Wenliang Li

The three-dimensional classical O ( N ) model with a boundary has received renewed interest due to the discovery of the extraordinary-log boundary universality class for 2 ≤ N < N c . The critical value N c and the exponent of the boundary correlation function are related to certain amplitudes in the normal universality class. To determine their precise values, we revisit the 3D O ( N ) boundary conformal field theory for N = 1 , 2, 3, 4, 5. After substantially improving the accuracy of the boundary bootstrap, our determinations are in excellent agreement with the Monte Carlo results, resolving the previous discrepancies due to low truncation orders. We also use the recent bulk bootstrap results to deduce highly accurate Ising data. Many bulk and boundary predictions are obtained for the first time. Our results demonstrate the great potential of the η minimization method for many unexplored bootstrap problems in which positivity constraints are absent.

Periodic structures are ubiquitous in quantum many-body systems and quantum field theories, ranging from lattice models, to compact spaces, to topological phenomena. However, previous bootstrap studies encountered technical challenges even for one-body periodic problems, such as a failure in determining the accurate dispersion relations for Bloch bands. In this work, we develop a new bootstrap procedure to resolve these issues, which does not make use of positivity constraints. We mainly consider a quantum particle in a periodic cosine potential. The same procedure also applies to a particle on a circle, where the role of the Bloch momentum k is played by the boundary condition or the θ angle. We unify the natural set of operators and the translation operator by a new set of operators { e i n x e i a p p s } . To extract the Bloch momentum k , we further introduce a set of differential equations for ⟨ e i n x e i a p p s ⟩ in the translation parameter a . At some fixed a , the boundary conditions can be determined accurately by analytic bootstrap techniques and matching conditions. After solving the differential equations, we impose certain reality conditions to determine the accurate dispersion relations, as well as the k dependence of other physical quantities. We also investigate the case of noninteger s using the Weyl integral in fractional calculus.

We investigate the ϕ 2 n deformations of the O ( N ) -symmetric (generalized) free theories with a flat boundary, where n ≥ 2 is an integer. The generalized free theories refer to the □ k free scalar theories with a higher derivative kinetic term, which is related to the multicritical generalizations of the Lifshitz type. We assume that the (generalized) free theories and the deformed theories have boundary conformal symmetry and O ( N ) global symmetry. The leading anomalous dimensions of some boundary operators are derived from the bulk multiplet recombination and analyticity constraints. We find that the ε 1 / 2 expansion in the ϕ 6 -tricritical version of the special transition extends to other multicritical cases with larger odd integer n , and most of the higher derivative cases involve a noninteger power expansion in ε . Using the analytic bootstrap, we further verify that the multiplet-recombination results are consistent with boundary crossing symmetry.

We investigate ϕ 2 n + 1 deformations of the generalized free theory in the ε expansion, where the canonical kinetic term is generalized to a higher-derivative version. For n = 1 , we use the conformal multiplet recombination method to determine the leading anomalous dimensions of the fundamental scalar operator ϕ and the bilinear composite operators J . Then, we extend the n = 1 analysis to the Potts model with S N + 1 symmetry and its higher-derivative generalization, in which ϕ is promoted to an N -component field. We further examine the Chew-Frautschi plots and their N dependence. However, for each integer n > 1 , the leading anomalous dimensions of ϕ and J are not fully determined and contain one unconstrained constant, which in the canonical cases can be fixed by the results from the diagrammatic method. In all cases, we verify that the multiplet-recombination results are consistent with crossing symmetry using the analytic bootstrap methods.

We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of ⟨ ϕ n ⟩ or ⟨ ( i ϕ ) n ⟩ in n . We first use the quantum harmonic oscillator to illustrate various aspects of the ϕ n trajectory bootstrap method, such as the large n expansion, matching conditions, exact quantization condition, and high energy asymptotic behavior. Then we derive highly accurate solutions for the anharmonic oscillators with the parity invariant potential V ( ϕ ) = ϕ 2 + ϕ m and the P T invariant potential V ( ϕ ) = − ( i ϕ ) m for a large range of integral m , showing the high efficiency and general applicability of this new bootstrap approach. For the Hermitian quartic and non-Hermitian cubic oscillators, we further verify that the noninteger n results for ⟨ ϕ n ⟩ or ⟨ ( i ϕ ) n ⟩ are consistent with those from the wave function approach. In the P T invariant case, the existence of ⟨ ( i ϕ ) n ⟩ with noninteger n allows us to bootstrap the non-Hermitian theories with noninteger powers, such as fractional and irrational m .

We revisit the large N two-matrix model with tr[A, B]² interaction and quartic potentials by the analytic trajectory bootstrap, where A and B represent the two matrices. In the large N limit, we can focus on the single trace moments associated with the words composed of the letters A and B. Analytic continuations in the lengths of the words and subwords lead to analytic trajectories of single trace moments and intriguing intersections of different trajectories. Inspired by the one-cut solutions of one-matrix models, we propose a simple ansatz for the singularity structure of the two-matrix generating functions and the corresponding single trace moments. Together with the self-consistent constraints from the loop equations, we determine the free parameters in the ansatz and obtain highly accurate solutions for the two-matrix model at a low computational cost. For a given length cutoff L _max, our results are within and more accurate than the positivity bounds from the relaxation method, such as about 6-digit accuracy for L _max = 18. The convergence pattern suggests that we achieve about 8-digit accuracy for L _max = 22. As the singularity structure is closely related to the eigenvalue distributions, we further present the results for various types of eigenvalue densities. In the end, we study the symmetry breaking solutions using more complicated ansatzes.

As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics. The Ising transition is realized by physical systems, such as the liquid-vapor transition. Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes. Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints. We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point. The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure. Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach. For instance, the results for the two relevant scaling dimensions (∆ _σ , ∆ _ϵ ) ≈ (0.518153, 1.41278) are close to the most precise rigorous bounds obtained at a much higher computational cost.