The recent discovery of the extraordinary-log (E-Log) criticality is a celebrated achievement in modern critical theory and calls for generalization. Using large-scale Monte Carlo simulations, we study the critical phenomena of plane defects in three- and four-dimensional O(n) critical systems. In three dimensions, we provide the first numerical proof for the E-Log criticality of plane defects. In particular, for n=2, the critical exponent q[over ^] of two-point correlation and the renormalization-group parameter α of helicity modulus conform to the scaling relation q[over ^]=(n-1)/(2πα), whereas the results for n≥3 violate this scaling relation. In four dimensions, it is strikingly found that the E-Log criticality also emerges in the plane defect. These findings have numerous potential realizations and would boost the ongoing advancement of conformal field theory.
The recent discovery of the extraordinary-log (E-Log) criticality is a celebrated achievement in modern critical theory and calls for generalization. Using large-scale Monte Carlo simulations, we study the critical phenomena of plane defects in three- and four-dimensional O(n) critical systems. In three dimensions, we provide the first numerical proof for the E-Log criticality of plane defects. In particular, for n=2, the critical exponent q[over ^] of two-point correlation and the renormalization-group parameter α of helicity modulus conform to the scaling relation q[over ^]=(n-1)/(2πα), whereas the results for n≥3 violate this scaling relation. In four dimensions, it is strikingly found that the E-Log criticality also emerges in the plane defect. These findings have numerous potential realizations and would boost the ongoing advancement of conformal field theory.
Recent advances in boundary critical phenomena have led to the discovery of a new surface universality class in the three-dimensional O ( N ) model. The newly found “extraordinary-log” phase can be realized on a two-dimensional surface for N < N c , with N c > 3 , and on a plane defect embedded into a three-dimensional system, for any N . One of the key features of the extraordinary-log phase is the presence of logarithmic violations of standard finite-size scaling. In this work we study finite-size scaling in the extraordinary-log universality class by means of Monte Carlo simulations of an improved lattice model. We simulate the model with open boundary conditions, realizing the extraordinary-log phase on the surface for N = 2 , 3 , as well as with fully periodic boundary conditions and in the presence of a plane defect for N = 2 , 3 , 4 . In line with theory predictions, renormalization-group invariant observables studied here exhibit a logarithmic dependence on the size of the system. We numerically access not only the leading term in the β function governing these logarithmic violations, but also the subleading term, which controls the evolution of the boundary phase diagram as a function of N .
We demonstrate the emergence of extraordinary-log criticality in a quantum many-body system by coupling a spin-1/2 Heisenberg chain to a two-dimensional quantum critical Heisenberg bilayer. Under the quantum-to-classical mapping, the chain acts as a plane defect in a three-dimensional critical environment for which the extraordinary-log universality class has recently been predicted. Using large-scale sign-problem-free quantum Monte Carlo simulations, we directly probe the renormalization-group flow of the effective defect coupling through the spin stiffness. We observe the characteristic logarithmic growth with linear system size L, corresponding to a stable weak-coupling fixed point and a running coupling g(L)∼1/logL. Furthermore, the extracted step-scaling function quantitatively follows the universal renormalization-group flow predicted for the extraordinary-log universality class, including the leading and subleading universal coefficients. Our results establish extraordinary-log criticality in a genuine quantum many-body setting and show that its universal behavior persists despite the topological θ-term present in the low-energy field theory of the isolated spin chain.
Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance r as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with d the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the O(N) system. In this logarithmic universality, g(r) decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional XY model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of g(r,L) has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with L as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the r-dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.
It was recently found that the classical 3d O (N) ( N ) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as \langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q} ⟨ S → ( x ) ⋅ S → ( 0 ) ⟩ ∼ 1 ( log x ) q . This universality class exists for a range 2 ≤ N < N_c 2 ≤ N < N c and Monte-Carlo simulations and conformal bootstrap indicate N_c > 3 N c > 3 . In this work, we extend this result to the 3d O (N) ( N ) model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N ≥2 N ≥ 2 . We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = ∞ N = ∞ is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N 1 / N corrections. We study the “central charge” a a for the O(N) O ( N ) model in the boundary and interface geometries and provide a non-trivial detailed check of an a a -theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O (N)
The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.
Criticality on Rényi Defects at (2+1)D O(3) Quantum Critical Points
At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)D quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index n, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.