Finite size effects on measures of critical exponents in d = 3 O(N) models

Ballesteros, H. G.; Fernández, L.A.; Martı́n-Mayor, V.; Sudupe, A. Muñoz

doi:10.1016/0370-2693(96)00984-7

Cited by 149 publications

(163 citation statements)

References 28 publications

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“…The unusually large value of η VBS is a direct consequence of fractionalization in the intermediate QSL phase and is often regarded as a smoking gun diagnostic of exotic critical points (see e.g., [33]). More quantitatively, our critical exponents are in rough agreement with the best estimate of η = 1.375(5) of the bilinear field and ν = 0.7525(10) in the O(4) model [34]. We note that the values for η VBS and ν VBS agree within the quoted errors for N = 10 and N = 11.…”

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confidence: 88%

Spin Nematics, Valence-Bond Solids, and Spin Liquids inSO(N)Quantum Spin Models on the Triangular Lattice

Kaul

2015

Phys. Rev. Lett.

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We introduce a simple model of SO(N ) spins with two-site interactions which is amenable to quantum Monte-Carlo studies without a sign problem on non-bipartite lattices. We present numerical results for this model on the two-dimensional triangular lattice where we find evidence for a spin nematic at small N , a valence-bond solid (VBS) at large N and a quantum spin liquid at intermediate N . By the introduction of a sign-free four-site interaction we uncover a rich phase diagram with evidence for both first-order and exotic continuous phase transitions.The destruction of magnetic order by quantum fluctuations in spin systems is frequently invoked as a route to exotic condensed matter physics such as spin liquid phases and novel quantum critical points [1][2][3]. The most commonly studied spin Hamiltonians have symmetries of the groups SO(3) and SU(2) which describe the rotational symmetry of 3-dimensional space. Motivated both by theoretical and experimental [4] interest, spin models with larger-N symmetries have been introduced, e.g. extensions of SU (2) to SU(N ) [5][6][7][8] or Sp(N ) [9].The extension of SO (3) to SO(N ) is an independant large-N enlargement of symmetry, with its own physical motivations [10]. While there have been many studies of SO(N ) spin models in one dimension [11][12][13], our understanding of their ground states and quantum phase transitions in higher dimension is in its infancy. To this end, we introduce here a simple SO(N ) spin model that surprisingly is sign free on any non-bipartite lattice. This model provides us with a new setting in which the destruction of magnetic order can be studied in higher dimensions using unbiased methods. As an example of interest, we present the results of a detailed study of the phase diagram of the our SO(N ) anti-ferromagnet on the two-dimensional triangular lattice.Models. -Consider a triangular lattice, each site of which has a Hilbert state of N states, we will denote the state of site j as |α j (1 ≤ α ≤ N ). Define the N (N −1)/2 generators of SO(N ) on site i asL αβ i with α < β; they will be chosen in the fundamental representation on all sites:L αβ j |γ j = iδ βγ |α j − iδ αγ |β j . Now consider the following SO(N ) [14] symmetric lattice model for N ≥ 3,where the "·" implies a summation over the N (N − 1)/2 generators and ij is the set of nearest neighbors. To see thatĤ J does not suffer from the sign problem, define a "singlet" state on a bond, |S ij ≡ 1 √ N α |αα ij and the singlet projectorP ij = |S ij S ij |. Using these operators and ignoring a constant shift we find the simple form [15],We make four observations: First, it is possible to create an SO(N ) spin singlet with only two spins for all N (in contrast to SU(N ) where N fundamental spins are required to create a singlet); Second Eq. (1) being a sum of projectors on this two-site singlet is the simplest SO(N ) coupling, despite it being a biquadratic interaction in the generatorsL αβ ; Third, since the singlet has a positive expansion,Ĥ J is Marshall positive on any lattice; ...

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confidence: 88%

Spin Nematics, Valence-Bond Solids, and Spin Liquids inSO(N)Quantum Spin Models on the Triangular Lattice

Kaul

2015

Phys. Rev. Lett.

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“…It is also clear that fluctuations lower the transition temperatures from their mean-field values, as it also happens in the three-dimensional Heisenberg model. 51 In addition, we find a helicoidal spin structure at sufficiently low temperatures.…”

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confidence: 61%

Discontinuous transitions in double-exchange materials

et al. 2001

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It is shown that the double-exchange Hamiltonian, with weak antiferromagnetic interactions, has a rich variety of first-order transitions between phases with different electronic densities and/or magnetizations. The paramagnetic-ferromagnetic transition moves towards lower temperatures, and becomes discontinuous as the relative strength of the double-exchange mechanism and antiferromagnetic coupling is changed. This trend is consistent with the observed differences between compounds with the same nominal doping, such as La 2/3 Sr 1/3 MnO 3 and La 2/3 Ca 1/3 MnO 3 . Our results suggest that, in the low doping regime, a simple magnetic mechanism suffices to explain the main features of the phase diagram.

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“…7, again plotting the h-dependence of the reduced percolation temperature t p = (T p − T c )/T c . The critical temperature T c of the magnetization transition for the 3D O(2) model without field was found to be J/kT c = 0.454165 in [17]. Again, the straight line in the log-log plot clearly indicates the power law behaviour of the Kertész line.…”

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confidence: 83%

Cluster percolation and pseudocritical behaviour in spin models

Fortunato

Satz

2001

Physics Letters B

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Abstract:The critical behaviour of many spin models can be equivalently formulated as percolation of specific site-bond clusters. In the presence of an external magnetic field, such clusters remain well-defined and lead to a percolation transition, even though the system no longer shows thermal critical behaviour. We investigate the 2-dimensional Ising model and the 3-dimensional O(2) model by means of Monte Carlo simulations. We find for small fields that the line of percolation critical points has the same functional form as the line of thermal pseudocritical points.Percolation theory [1,2] provides in many cases an elegant interpretation of the mechanism of second order phase transitions: the existence of an infinite spanning cluster represents the new order of the microscopic constituents of the system due to spontaneous symmetry breaking.In the Ising model, a rigorous correspondence between thermal critical behaviour and percolation was established by Coniglio and Klein [3]; the clusters are constructed by linking like-sign spins through a temperature-dependent bond probability p = 1 − exp(−2J/kT ), where J is the Ising coupling and T the temperature. This result has been recently extended to a wide class of models, from continuous spin Ising-like models [4,5] to O(n) models [6]. In all cases the system was considered in the absence of any external field. The reason for this is clear: by introducing an external field H, we explicitly break the symmetry of the Hamiltonian of the system, thus eliminating the thermal critical behaviour of the model. None of the thermodynamic potentials exhibits discontinuities of any kind, since the partition function is analytical for H = 0.On the other hand, the Coniglio-Klein clusters can be built as well when H =0. Because of the field, the system has a non-vanishing magnetization m parallel to the direction of H for any finite value of the temperature T . For T → ∞, m→ 0, and for T = 0, m = 1. This suggests that for a fixed value of H, the clusters will start to form an infinite network at some temperature T p (H). Varying the field H, one thus obtains a curve T p (H) in the T − H plane, the Kertész line [7].The Kertész line specifies the usual percolation threshold, just as in the case H = 0, and the percolation variables exibit the usual singularities, leading to a set of critical exponents. In particular, the percolation strength, which is the relative size of the percolation cluster compared to the total volume of the system, remains the order parameter of the percolation transition.

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Finite size effects on measures of critical exponents in d = 3 O(N) models

Cited by 149 publications

References 28 publications

Spin Nematics, Valence-Bond Solids, and Spin Liquids inSO(N)Quantum Spin Models on the Triangular Lattice

Spin Nematics, Valence-Bond Solids, and Spin Liquids inSO(N)Quantum Spin Models on the Triangular Lattice

Discontinuous transitions in double-exchange materials

Cluster percolation and pseudocritical behaviour in spin models

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