scholarly journals Analytic and numerical bootstrap of CFTs with $O(m)\times O(n)$ global symmetry in 3D

2020 ◽  
Vol 9 (3) ◽  
Author(s):  
Johan Henriksson ◽  
Stefanos R. Kousvos ◽  
Andreas Stergiou
Keyword(s):  
Power Series ◽  
Fixed Points ◽  
Parameter Space ◽  
Field Theories ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Bootstrap Techniques ◽  
Perturbative Methods ◽  
Anomalous Dimensions

Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with O(m)\times O(n)O(m)×O(n) global symmetry in d=3d=3 spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in \varepsilon=4-dε=4−d and in 1/n1/n, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for O(2)\times O(n)O(2)×O(n) theories for various values of nn. Some of these islands can be attributed to fixed points predicted by perturbative methods like the \varepsilonε and large-nn expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.

scholarly journals Bounds in 4D conformal field theories with global symmetry

2010 ◽  
Vol 44 (3) ◽  
pp. 035402 ◽  
Author(s):  
Riccardo Rattazzi ◽  
Slava Rychkov ◽  
Alessandro Vichi
Keyword(s):  
Field Theories ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field

scholarly journals Perturbative and nonperturbative studies of CFTs with MN global symmetry

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Johan Henriksson ◽  
Andreas Stergiou
Keyword(s):  
Fixed Points ◽  
Numerical Data ◽  
Three Dimensions ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
State Of Affairs ◽  
Perturbative Limit ◽  
Good Agreement

Fixed points in three dimensions described by conformal field theories with \ensuremath{M N}_{m,n} = O(m)^n\rtimes S_nMNm,n=O(m)n⋊Sn global symmetry have extensive applications in critical phenomena. Associated experimental data for m=n=2m=n=2 suggest the existence of two non-trivial fixed points, while the \varepsilonε expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters mm and nn, with critical exponents in good agreement with experimental determinations in the m=n=2m=n=2 case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters mm and nn. We find that one family of kinks approaches a perturbative limit as mm increases, and using large spin perturbation theory we construct a large mm expansion that fits well with the numerical data. This new expansion, akin to the large NN expansion of critical O(N)O(N) models, is compatible with the fixed point found in the \varepsilonε expansion. For the other family of kinks, we find that it persists only for n=2n=2, where for large mm it approaches a non-perturbative limit with \Delta_\phi\approx 0.75Δϕ≈0.75. We investigate the spectrum in the case \ensuremath{M N}_{100,2}MN100,2 and find consistency with expectations from the lightcone bootstrap.

scholarly journals ON SOLITON AUTOMORPHISMS IN MASSIVE AND CONFORMAL THEORIES

1999 ◽  
Vol 11 (03) ◽  
pp. 337-359 ◽  
Author(s):  
MICHAEL MÜGER
Keyword(s):  
Field Theories ◽  
Global Symmetry ◽  
Subsequent Paper ◽  
Quantum Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Symmetry Transformations ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Quasilocal Algebra

For massive and conformal quantum field theories in 1+1 dimensions with a global gauge group we consider soliton automorphisms, viz. automorphisms of the quasilocal algebra which act like two different global symmetry transformations on the left and right spacelike complements of a bounded region. We give a unified treatment by providing a necessary and sufficient condition for the existence and Poincaré covariance of soliton automorphisms which is applicable to a large class of theories. In particular, our construction applies to the QFT models with the local Fock property — in which case the latter property is the only input from constructive QFT we need — and to holomorphic conformal field theories. In conformal QFT soliton representations appear as twisted sectors, and in a subsequent paper our results will be used to give a rigorous analysis of the superselection structure of orbifolds of holomorphic theories.

scholarly journals Floquet conformal field theories with generally deformed Hamiltonians

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Ruihua Fan ◽  
Yingfei Gu ◽  
Ashvin Vishwanath ◽  
Xueda Wen
Keyword(s):  
Fixed Points ◽  
Entanglement Entropy ◽  
Momentum Density ◽  
Field Theories ◽  
Geometrical Approach ◽  
Heating Regime ◽  
Conformal Field Theories ◽  
Arbitrary Smooth Function ◽  
Conformal Field ◽  
Heating Phase

In this work, we study non-equilibrium dynamics in Floquet conformal field theories (CFTs) in 1+1D, in which the driving Hamiltonian involves the energy-momentum density spatially modulated by an arbitrary smooth function. This generalizes earlier work which was restricted to the sine-square deformed type of Floquet Hamiltonians, operating within a \mathfrak{sl}_2𝔰𝔩2 sub-algebra. Here we show remarkably that the problem remains soluble in this generalized case which involves the full Virasoro algebra, based on a geometrical approach. It is found that the phase diagram is determined by the stroboscopic trajectories of operator evolution. The presence/absence of spatial fixed points in the operator evolution indicates that the driven CFT is in a heating/non-heating phase, in which the entanglement entropy grows/oscillates in time. Additionally, the heating regime is further subdivided into a multitude of phases, with different entanglement patterns and spatial distribution of energy-momentum density, which are characterized by the number of spatial fixed points. Phase transitions between these different heating phases can be achieved simply by changing the duration of application of the driving Hamiltonian. %In general, there are rich internal structures in the heating phase characterized by different numbers of spatial fixed points, which result in different entanglement patterns and distribution of energy-momentum density in space. %Interestingly, after each driving cycle, these spatial fixed points will shuffle to each other in the array, and come back to the original locations after pp (p\ge 1p≥1) driving cycles. We demonstrate the general features with concrete CFT examples and compare the results to lattice calculations and find remarkable agreement.

scholarly journals SIMPLE CURRENTS, MODULAR INVARIANTS AND FIXED POINTS

1990 ◽  
Vol 05 (15) ◽  
pp. 2903-2952 ◽  
Author(s):  
A.N. SCHELLEKENS ◽  
S. YANKIELOWICZ
Keyword(s):  
Field Theory ◽  
Fixed Point ◽  
Fixed Points ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Conformal Field ◽  
Modular Invariants ◽  
Simple Currents

We review the use of simple currents in constructing modular invariant partition functions and the problem of resolving their fixed points. We present some new results, in particular regarding fixed point resolution. Additional empirical evidence is provided in support of our conjecture that fixed points are always related to some conformal field theory. We complete the identification of the fixed point conformal field theories for all simply laced and most non-simply laced Kac-Moody algebras, for which the fixed point CFT’s turn out to be Kac-Moody algebras themselves. For the remaining non-simply laced ones we obtain spectra that appear to correspond to new non-unitary conformal field theories. The fusion rules of the simplest unidentified example are computed.

scholarly journals Applications of dispersive sum rules: $ε$-expansion and holography

2021 ◽  
Vol 10 (6) ◽  
Author(s):  
Dean Carmi ◽  
Joao Penedones ◽  
Joao A. Silva ◽  
Alexander Zhiboedov
Keyword(s):  
Sum Rules ◽  
Dispersion Relations ◽  
Field Theories ◽  
Tree Level ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Anomalous Dimensions ◽  
Fisher Model ◽  
Twist Operators ◽  
Space Dispersion

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in d=4-\epsilond=4−ϵ dimensions. We re-derive many of the known results to order \epsilon^4ϵ4 and we make new predictions. No assumption of analyticity down to spin 00 was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.

scholarly journals Exclusion inside or at the border of conformal bootstrap continent

2020 ◽  
Vol 35 (06) ◽  
pp. 2050036
Author(s):  
Yu Nakayama
Keyword(s):  
Bound States ◽  
Pauli Exclusion Principle ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Field Theories ◽  
Exclusion Principle ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Anomalous Dimensions

How large can anomalous dimensions be in conformal field theories? What can we do to attain larger values? One attempt to obtain large anomalous dimensions efficiently is to use the Pauli exclusion principle. Certain operators constructed out of constituent fermions cannot form bound states without introducing nontrivial excitations. To assess the efficiency of this mechanism, we compare them with the numerical conformal bootstrap bound as well as with other interacting field theory examples. In two dimensions, it turns out to be the most efficient: it saturates the bound and is located at the (second) kink. In higher dimensions, it more or less saturates the bound but it may be slightly inside.

scholarly journals Bootstrapping mixed correlators in three-dimensional cubic theories II

2020 ◽  
Vol 8 (6) ◽  
Author(s):  
Stefanos R. Kousvos ◽  
Andreas Stergiou
Keyword(s):  
Condensed Matter ◽  
Three Dimensional ◽  
Global Symmetry ◽  
Energy Tensor ◽  
Conformal Field Theories ◽  
Cubic Symmetry ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Perturbative Methods ◽  
Stress Energy

Conformal field theories (CFTs) with cubic global symmetry in 3D are relevant in a variety of condensed matter systems and have been studied extensively with the use of perturbative methods like the \varepsilonε expansion. In an earlier work, we used the nonperturbative numerical conformal bootstrap to provide evidence for the existence of a previously unknown 3D CFT with cubic symmetry, dubbed “Platonic CFT”. In this work, we make further use of the numerical conformal bootstrap to perform a three-dimensional scan in the space of scaling dimensions of three low-lying operators. We find a three-dimensional isolated allowed region in parameter space, which includes both the 3D (decoupled) Ising model and the Platonic CFT. The essential assumptions on the spectrum of operators used to provide the isolated allowed region include the existence of a stress-energy tensor and the irrelevance of certain operators (in the renormalization group sense).

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