villain approximation, self-dual, …) x x 5 ≤ p < ∞ BKT BKT critical disordered ; see Borisenko et al. , PRE 83, 041120 (2011). x p → ∞ BKT review: “40 years of BKT theory”, ed. by J. V. Jose Numerical results with the “exact” p-state clock model: (mainly for the high-T transition) Baek/Minnhagen/Kim PRE 2010: ( helicity modulus , more rigorously) Lapilli/Pfeifer/Wexler, PRL 2006: Noop, IT IS the BKT for p = 6! ( helicity modulus ) but… It’s not the BKT transition for p < 8! Baek/Minnhagen PRE 2010: Hwang, PRE 80, 042103 (2009): p = 5 looks strange… Borisenko et al, PRE 2011. (Fisher zero study) ? Baek et al, PRE 2013. Indeed, it doesn’t look like BKT for p=6. Kumano et al, PRE 2013. Chatelain, JSM 2014: DMRG Department of Physics & Photon Science
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Disagreements & Questions p-state clock model Helicity modulus Fisher zero 2nd. order ? p=6 BKT (WL calc. - Hwang 2009) BKT ? p=5 (with a new definition of helicity modulus) The only previous Fisher calculations in the p-state clock models: Hwang, PRE 80, 042103 (2009): p=6 , up to L=28 with W ang- L andau method It disagreed with the helicity modulus results but has not been re-examined. - Wang-Landau method gives very accurate results, usually. - L=28: too small. WL simulations can be done for much larger ones for p=6; - cf. Lee-Yang zeros in the XY model: up to L=256 with MC + histogram reweighting. Q. Are there any fundamental issues with the BKT transition? Department of Physics & Photon Science
In this talk, … What was wrong with Fisher zeros at the BKT transitions? Monte Carlo noises become unbearable, very quickly. This is a “ feature ” of BKT; it’s unavoidable within MC . Can we improve the situation? Yes, with HOTRG , to some extent. Logarithmic subleading-order corrections are essential. Better finite-size-scaling analysis can be done. Department of Physics & Photon Science
<latexit sha1_base64="M9TB/IMcCWYruLDz1GY61LE/xtw=">ACFXicbVDLSgMxFM3UV62vUZdugkUQsXWmCrosuHhop9QDsdMmnahiaZIckI7difcOvuHGhiFvBnX9j+lho64HA4Zx7uTkniBhV2nG+rdTC4tLySno1s7a+sblb+9UVBhLTMo4ZKGsBUgRgUpa6oZqUWSIB4wUg16lyO/ek+koqG40/2IeBx1BG1TjLSRfPv4ATY40l3Jk1syrA9814M5OPAxNIaiHF43k5x70hBx82jo21kn74wB54k7JVkwRcm3vxqtEMecCI0ZUqruOpH2EiQ1xYwM41YkQjhHuqQuqECcaK8ZJxqCA+M0oLtUJonNByrvzcSxJXq8BMjhKoW8k/ufVY92+8BIqolgTgSeH2jGDOoSjimCLSoI16xuCsKTmrxB3kURYmyIzpgR3NvI8qRTy7m+cHOWLRandaTBHtgHh8AF56AIrkAJlAEGj+AZvI368l6sd6tj8loypru7I/sD5/A5KnXc=</latexit> <latexit sha1_base64="NHtLA7FJzW0iyCzrxwOl/EROha0=">ACnicbVDLSsNAFJ3UV62vqks3o0VwY02qoMuCGwUXFewDkhgm02k7dGYSZiZCDV278VfcuFDErV/gzr9x0mah1QMXDufcy73hDGjStv2l1WYm19YXCoul1ZW19Y3yptbLRUlEpMmjlgkOyFShFBmpqRjqxJIiHjLTD4Xnmt+IVDQSN3oUE5+jvqA9ipE2UlDe9TjSA8nTSz6G7n3g+NBTlMOr2/TQOfJEMg7KFbtqTwD/EicnFZCjEZQ/vW6E06Exgwp5Tp2rP0USU0xI+OSlygSIzxEfeIaKhAnyk8nr4zhvlG6sBdJU0LDifpzIkVcqREPTWd2uJr1MvE/z01078xPqYgTQSeLuolDOoIZrnALpUEazYyBGFJza0QD5BEWJv0SiYEZ/blv6RVqzrH1dr1SaVez+Mogh2wBw6A05BHVyABmgCDB7AE3gBr9aj9Wy9We/T1oKVz2yDX7A+vgHrGpm/</latexit> Zeros of Partition Function: the “Fisher” zeros M. E. Fisher (1965) ( complex temperature , no external field) singular free energy! Phase transition: F = − k B T ln Z X At, Z ( β ) = g ( E ) exp[ − β E ] = 0 β = β c E No real solution exists in finite-size systems, but … Finite-Size-Scaling behavior of the “leading” Fisher zero as L increases Im[ z 1 ] ∼ L − 1 / ν | Re[ z 1 ] − z c | ∼ L − 1 / ν ∗ Re[ 𝞬 ] A tool to study a phase transition without an order parameter x 𝞬 c Lots of works have done. For a review, see Bena et al., Int. J. Mod. Phys. B 19, 4269 (2005). Im[ 𝞬 ] Department of Physics & Photon Science
Numerical strategy to compute Fisher zeros Given that we have g(E) … 1. solving polynomial equation from exact counting, histogram reweighing, for equally spaced discrete energies Wang-Landau, or … X X Y g n e − � n ✏ = ( z − z i ) g n z n Z = polynomial solver i n quick and easy, and works the best with exact g(E). 2. graphical solution + minimization Z ( β ) = Z R ( β ) + i Z I ( β ) slow, manual-geared, cumbersome, but allows error analysis! map of zeros map of zeros fine tuning find intersections minimize |Z| Department of Physics & Photon Science
<latexit sha1_base64="Qd13sKsTAQaT+iRA1XhD7wxgPq8=">ACM3icbVDLSgNBEJz1GeMr6tHLYBDiJez6QI8BL+Ipiolidgmzk14dnH040xsIy/6TF3/EgyAeFPHqPzgbc9DEgoaiqpvuLj+RQqNtv1hT0zOzc/OlhfLi0vLKamVtva3jVHFo8VjG6spnGqSIoIUCJVwlCljoS7j074L/7IPSos4usBAl7IbiIRCM7QSN3KqYtC9iBzQ4a3nMnsOs+pC/ep6FM3UIz/dmquD8h28kmte76TdytVu24PQSeJMyJVMkKzW3lyezFPQ4iQS6Z1x7ET9DKmUHAJedlNSM37Eb6BgasRC0lw1/zum2UXo0iJWpCOlQ/T2RsVDrQeibzuJYPe4V4n9eJ8XgyMtElKQIEf9ZFKSYkyLAGlPKOAoB4YwroS5lfJbZpJCE3PZhOCMvzxJ2rt1Z69+cLZfbTRGcZTIJtkiNeKQ9IgJ6RJWoSTB/JM3si79Wi9Wh/W50/rlDWa2SB/YH19A1jvrIM=</latexit> Numerical strategy to compute Fisher zeros 1. solving polynomial equation X X Y g n e − � n ✏ = ( z − z i ) g n z n Z = i n 2. Graphical search + refinement Z R ( β ) X Find zeros. Z ( β R ) = P ( β R ; E ) cos( β I E ) Z ≡ Z ( β ) ˜ E Z ( β ) = Z R ( β ) + i Z I ( β ) Z ( β R ) Z I ( β ) X Find zeros. Z ( β R ) = P ( β R ; E ) sin( β I E ) E 0.5 We need the energy distribution at a real T! 0.4 Monte Carlo with histogram reweighting 0.3 Im[ β ] Wang-Landau sampling method 0.2 0.1 1. Search for the cross point 2. Minimize |Z| for refinement 0 0.7 0.75 0.8 0.85 0.9 Re[ β ] Department of Physics & Photon Science
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