We study the conformality loss of theories with long-range interactions. We consider the $O(2)\times O(N)$ multiscalar model with coupling $r^{-d-δ}$ in $d=4-ε$ dimension. We compute the critical exponents of the long-range fixed points (LRFPs) to three loops. The phase diagram of the model is dominated by two processes: the short-range crossover and merger-annihilation of LRFPs. The two processes intersect at the lower edge of the conformal window, below which the LRFPs disappear into the complex plane. We propose a novel scenario for the short-range crossover of complex LRFPs, in which the short-range crossover occurs on a vertical line in the complex plane of $δ$. The complex marginal operator generates renormalization group flow on the transition line and significantly enriches the short-range crossover of complex LRFPs.
We study the conformality loss of theories with long-range interactions. We consider the O(2)×O(N) multiscalar model with coupling r−d−δ in d=4−ε dimension. We compute the critical exponents of the long-range fixed points (LRFPs) to three loops. The phase diagram of the model is dominated by two processes: the short-range crossover and merger-annihilation of LRFPs. The two processes intersect at the lower edge of the conformal window, below which the LRFPs disappear into the complex plane. We propose a novel scenario for the short-range crossover of complex LRFPs, in which the short-range crossover occurs on a vertical line in the complex plane of δ. The complex marginal operator generates renormalization group flow on the transition line and significantly enriches the short-range crossover of complex LRFPs.
A general mechanism for the loss of conformal invariance is the merger and disappearance of an infrared fixed point and an ultraviolet fixed point of a renormalization group flow. We show explicitly how this mechanism works in the case of identical bosons at unitarity as the spatial dimension d is varied. For d between the critical dimensions d1=2.30 and d2=3.76, there is loss of conformality as evidenced by the Efimov effect in the three-body sector. The $\ensuremath{\beta}$ function for an appropriate three-body coupling is a quadratic polynomial in that coupling. For d<d1 and for d>d2, the $\ensuremath{\beta}$ function has two real roots that correspond to infrared and ultraviolet fixed points. As d approaches d1 from below and as d approaches d2 from above, the fixed points merge and disappear into the complex plane. For d1<d<d2, the $\ensuremath{\beta}$ function has complex roots and the renormalization group flow for the three-body coupling is a limit cycle.
Abstract We study the second-order phase transition in the d -dimensional Ising model with long-range interactions decreasing as a power of the distance 1 / r d + s . For s below some known value s ∗ , the transition is described by a conformal field theory without a local stress tensor operator, with critical exponents varying continuously as functions of s . At s = s ∗ , the phase transition crosses over to the short-range universality class. While the location s ∗ of this crossover has been known for 40 years, its physics has not been fully understood, the main difficulty being that the standard description of the long-range critical point is strongly coupled at the crossover. In this paper we propose another field-theoretic description which, on the contrary, is weakly coupled near the crossover. We use this description to clarify the nature of the crossover and make predictions about the critical exponents. That the same long-range critical point can be reached from two different UV descriptions provides a new example of infrared duality.
The critical exponents for two ${\ensuremath{\varphi}}^{3}$-field theories with long-range (LR) interactions decaying as 1/${R}^{d+\ensuremath{\sigma}}$, \ensuremath{\sigma}>0, are calculated to two-loop order in renormalized perturbation theory in d=3\ensuremath{\sigma}-\ensuremath{\epsilon}' dimensions. One is the continuum version of the p-state Potts model and the other is the scalar field theory with imaginary coupling that describes the Yang-Lee edge-singularity problem. The two crossover exponents for quadratic symmetry breaking discussed by Wallace and Young and in recent work by the present authors are also calculated in the first case. By means of renormalization-group recursion relations to one-loop order, it is shown that the LR fixed point is stable for all \ensuremath{\sigma}2 whenever ${\ensuremath{\eta}}_{\mathrm{SR}}$, the critical correlation-function exponent for short-range (SR) interactions, is negative, implying a discontinuity of critical exponents at \ensuremath{\sigma}=2. This is the case for the (p2)-state Potts model and for the Yang-Lee edge-singularity problem, and is in agreement with recent results by Chang and Sak for the Ising spin-glass problem. For the (p>2)-state Potts model there is an indication of a continuous crossover to SR behavior at \ensuremath{\sigma}=2-${\ensuremath{\eta}}_{\mathrm{SR}}$, with ${\ensuremath{\eta}}_{\mathrm{SR}}$>0. It is pointed out that a number of exact results [\ensuremath{\beta}=(1-\ensuremath{\epsilon}'/2\ensuremath{\sigma})\ensuremath{\sigma}\ensuremath{\nu}, \ensuremath{\sigma}^=(d-\ensuremath{\sigma})/(d+\ensuremath{\sigma}), in which \ensuremath{\sigma}^ is the Yang-Lee edge-singularity exponent, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{-}}1}$=(d-\ensuremath{\sigma})/2 for a scalar theory] may apply within the LR expansion depending on \ensuremath{\eta}=2-\ensuremath{\sigma} (shown here to hold at least to two-loop order) being exact, to all orders.
The 6−ε Expansion for Long-Range Lee--Yang and Percolation Criticality
The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the ε′=3σ−d expansion fixes the anomalous dimension at η=2−σ, whereas SR percolation has ηSR<0 near d=6. Sak's matching condition then places the crossover above σ=2, outside the regime in which the LR interaction dominates. In spatial dimension d=6−ε, we formulate a perturbative expansion for the LR ϕ3 field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime 0<δ<ε/3, where δ=2−σ. We derive the one-loop corrections to the critical exponents η and ν, which acquire nontrivial dependence on ε and δ. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR 6−ε results as σ→2. These results support a crossover threshold σ∗=2 and remove the apparent discontinuity of η between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and q-state Potts universality classes with q<2.
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. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.
The establishment of the Wilson-Fisher fixed point (WFP) for O(n) spin models in d=4−ε dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as ∝1/rd+σ, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the ε-expansion calculations up to the two-loop level. We show that, as long as σ<2, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with d/2<σ<2, the critical exponents, including the anomalous dimension, are functions of ε, δ=2−σ and n, which reduce to the exact results in the limiting cases ε→0, δ→0 or n→∞. Our (4−ε)-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at σ∗=2, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.