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The $d$-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power $1/r^{d+s}$, admits a second order phase transition with continuously varying critical exponents.
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Formally our considerations apply also to non-integer dimensions in the range 1 < d < 4 1<d<4
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We reserve the letter σ \sigma for the short-range Ising spin field. What we have called s s is usually denoted σ \sigma
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The action ( 1
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Throughout the paper scaling dimensions of various fields X X are denoted interchangeably by Δ X \Delta_{X} or [ X ] [X]
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The long-distance behavior is again mean-field with powerlike (as opposed to exponential) correlations when we are away from the transition, in the disordered phase. Nonalyticity of the ϕ \phi propagator in momentum space, 1 / ( | p | s + t ) 1/(|p|^{s}+t) , at p = 0 p=0 is what causes this powerlike falloff. The IR value of the ϕ \phi dimension can be read off as [ ϕ ] dis = d − [ ϕ ] UV [\phi]_{\rm dis}=d-[\phi]_{\rm UV} . This also holds for fluctuations of ϕ \phi around the mean value in the ordered phase
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The puzzle can be stated more formally in terms of “recombination rules” of unitary representations of the conformal algebra 𝔰 o ( d + 1 , 1 ) {\mathfrak{s}o}(d+1,1) . (The enhancement of scale invariance to conformal invariance at the long-range fixed point – even in the absence of a local stress tensor – has been recently demonstrated in [ 10 ] .) The standard stress tensor of the SRFP is the lowest weight (conformal primary) state of the shortened
Cited in the paper.
Z. Komargodski and D. Simmons-Duffin, J. Phys. A: Math. Theor. 101
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D. Simmons-Duffin, (2016), arXiv:1612.08471 [hep-th]
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C. Behan, L. Rastelli, S. Rychkov, and B. Zan, (2017), arXiv:1703.05325 [hep-th]
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2017
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