Three-dimensional tricritical spins and polymers

Bauerschmidt, Roland; Lohmann, Martin; Slade, Gordon

doi:10.1063/1.5110277

Cited by 5 publications

(6 citation statements)

References 39 publications

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“…Jensen estimated M e = 0.8857(1), M g = 0.12424(4), and M m = 0.3894 (1). The quantity A ∞ in (13) is just M g /M e , and B ∞ is M m /M e .…”

Section: Honeycomb Latticementioning

confidence: 97%

“…The polymer model is exactly equivalent to a supersymmetric spin model which corresponds to the n = 0 version of the |φ| 6 model. For both models the authors of [1] identify the tricritical point, and prove that the tricritical two-point function has Gaussian long-distance decay, namely 1/|x|.…”

Section: Isaws In Three Dimensionsmentioning

confidence: 99%

“…One interesting aspect of ISAWs is that d = 3 is the critical dimension. In [1] two related models on Z 3 are studied. One is a model of classical unbounded n-component continuous spins with a triple-well single-spin potential (the |φ| 6 model), and the other is a random walk model of linear polymers with a three-body repulsion and two-body attraction at the tricritical theta point.…”

Section: Isaws In Three Dimensionsmentioning

confidence: 99%

“…We shall use the symbol Ω n to mean the set of all SAWs of length n, and denote c n = |Ω n |. c n = Cn γ−1 µ n (1 + o(1)) (1) where the constant C is lattice-dependent but the exponent γ is universal, and depends only on the dimension. 1 Without loss of generality we will orient our honeycomb lattice so that it has vertical edges, and assume that SAWs start at a vertex at the bottom of a vertical edge (see Figure 1).…”

Section: Introductionmentioning

confidence: 99%

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Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

Beaton¹,

Guttmann²,

Jensen³

2020

J. Phys. A: Math. Theor.

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We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the θ-point to be at u c = 2.767 ± 0.002. The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is 2 + √ 2 [14], while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is 1 + √ 2 [3]. We could not help but notice that 2 + 4 √ 2 = 2.767 . . . . We discuss the difficulties of trying to prove, or disprove, this possibility.For square lattice ISAWs we find u c = 1.9474 ± 0.001, which is consistent with the best Monte Carlo analysis.We also study bridges and terminally-attached walks (TAWs) on the square lattice at the θ-point. We estimate the exponents to be γ b = 0.00 ± 0.03, and γ 1 = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction [18, 37, 38] γ 1 (θ) = ν = 4 7 , albeit for a modified version of the problem, while the former estimate is predicted in [16] to be zero.

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“…Jensen estimated M e = 0.8857(1), M g = 0.12424(4), and M m = 0.3894 (1). The quantity A ∞ in (13) is just M g /M e , and B ∞ is M m /M e .…”

Section: Honeycomb Latticementioning

confidence: 97%

Section: Isaws In Three Dimensionsmentioning

confidence: 99%

Section: Isaws In Three Dimensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

Beaton¹,

Guttmann²,

Jensen³

2020

J. Phys. A: Math. Theor.

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“…Nonrigorous methods were used in the physics literature to study the density transition in dimensions 2 and 3, and in particular its tricritical behaviour, in the 1980s [9][10][11][12]. In recent work with Lohmann, we applied a rigorous renormalisation group method to study the 3-dimensional tricritical point [3], and proved that the tricritical two-point function has Gaussian |x| −1 decay for the model on Z 3 . In [14], the transition across the second-order phase boundary was studied on a 4-dimensional hierarchical lattice, where a logarithmic correction to the mean-field behaviour of the density was proved.…”

Section: Introductionmentioning

confidence: 99%

Mean-field tricritical polymers

Bauerschmidt,

Slade

2019

Preprint

Self Cite

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We provide an introductory account of a tricritical phase diagram, in the setting of a mean-field random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices N in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter g. A chemical potential ν controls the walk length. We determine the phase diagram in the (g, ν) plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order N ) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.

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Statistical Mechanics of Confined Polymer Networks

Duplantier

Guttmann

2020

J Stat Phys

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We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical Θ-point. In the Θ-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, γ Θ b , to that of terminally-attached arches, γ Θ 11 , and to the correlation length exponent ν Θ . We find γ Θ b = γ Θ 11 + ν Θ . In the case of the special transition, we find γ Θ b (sp) = 1 2 [γ Θ 11 (sp) + γ Θ 11 ] + ν Θ . For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the Θ-point. We provide compelling numerical evidence for some of these results both in two-and three-dimensions.

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Three-dimensional tricritical spins and polymers

Cited by 5 publications

References 39 publications

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

Mean-field tricritical polymers

Statistical Mechanics of Confined Polymer Networks

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