We investigate the one-dimensional Ising model with long-range interactions decaying as 1/r^{1+s} 1 / r 1 + s . In the critical regime, for 1/2 ≤ s ≤ 1 1 / 2 ≤ s ≤ 1 , this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with s s . For s;gt1 s > 1 the model has instead no phase transition at finite temperature, as in the short-range case. In the standard field-theoretic description, involving a generalized free field with quartic interactions, the critical model is weakly coupled near s=1/2 s = 1 / 2 but strongly coupled in the vicinity of the short-range crossover at s=1 s = 1 . We introduce a dual formulation that becomes weakly coupled as s \to 1 s → 1 . Precisely at s=1 s = 1 , the dual description becomes an exactly solvable conformal boundary condition of the two-dimensional free scalar. We present a detailed study of the dual model and demonstrate its effectiveness by computing perturbatively the CFT data near s=1 s = 1 , up to next-to-next-to-leading order in 1-s 1 − s , by two independent approaches: (i) standard renormalization of our dual field-theoretic description and (ii) the analytic conformal bootstrap. The two methods yield complete agreement.
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.
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 long-range Ising model is the prototype of criticality with power-law interactions. In each dimension its interacting critical points form a one-parameter family of nonlocal conformal field theories, weakly coupled at both ends but not in between. We study this family in one, two and three dimensions from several independent directions. We extend the expansion about the generalized free field from three to eight loops and expand about the short-range end. We show that Hamiltonian truncation applies to this nonlocal theory once a truncation artifact is corrected. Where they can be compared, the resummed expansions, Hamiltonian truncation, the numerical renormalization group and Monte Carlo simulations agree on the dimension of the leading
Z2-even scalar at the sub-percent level, as universality requires.
The statistical long-range Ising (LRI) model in one dimension provides a family of some of the simplest non-exactly solvable critical points. These critical points are described by long-range conformal field theories (CFTs) that are well understood analytically with perturbative expansions and some exact results. Guided by this, we revisit the Monte Carlo (MC) approach to the corresponding lattice model which has many favorable features compared to other systems studied with MC methods. In particular, with very modest computational cost, we are able to estimate scaling dimensions of leading and subleading operators as well as the associated three-point functions, and even certain four-point functions. These results help highlight the fact that 1d LRI CFTs are a phenomenal playground to more generally test non-perturbative methods and ideas for strongly coupled CFTs.
We investigate the critical behavior of the one-dimensional Ising model with long-range interactions using the functional renormalization group in the local potential approximation (LPA), and compare our findings with Dyson's hierarchical model (DHM). While the DHM lacks translational invariance, it admits a field-theoretical description closely resembling the LPA, up to minor but nontrivial differences. After reviewing the real-space renormalization group approach to the DHM, we demonstrate a remarkable agreement in the critical exponent
ν between the two methods across the entire range of power-law decays
1/2<σ<1. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime,
σ≲1.
We study a one-dimensional spin-
1/2 model with three-spin interactions and a transverse magnetic field
h. The model is known to have a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry and a duality between
h and
1/h. The self-dual point at
h=1 is a quantum critical point with a continuous phase transition. We compute the critical exponents $z, \ensuremath{\beta}, \ensuremath{\gamma}$, and $\ensuremath{\nu}$, and the central charge
c numerically using exact diagonalization (ED) for systems with periodic boundary conditions. We find that both
z and
c are equal to 1, implying that the critical point is governed by a conformal field theory with a marginal operator. The values obtained for $\ensuremath{\beta}/\ensuremath{\nu}, \ensuremath{\gamma}/\ensuremath{\nu}$, and $\ensuremath{\nu}$ from ED suggest that the model exhibits Ashkin-Teller criticality with an effective coupling that is intermediate between the four-state Potts model and two decoupled transverse field Ising models. A more careful analysis on much larger systems but with open boundaries using density-matrix renormalization group (DMRG) calculations, however, reveals important additive and multiplicative logarithmic corrections at and near criticality, and we present evidence that the self-dual point may be in the same universality class as the four-state Potts model. An energy level spacing analysis shows that the model is not integrable. For a system with an even number of sites and periodic boundary conditions, there are exact mid-spectrum zero-energy eigenstates whose number grows exponentially with the system size. A subset of these eigenstates have wave functions that are independent of the value of
h and have unusual entanglement structures; hence these can be considered to be quantum many-body scars. The number of such quantum scars scales at least linearly with system size. Finally, we study the infinite-temperature autocorrelation functions at sites close to one end of an open system. We find that some of the autocorrelators relax anomalously in time, with pronounced oscillations and very small decay rates if $h\ensuremath{\gg}1$ or $h\ensuremath{\ll}1$. If
h is close to the critical point, the autocorrelators decay quickly to zero except for an autocorrelator at the end site.