<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtext>SU</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Heisenberg model on the square lattice: A continuous-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>quantum Monte Carlo study

Beach, K. S. D.; Alet, Fabien; Mambrini, Matthieu; Capponi, Sylvain

doi:10.1103/physrevb.80.184401

Cited by 86 publications

(101 citation statements)

References 30 publications

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“…IV, using an expansion around n = 2 and dimension d = 4, that fluctuations lead to a continuous transition for n < n c , with n c > 2 if d < 4. At n = 3 in 3D we find behaviour consistent with a continuous transition, and obtain the exponent values ν = 0.536 (13) and γ = 0.97 (2). If this is indeed a new critical point, it implies the possibility of similar behaviour in two-dimensional quantum SU (3) magnets.…”

Section: 20mentioning

confidence: 62%

“…Combining the results from our analysis of data on the K-lattice for χ, C, the order parameter (not shown), and using bootstrap methods to determine errors, our best estimates for the two independent critical exponents are ν = 0.536 (13) and η = 0.23 (2). The scaling relations γ = (2 − η)ν and β = ν(1 + η)/2 imply γ = 0.97 (2) and β = 0.33 (1).…”

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confidence: 99%

“…Indeed, the models we discuss here are closely related to loop algorithms that have been developed for Monte Carlo simulation of quantum spin systems. [11][12][13] A feature specific to our three-dimensional lattice model is that, since the 'time-like' axis is microscopically equivalent to the two 'space-like' ones, at a continuous quantum phase transition the dynamical exponent value z = 1 is guaranteed, rather than a matter for calculation. Loop models with 'deconfined' transitions to valence bond solid phases may also be constructed by introducing extra couplings, and will be discussed in a separate paper.…”

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confidence: 99%

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Phase transitions in three-dimensional loop models and theCPn−1sigma model

et al. 2013

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We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretisations of CP n−1 σ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the σ model, and we discuss the relationship between loop properties and σ model correlators. On large scales, loops are Brownian in an ordered phase and have a non-trivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1, 2, 3 and first order transitions for n ≥ 4. We also give a renormalisation group treatment of the CP n−1 model that shows how a continuous transition can survive for values of n larger than (but close to) two, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU (n) quantum magnets in (2+1) dimensions, Anderson localisation in symmetry class C, and the statistics of random curves in three dimensions.

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Section: 20mentioning

confidence: 62%

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confidence: 99%

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confidence: 99%

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Phase transitions in three-dimensional loop models and theCPn−1sigma model

et al. 2013

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“…the obvious SO(N ) of Eq. (1) is enlarged to an SU(N ) symmetry.Since the bipartite SU(N ) case has been studied in great detail in past work on various lattices [7,[16][17][18][19][20][21][22][23], we shall concern ourselves here with the non-bipartite SO(N ) case which is relatively unexplored. Phases ofĤ J : Starting at N = 3, Eq.…”

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confidence: 99%

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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“…25,26 This critical value depends on N , and from the numerical work [28][29][30] it is known that the two-dimensional CP N −1 model on a square lattice is disordered for N/n c ≥ 5.…”

Section: Effective Low-energy Field Theory For the Sp(n ) Heisenbmentioning

confidence: 99%

Magnetic phases of spin-32fermions on a spatially anisotropic square lattice

Kolezhuk¹,

Vekua²

2011

Phys. Rev. B

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We study the magnetic phase diagram of spin-3 2 fermions in a spatially anisotropic square optical lattice at quarter filling (corresponding to one particle per lattice site). In the limit of the large onsite repulsion the system can be mapped to the so-called Sp(N ) Heisenberg spin model with N = 4. We analyze the Sp(N ) spin model with the help of the large-N field-theoretical approach and show that the effective theory corresponds to the Sp(N ) extension of the CP N−1 model, with the Lorentz invariance generically broken. We obtain the renormalization flow of the model couplings and show that although the Sp(N ) terms are seemingly irrelevant, their presence leads to a renormalization of the CP N−1 part of the action, driving a phase transition. We further consider the influence of the external magnetic field (the quadratic Zeeman effect), and present the qualitative analysis of the ground state phase diagram.

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SU(N)Heisenberg model on the square lattice: A continuous-Nquantum Monte Carlo study

Cited by 86 publications

References 30 publications

Phase transitions in three-dimensional loop models and theCPn−1sigma model

Phase transitions in three-dimensional loop models and theCPn−1sigma model

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

Magnetic phases of spin-32fermions on a spatially anisotropic square lattice

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