Abstract
We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for O(N )-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as N varies, between the first two classes. For positive N , it contains a single line of infrared fixed points spanning the values of N from \( \sqrt{2}-1 \) to 1. The symmetry sector of the energy density operator is superuniversal (i.e. N -independent) along this line. For N = 2 a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder.
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References
A.B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C 7 (1974) 1671.
J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge U.K. (1996).
H. Nishimori, Internal energy, specific heat and correlation function of the bond-random Ising model, Prog. Theor. Phys. 66 (1981) 1169.
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1997).
G. Delfino, Exact results for quenched bond randomness at criticality, Phys. Rev. Lett. 118 (2017) 250601 [arXiv:1701.01816] [INSPIRE].
G. Delfino, Parafermionic excitations and critical exponents of random cluster and O(n) models, Annals Phys. 333 (2013) 1 [arXiv:1212.3178] [INSPIRE].
G. Delfino, Fields, particles and universality in two dimensions, Annals Phys. 360 (2015) 477 [arXiv:1502.05538] [INSPIRE].
G. Delfino and E. Tartaglia, On superuniversality in the q-state Potts model with quenched disorder, J. Stat. Mech. 12 (2017) 123303 [arXiv:1709.00364].
V.S. Dotsenko and Vl. S. Dotsenko, Phase transition in the 2D Ising model with impurity bonds, Sov. Phys. JETP Lett. 33 (1981) 37.
V.S. Dotsenko and Vl. S. Dotsenko, Critical behavior of the phase transition in the 2 dimensional Ising model with impurities, Adv. Phys. 32 (1983) 129.
A.W.W. Ludwig, Infinite hierarchies of exponents in a diluted ferromagnet and their interpretation, Nucl. Phys. B 330 (1990) 639 [INSPIRE].
V. Dotsenko, M. Picco and P. Pujol, Renormalization group calculation of correlation functions for the 2D random bond Ising and Potts models, Nucl. Phys. B 455 (1995) 701 [hep-th/9501017] [INSPIRE].
H. Shimada, Disordered O(n) loop model and coupled conformal field theories, Nucl. Phys. B 820 (2009) 707 [arXiv:0903.3787] [INSPIRE].
H. Shimada, J.L. Jacobsen and Y. Kamiya, Phase diagram and strong-coupling fixed point in the disordered O(n) loop model, J. Phys. A 47 (2014) 122001 [arXiv:1308.4333] [INSPIRE].
M. Picco, A. Honecker and P. Pujol, Strong disorder fixed points in the two-dimensional random-bond Ising model, J. Stat. Mech. 09 (2006) P09006 [cond-mat/0606312].
M. Hasenbusch, F. Parisen Toldin, A. Pelissetto and E. Vicari, Multicritical Nishimori point in the phase diagram of the ±J Ising model on a square lattice, Phys. Rev. E 77 (2008) 051115.
A.B. Zamolodchikov, Exact S matrix associated with selfavoiding polymer problem in two-dimensions, Mod. Phys. Lett. A 6 (1991) 1807 [INSPIRE].
R.J. Eden et al., The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).
V.L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems, Sov. Phys. JETP 32 (1971) 493 [Zh. Eksp. Teor. Fiz. 59 (1971) 907].
J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [INSPIRE].
Z. Komargodski and D. Simmons-Duffin, The random-bond Ising model in 2.01 and 3 Dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
G. Delfino and N. Lamsen, in preparation.
P.G. De Gennes, Exponents for the excluded volume problem as derived by the Wilson method, Phys. Lett. A 38 (1972) 339.
B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys. 34 (1984) 731 [INSPIRE].
W. Guo, H.W.J. Blote and F.Y. Wu, Phase transition in the n > 2 honeycomb O(n) model, Phys. Rev. Lett. 85 (2000) 3874.
V. Alba, A. Pelissetto and E. Vicari, Magnetic and glassy transitions in the square-lattice XY model with random phase shifts, J. Stat. Mech. 03 (2010) P03006.
Y. Ozeki, S. Yotsuyanagi, T. Sakai and Y. Echinaka, Numerical studies on critical properties of the Kosterlitz-Thouless phase for the gauge glass model in two dimensions, Phys. Rev. E 89 (2014) 022122.
S. Chen, A.M. Ferrenberg and D.P. Landau, Randomness-induced second-order transition in the two-dimensional eight-state Potts model: a Monte Carlo study, Phys. Rev. Lett. 69 (1992) 1213.
S. Chen, A.M. Ferrenberg and D.P. Landau, Monte Carlo simulation of phase transitions in a two-dimensional random-bond Potts model, Phys. Rev. E 52 (1995) 1377.
S. Wiseman and E. Domany, Critical behavior of the random-bond Ashkin-Teller model: a Monte Carlo study, Phys. Rev. E 51 (1995) 3074 [INSPIRE].
M. Kardar, A.L. Stella, G. Sartoni and B. Derrida, Unusual universality of branching interfaces in random media, Phys. Rev. E 52 (1995) R1269.
J. Cardy and J.L. Jacobsen, Critical behavior of random-bond Potts models, Phys. Rev. Lett. 79 (1997) 4063.
C. Chatelain and B. Berche, Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries, Phys. Rev. E 60 (1999) 3853.
T. Olson and A.P. Young, Monte Carlo study of the critical behavior of random bond Potts models, Phys. Rev. B 60 (1999) 3428 [cond-mat/9903068] [INSPIRE].
J.L. Jacobsen and M. Picco, Large-q asymptotics of the random-bond Potts model, Phys. Rev. E 61 (2000) R13.
J.L. Jacobsen, Multiscaling of energy correlations in the random-bond Potts model, Phys. Rev. E 61 (2000) R6060.
J.-Ch. Anglès d’Auriac and F. Igloi, Phase Transition in the 2D random Potts model in the large-q Limit, Phys. Rev. Lett. 90 (2003) 190601.
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Delfino, G., Lamsen, N. Exact results for the O(N ) model with quenched disorder. J. High Energ. Phys. 2018, 77 (2018). https://doi.org/10.1007/JHEP04(2018)077
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DOI: https://doi.org/10.1007/JHEP04(2018)077