arXiv subjects

A. Liam Fitzpatrick

Publications and source records attributed to A. Liam Fitzpatrick.

At least 19 recordsLinked to original sources

arXiv2026

3d Ising Field Theory with Magnetic Deformation: Fuzzy Sphere Meets TCSA

We study the magnetic deformation of the $(2+1)$d Ising CFT, also known as Ising Field Theory (IFT), on a spatial sphere. We use the Fuzzy Sphere (FS) Ising setup and compare it to the Truncated Conformal Space Approach (TCSA). Universal, infinite volume, IFT quantities are extracted by modeling the effects of curvature, as expected from a local effective theory on the sphere. We find evidence that IFT contains a bound-state with a small binding energy. Locality also allows us to extract the same IFT quantities from different angular momentum sectors, providing additional consistency checks.

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arXiv2026

Descending into the Modular Bootstrap

In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near $c = 1$ than the existing bound of $\Delta_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$.

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arXiv2026

Improving 3d Ising OPE Coefficients with Fuzzy Sphere Conformal Generators

We use the $K$ special conformal generator in the Fuzzy sphere setup of the Ising CFT to determine primary states. For $\Delta \lesssim 8$, we recover the known primaries and find several new ones, including in the parity-odd sector. We then use these primaries to compute OPE coefficients. We find that using primaries constructed from special-$K$ allows for better extrapolation of OPE coefficients to the CFT limit, because of the existence of an $O(1)$ gap between primaries and descendants in the spectrum of eigenvalues of $|K|^2$ which protects the primaries from strongly mixing with descendants. We compare the CFT data we obtain with the Eigenstate Thermalization Hypothesis.

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arXiv2025

Towards Large-Spin Effective Theory I: Three-Particle States in AdS $\phi^4$ Theory

We describe how to construct an effective Hamiltonian for leading twist states in $d\ge 3$ CFTs based on the separation of scales that emerges at large spin $J$ between the AdS radius $\ell_{\rm AdS}$ and the characteristic distance $\sim \ell_{\rm AdS} \log J$ between particles rotating in AdS with angular momentum $J$. As a controlled example, we work specifically with the toy model of a bulk complex scalar field $\phi$ with a $\lambda |\phi|^4$ coupling in AdS, up to $O(\lambda^2)$. For a given choice of twist cutoff $\Lambda_\tau$ in the effective theory, interactions are separated into long-distance nonlocal potential terms, arising from $t$-channel exchange of states with twist $\le \Lambda_\tau$, and short-distance local terms fixed by matching to low spin CFT data. At $O(\lambda^2)$, the effective Hamiltonian for the toy model has two-body nonlocal potential terms from one-loop bulk diagrams as well as three-body nonlocal potential terms from tree-level exchange of $\phi$. We describe in detail how these contributions are evaluated and how they are related to the CFT data entering in the large spin expansion. We discuss how to apply the construction of such effective Hamiltonians for models which do not have a large central charge or a sparse spectrum and are not typically considered holographic.

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arXiv2025

Towards Large-Spin Effective Theory II: $O(2)$ model in $d=4-\epsilon$

We show how to construct a holographic effective theory for the leading-twist operators in the $O(2)$ model in the $4-d=\epsilon$ expansion up to $O(\epsilon^2)$, based on the separation of short-distance and long-distance effects that arises as a function of spin $J$. We obtain the Hamiltonian of the theory and show that it correctly reproduces all the dimensions at $O(\epsilon^2)$ of the leading twist operators for all values of the charge $Q$ and spin $J$. The holographic Hamiltonian is given by the bulk exchange of a charged scalar $\phi$, neutral scalar $s \sim \phi \phi^*$, and a `ghost' field $c$, as well as a single local bulk interaction $(\phi \phi^*)^2$. We analyze various aspects of the spectrum and discuss their interpretation in light of the bulk description.

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arXiv2025

Hamiltonian Truncation of Large $N_f$ QED and Large $N$ Vector-like Theories in $d=2+1$

We consider a general class of large $N$ vector-like theories in $d=2+1$ in a Hamiltonian approach. We show that by using lightcone quantization and the $N\to\infty$ limit, we can diagonalize the Hamiltonian exactly and construct the eigenstates, spectral density, S-matrix, and form factors for the theory. For concreteness, we mainly focus on QED$_3$ at large $N_f$ as an explicit example. We comment on extending the approach to finite $N$ calculations.

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arXiv2025

Properties of scalar partition functions of 2d CFTs

We study the spectrum of scalar primary operators in any two-dimensional conformal field theory. We show that the scalars alone obey a nontrivial crossing equation. This extends previous work that derived a similar equation for Narain conformal field theories. Additionally, we show that at high temperature, the difference between the true scalar partition function and the one predicted from a semiclassical gravity calculation is controlled by: the modular integral of the partition function, the light states of the theory, and an infinite series terms directly related to the nontrivial zeros of the Riemann zeta function. We give several numerical examples and compute their modular integrals.

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arXiv2024

Toolkit for General 2d Scalar Potential in LCT

We present efficient algorithms for obtaining the Hamiltonian in Lightcone Conformal Truncation (LCT) for a 2d scalar field with a generic potential. We apply this method to the sine-Gordon and sinh-Gordon models in $1+1d$, and find precise agreement with integrability results when the scaling dimension $\Delta$ of the deforming cosine/cosinh potential is in the range $ \Delta \leq 1$. The agreement provides additional evidence for a recent conjecture for how to compute the effective lightcone Hamiltonian in this class of models. In addition, to high precision, we provide the first direct confirmation for the conjectured self-duality of the sinh-Gordon model ($\Delta < 0)$, which relates $\Delta \leftrightarrow 4/\Delta$. As the dimension approaches the upper limit $\Delta=1$ from below, we show analytically that the Hamiltonian matrix elements exactly reproduce those of a free Majorana fermion, demonstrating how bosonization is manifested in the LCT basis. We comment on the possible extension of the approach to $\Delta > 1$.

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arXiv2024

Constructing the Infrared Conformal Generators on the Fuzzy Sphere

We investigate the conformal algebra on the fuzzy sphere, and in particular the generators of translations and special conformal transformations which are emergent symmetries in the infinite IR but are broken along the RG flow. We show how to extract these generators using the energy momentum tensor, which is complicated by the fact that one does not have a priori access to the energy momentum tensor of the CFT limit but rather must construct it numerically. We discuss and quantitatively analyze the main sources of corrections to the conformal generators due to the breaking of scale-invariance at finite energy, and develop efficient methods for removing these corrections. The resulting generators have matrix elements that match CFT predictions with accuracy varying from sub-percent level for the lowest-lying states up to several percent accuracy for states with dimension $\sim 5$ with $N=16$ fermions. We show that the generators can be used to accurately identify primary operators vs descendant operators in energy ranges where the spectrum is too dense to do the identification solely based on the approximate integer spacing within conformal multiplets.

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arXiv2024

Holography and Regge Phases with $U(1)$ Charge

We use holography to study the large spin $J$ limit of the spectrum of low energy states with charge $Q$ under a $U(1)$ conserved current in CFTs in $d>2$ dimensions, with a focus on $d=3$ and $d=4$. For $Q=2$, the spectrum of such states is known to be universal and properly captured by the long-distance limit of holographic theories, regardless of whether the CFT itself is holographic. We study in detail the holographic description of such states at $Q>2$, by considering the contribution to the energies of $Q$ scalar particles coming from single photon and graviton exchange in the bulk of AdS; in some cases, scalar exchange and bulk contact terms are also included. For a range of finite values of $Q$ and $J$, we numerically diagonalize the Hamiltonian for such states and examine the resulting spectrum and wavefunctions as a function of the dimension $Δ$ of the charge-one operator and the central charges $c_{\mathcal{T}}, c_{\mathcal{J}}$ of the stress tensor and U(1) current, finding multiple regions in parameter space with qualitatively different behavior. We discuss the extension of these results to the regime of parametrically large charge $Q$, as well as to what extent such results are expected to hold universally, beyond the limit of holographic CFTs. We compare our holographic computations to results from the conformal bootstrap for the $3d$ O(2) model at $Q=3$ and $Q=4$ and find excellent agreement.

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arXiv2024

CFT and Lattice Correlators Near an RG Domain Wall between Minimal Models

Conformal interfaces separating two conformal field theories (CFTs) provide maps between different CFTs, and naturally exist in nature as domain walls between different phases. One particularly interesting construction of a conformal interface is the renormalization group (RG) domain wall between CFTs. For a given Virasoro minimal model $\mathcal{M}_{k+3,k+2}$, an RG domain wall can be generated by a specific deformation which triggers an RG flow towards its adjacent Virasoro minimal model $\mathcal{M}_{k+2,k+1}$ with the deformation turned on over part of the space. An algebraic construction of this domain wall was proposed by Gaiotto in \cite{Gaiotto:2012np}. In this paper, we will provide a study of this RG domain wall for the minimal case $k=2$, which can be thought of as a nonperturbative check of the construction. In this case the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising Model (IM) CFT. We will check the analytical results of correlation functions from the RG brane construction with the numerical density matrix renormalization group (DMRG) calculation using a lattice model proposed in \cite{Grover:2012bm,Grover:2013rc}, and find a perfect agreement. We comment on possible experimental realizations of this RG domain wall.

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arXiv2023

Lightcone Hamiltonian for Ising Field Theory I: T < T_c

We study 2d Ising Field Theory (IFT) in the low-temperature phase in lightcone quantization, and show that integrating out zero modes generates a very compact form for the effective lightcone interaction that depends on the finite volume vacuum expectation value of the $\sigma$ operator. This form is most naturally understood in a conformal basis for the lightcone Hilbert space. We further verify that this simple form reproduces to high accuracy results for the spectra, the $c$-function, and the form-factors from integrability methods for the magnetic deformation of IFT. For generic non-integrable values of parameters we also compute the above observables and compare our numeric results to those of equal-time truncation. In particular, we report on new measurements of various bound-state form-factors as well as the stress-tensor spectral density. We find that the stress tensor spectral density provides additional evidence that certain resonances of IFT are surprisingly narrow, even at generic strong coupling. Explicit example code for constructing the effective Hamiltonian is included in an appendix.

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arXiv2023

Higher $d$ Eisenstein Series and a Duality-Invariant Distance Measure

The Petersson inner product is a natural inner product on the space of modular invariant functions. We derive a formula, written as a convergent sum over elementary functions, for the inner product $E_s(G,B)$ of the real analytic Eisenstein series $E_s(τ, \barτ)$ and a general point in Narain moduli space. We also discuss the utility of the Petersson inner product as a distance measure on the space of 2d CFTs, and apply our procedure to evaluate this distance in various examples.

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arXiv2023

LSZ in Action: Extracting Form Factors from Correlators Nonperturbatively in 2d $ϕ^4$ Theory

In this paper, we compute multiparticle form factors of local operators in 2d $ϕ^4$ theory using a recently proposed method [1] for efficiently implementing the LSZ prescription with Hamiltonian Truncation methods, and we adopt Lightcone Conformal Truncation (LCT) in particular for our calculations. We perform various checks of our results at weak and strong coupling, and elucidate the parametric behavior of truncation errors. This opens up the possibility to compute S-matrix in various strongly coupled models using the LSZ method in LCT.

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arXiv2023

Improving Modular Bootstrap Bounds with Integrality

We implement methods that efficiently impose integrality -- i.e., the condition that the coefficients of characters in the partition function must be integers -- into numerical modular bootstrap. We demonstrate the method with a number of examples where it can be used to strengthen modular bootstrap results. First, we show that, with a mild extra assumption, imposing integrality improves the bound on the maximal allowed gap in dimensions of operators in theories with a $U(1)^c$ symmetry at $c=3$, and reduces it to the value saturated by the $SU(4)_1$ WZW model point of $c=3$ Narain lattices moduli space. Second, we show that our method can be used to eliminate all but a discrete set of points saturating the bound from previous Virasoro modular bootstrap results. Finally, when central charge is close to $1$, we can slightly improve the upper bound on the scaling dimension gap.

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arXiv2023

Large Momentum EFT and Lightcone Quantization

We develop methods for computing the effective action at infinite momentum for $1+1d$ QFTs at finite volume which do not rely on the theory having a Lagrangian description. We do this by taking the infinite momentum limit of equal-time quantization and integrating out all except for the chiral modes of the theory. Our main application of this method is to the Ising Field Theory (IFT), with an energy and magnetic deformation, where we compute the effective lightcone Hamiltonian numerically and check it against results from TCSA. Remarkably, in the low-temperature phase, the Lorentz invariant effective Hamiltonian at infinite momentum takes a very compact form and depends on the volume only through the finite volume vacuum expectation value of $\langle\sigma\rangle$, the spin operator.

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arXiv2023

Thermalization and chaos in a 1+1d QFT

We study aspects of chaos and thermodynamics at strong coupling in a scalar model using LCT numerical methods. We find that our eigenstate spectrum satisfies Wigner-Dyson statistics and that the coefficients describing eigenstates in our basis satisfy Random Matrix Theory (RMT) statistics. At weak coupling, though the bulk of states satisfy RMT statistics, we find several scar states as well. We then use these chaotic states to compute the equation of state of the model, obtaining results consistent with Conformal Field Theory (CFT) expectations at temperatures above the scale of relevant interactions. We also test the Eigenstate Thermalization Hypothesis by computing the expectation value of local operators in eigenstates, and check that their behavior is consistent with thermal CFT values at high temperatures. Finally, we compute the Spectral Form Factor (SFF), which has the expected behavior associated with the equation of state at short times and chaos at long times. We also propose a new technique for extracting the connected part of the SFF without the need of disorder averaging by using different symmetry sectors.

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arXiv2022

Snowmass Theory Frontier: Effective Field Theory

We summarize recent progress in the development, application, and understanding of effective field theories and highlight promising directions for future research. This Report is prepared as the TF02 "Effective Field Theory" topical group summary for the Theory Frontier as part of the Snowmass 2021 process.

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