Euclidean Lattice Research Articles

Non-chiral vertex operator algebra associated to Lorentzian lattices and Narain CFTs

Frenkel, Lepowsky, and Meurman constructed a vertex operator algebra (VOA) associated to any even, integral, Euclidean lattice. In the language of physics, these are examples of chiral conformal field theories (CFT). In this paper, we define non-chiral vertex operator algebra and some associated notions. We then give a construction of a non-chiral VOA associated to an even, integral, Lorentzian lattice and construct their irreducible modules. We obtain the moduli space of such modular invariant non-chiral CFTs based on even, self-dual Lorentzian lattices of signature (m,n)(m,n) assuming the validity of a technical result about automorphisms of the lattice. We finally show that Narain conformal field theories in physics are examples of non-chiral VOA. Our formalism helps us to identify the chiral algebra of Narain CFTs in terms of a particular sublattice and give us the decomposition of its partition function into sum of characters.

Operator product expansion for radial lattice quantization of 3D ϕ4 theory

At its critical point, the three-dimensional lattice Ising model is described by a conformal field theory (CFT), the 3D Ising CFT. Instead of carrying out simulations on Euclidean lattices, we use the quantum finite elements method to implement radially quantized critical ϕ4 theory on simplicial lattices approaching R×S2. Computing the four-point function of identical scalars, we demonstrate the power of radial quantization by the accurate determination of the scaling dimensions Δε and ΔT as well as ratios of the operator product expansion coefficients fσσε and fσσT of the first spin-0 and spin-2 primary operators ε and T of the 3D Ising CFT. Published by the American Physical Society 2024

An overview of lattice results for parton distribution functions

Following a ground-breaking proposal by Ji [Phys. Rev. Lett. 110, 262002 (2013)], numerical simulations of Quantum Chromo Dynamics (QCD) on a Euclidean lattice have provided new, valuable information on the structure of hadrons. In this talk, we briefly review the lattice approach to the reconstruction of parton densities, highlighting the connection between lattice observables and parton densities, with a focus on theoretical issues. Since parton distributions are extracted from lattice data by solving an inverse problem, we discuss some of the difficulties that affect these determinations and how they can be formulated in a Bayesian framework.

Deciding whether a lattice has an orthonormal basis is in co-NP

AbstractWe show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix $$G \in \mathbb {Q}^{n \times n}$$ G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix $$B \in \mathbb {Q}^{4n \times n}$$ B ∈ Q 4 n × n such that $$B^\intercal B = G$$ B ⊺ B = G .

Teaching to extract spectral densities from lattice correlators to a broad audience of learning-machines

We present a new supervised deep-learning approach to the problem of the extraction of smeared spectral densities from Euclidean lattice correlators. A distinctive feature of our method is a model-independent training strategy that we implement by parametrizing the training sets over a functional space spanned by Chebyshev polynomials. The other distinctive feature is a reliable estimate of the systematic uncertainties that we achieve by introducing several ensembles of machines, the broad audience of the title. By training an ensemble of machines with the same number of neurons over training sets of fixed dimensions and complexity, we manage to provide a reliable estimate of the systematic errors by studying numerically the asymptotic limits of infinitely large networks and training sets. The method has been validated on a very large set of random mock data and also in the case of lattice QCD data. We extracted the strange-strange connected contribution to the smeared R-ratio from a lattice QCD correlator produced by the ETM Collaboration and compared the results of the new method with the ones previously obtained with the HLT method by finding a remarkably good agreement between the two totally unrelated approaches.

Euclidean lattices: theory and applications

In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the intersection of theory and applications, providing the context for the articles showcased in this special issue.

(Invited) Controlling the Layer Stacking and Twisting of 2D Quantum Materials

Rationally and precisely controlling the structures of two-dimensional (2D) layered materials allows us to tune the electronic structures and quantum states of matter, discover new physical properties, thus enable new applications. For example, tuning the twist angles between 2D materials results in the formation of moiré patterns and the manipulation of electronic states, which have led to novel quantum phenomena, including unconventional superconductivity, moiré excitons, and tunable Mott insulators. We have shown how the phase, layer stacking and thus physical properties of 2D MX2 materials can be influenced by screw dislocations. Moreover, we achieved systematic interlayer twisting of MX2 spiral layers grown via by screw dislocations due to mismatched geometry between Euclidean crystal lattices and non-Euclidean (curved) surfaces (Science 2020, 370, 442). We experimentally demonstrated the growth of twisted spirals of WS2 and WSe2 draped over nanoparticles near their centers and microscopic structural analysis confirmed the moiré superlattices between the atomic layers. Tuning the stacking and twisting of 2D materials opens up new dimensions of 2D quantum materials (Acc. Mater. Res. 2022, 3, 369) for the study of moiré excitons, nonlinear optical properties, chiral optoelectronics, valleytronics and twistronics.

Optimal Hardy weights on the Euclidean lattice

We investigate the large-distance asymptotics of optimal Hardy weights on Z d \mathbb Z^d , d ≥ 3 d\geq 3 , via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar ( d − 2 ) 2 4 | x | − 2 \frac {(d-2)^2}{4}|x|^{-2} as | x | → ∞ |x|\to \infty . We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on Z d \mathbb Z^d : (1) averages over large sectors have inverse-square scaling, (2) for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3) for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply | x | − 4 |x|^{-4} -scaling for Rellich weights on Z d \mathbb Z^d . Analogous results are also new in the continuum setting. The proofs leverage Green’s function estimates rooted in homogenization theory.

Pion observables calculated in Minkowski and Euclidean spaces with Ansätze for quark propagators

We study two quark--propagator meromorphic Ans\"atze that admit clear connection between calculations in Euclidean space and Minkowski spacetime. The connection is established through a modified Wick rotation in momentum space, where the integration contour along the imaginary axis is adequately deformed. The Ans\"atze were previously proposed in the literature and fitted to Euclidean lattice QCD data. The generalized impulse approximation is used to calculate the pion transition form factor and electromagnetic form factor, correcting an earlier result. The pion decay constant and distribution amplitude are also calculated. The latter is used to deduce the asymptotic behavior of the form factors. Such an asymptotic behavior is compared with those obtained directly from the generalized impulse approximation and the causes of differences are pointed out.

Vacuum decay and Euclidean lattice Monte Carlo

The decay rate of a metastable vacuum is usually calculated using a semiclassical approximation to the Euclidean path integral. The extension to a complete Euclidean lattice Monte Carlo computation, however, is hampered by analytic continuations that are ill-suited to numerical treatment, and the nonequilibrium nature of a metastable state. In this paper we develop a new methodology to compute vacuum decay rates from Monte Carlo simulations of Euclidean lattice theories. To test the new method, we consider simple quantum mechanical systems systems with metastable vacua. This work can be extended to Euclidean field theories, which we discuss in the Conclusions.

Fermionic CFTs from classical codes over finite fields

We construct a class of chiral fermionic CFTs from classical codes over finite fields whose order is a prime number. We exploit the relationship between classical codes and Euclidean lattices to provide the Neveu–Schwarz sector of fermionic CFTs. On the other hand, we construct the Ramond sector using the shadow theory of classical codes and Euclidean lattices. We give various examples of chiral fermionic CFTs through our construction. We also explore supersymmetric CFTs in terms of classical codes by requiring the resulting fermionic CFTs to satisfy some necessary conditions for supersymmetry.

Large-time correlation functions in bosonic lattice field theories

Large-time correlation functions have a pivotal role in extracting particle masses from Euclidean lattice field theory calculations, however little is known about the statistical properties of these quantities. In this work, the asymptotic form of the distributions of the correlation functions at vanishing momentum is determined for bosonic interacting lattice field theories with a unique gapped vacuum. It is demonstrated that the deviations from the asymptotic form at large Euclidean times can be utilized to determine the spectrum of the theory.

Gapped lineon and fracton models on graphs

We introduce a ${\mathbb{Z}}_{N}$ stabilizer code that can be defined on any spatial lattice of the form $\mathrm{\ensuremath{\Gamma}}\ifmmode\times\else\texttimes\fi{}{C}_{{L}_{z}}$, where $\mathrm{\ensuremath{\Gamma}}$ is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic ${\mathbb{Z}}_{N}$ Laplacian model. It is gapped, robust (i.e., stable under small deformations), and has lineons. Its ground-state degeneracy (GSD) is expressed in terms of a ``mod $N$-reduction'' of the Jacobian group of the graph $\mathrm{\ensuremath{\Gamma}}$. In the special case when space is an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{z}$ cubic lattice, the logarithm of the GSD depends on $L$ in an erratic way and grows no faster than $O(L)$. We also discuss another gapped model, the ${\mathbb{Z}}_{N}$ Laplacian model, which can be defined on any graph. It has fractons and a similarly strange GSD.

Variational quantum solutions to the Shortest Vector Problem

A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp. estimates) for the number of qubits required per dimension for any lattices (resp. random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately 103 noisy qubits such instances can be tackled.

Extension procedures for lattice Lipschitz operators on Euclidean spaces

We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.

Guest Editorial: Guest Editorial on Cryptanalysis of (NIST PQC) post‐quantum proposals

Public-key cryptography is now ubiquitous in our lives, protecting everyday interactions ranging from mobile and wireless communications to banking transactions, over-the-air software updates, electronic voting or digital car keys. However, almost the entirety of currently deployed public-key cryptography implementations rely on the hardness of integer factorisation or computing discrete logarithms, and both problems are known to be easy to solve using large-scale quantum computers. Whether such large-scale computers are years away, decades away or more remains to be seen, but since they would break the security of so many critical systems essentially overnight, preparing the transition from classical cryptography to so-called post-quantum primitives, which are believed to be secure even against quantum computers, is seen as necessary and urgent—particularly as updating existing systems is likely to be a lengthy process, and some systems rely on the security of cryptographic keys over long periods of time. In view of these challenges, starting in 2016, US standards institute NIST has been running a standardisation effort in order to come up with post-quantum encryption and signature schemes ready for deployment. The first four selected primitives have been announced in July 2022, with other schemes undergoing further analysis. Concurrently, other countries, such as the Republic of Korea, have launched similar standardisation processes. One of the difficulties involved in running such standardisation processes and selecting primitives is security estimation: in order to set parameters for candidate schemes and to make apple-to-apple comparisons between them, consistent security levels need to be defined, and evidence that the schemes achieve those levels needs to be provided. This is typically done by estimating the cost of the best attacks (both classical and quantum) against the proposed constructions. Cryptanalysis is thus an essential tool for design and standardisation. Not only does it allow to provide and progressively refine security estimates, it also sometimes eliminates entire schemes, plain and simple, by uncovering serious security flaws. Both applications of cryptanalysis have been extensively represented in the NIST standardisation process in particular. ‘Quantum Algorithms for Attacking Hardness Assumptions in Classical and Post-Quantum Cryptography’, by J.-F. Biasse, X. Bonnetain, E. Martin, E. Kirshanova, A. Schrottenloher and F. Song, on quantum cryptanalysis; ‘Recent Progress in the Security Evaluation of Multivariate Public-Key Cryptography’, by Y. Ikematsu, S. Nakamura and T. Takagi, on multivariate cryptography; ‘Lattice-Based Cryptosystems in Standardization Processes: A Survey’, by A. Wang, D. Xiao and Y. Yu, on lattice-based cryptography; and ‘Torsion point attacks on `SIDH-like' cryptosystems’, by P. Kutas and C. Petit, on isogeny-based cryptography. Since some post-quantum proposals rely on relatively new assumptions that have only received much scrutiny as part of the standardisation processes, their cryptanalysis is a fast-changing landscape. Very recent, dramatic developments include W. Beullens' cryptanalysis of the Rainbow multivariate signature, and W. Castryk and T. Decru's cryptanalysis of the SIKE isogeny-based KEM (along with several follow-ups). Both Rainbow and SIKE were serious contenders for eventual standardisation. Those surprising results, which appeared too late to be captured in this special issue, are thus further testimonies to the utmost importance of cryptanalytic work as part of standardisation efforts. We therefore hope that the contributions included in this special issue will be of great value to the community, insofar as they record, and present in an accessible way, important results towards the goal of obtaining secure and dependable standards for post-quantum cryptography. Data sharing not applicable to this article as no data sets were generated or analysed during the current study. Ayoub Otmani obtained a Ph.D. degree in Mathematics and its Applications from the University of Limoges (France) in 2002. He is currently Professor in Computer Science at University of Rouen Normandie. His research lies in coding theory and cryptography. Christophe Petit obtained a PhD in Cryptography at Université catholique de Louvain. He is now Associate Professor at the University of Birmingham and Université libre de Bruxelles. His research focuses on cryptanalysis and mathematical aspects of cryptography, particularly on isogeny-based cryptography. An alumni of ENS (Paris, France), Mehdi Tibouchi obtained his Ph.D. in computer science from Univ. Paris VII and Univ. Luxembourg in 2011. He is now distinguished researcher at NTT Corporation (Tokyo, Japan) and guest associate professor at Kyoto University (Kyoto, Japan). His research interests cover various mathematical aspects of public-key cryptography and cryptanalysis, particularly related to elliptic curves and Euclidean lattices.

The sphaleron rate from 4D Euclidean lattices

We develop a new method to determine thermal activation rates, such as for bubble nucleation, topology change, etc., using 4-dimensional Euclidean methods. This allows nonperturbative study on the lattice. We then investigate the strong sphaleron rate in pure-glue QCD at temperatures between 1.3 Tc and 1000 Tc, making contact with previous results but extending them down close to the critical temperature. The extension to full QCD will be straightforward. Limitations of the proposal (the inability to compute a certain dynamical prefactor, puzzling large-volume behavior, and the inability to treat temperatures T < 1.3 Tc) are also discussed.

Probing non-perturbative supersymmetry breaking through lattice path integrals

We investigate non-perturbative supersymmetry breaking in various models of quantum mechanics, including an interesting class of $PT$-invariant models, using lattice path integrals. These theories are discretized on a temporal Euclidean lattice with anti-periodic boundary conditions. Hybrid Monte Carlo algorithm is used to update the field configurations to their equilibrium values. We used the Ward identities and expectation values of superpotentials as tools for probing supersymmetry breaking.

Persistence length of semi-flexible polymer chains on Euclidean lattices

We have studied the persistence length behavior of semi-flexible linear polymers, represented by self-avoiding random walks (SAWs) on the square (d=2) and simple cubic (d=3) lattice. By employing the PERM Monte Carlo algorithm we have generated SAWs, changing the chain stiffness (characterized by the parameter s, related to each bend of a SAW). We have examined two quantities that measure persistence length of N-step SAWs: (1) ℓN, which is defined as an average length of straight parts of polymer chains, and (2) λN, defined as the mean projection of the end-to-end distance vector on the SAW’s first step. For each particular s, we have found that ℓN is a linear function of 1/N in both dimensions, d=2 and d=3, while λN is a linear function of 1/NΦeff, where Φeff≈0.3 in d=2 and Φeff≈0.82 in d=3. Sequences of ℓN and λN have been extrapolated in the range of very long chains, and we have established that, for each examined s, they converge to s dependent constants ℓp=ℓN→∞ and λp=λN→∞. We analyze dependence of ℓp and λp on the polymer stiffness s, and make a comparison between our findings and former results for studied quantities.

Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices.

Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potential for a new class of topological metamaterials. One important feature of hyperbolic lattices is that their translation group is nonabelian, permitting high-dimensional irreducible representations (irreps), in contrast to abelian translation groups in Euclidean lattices. Here we introduce a general framework to construct wave eigenstates of high-dimensional irreps of infinite hyperbolic lattices, thereby generalizing Bloch's theorem, and discuss its implications on unusual mode counting and degeneracy, as well as bulk-edge correspondence in hyperbolic lattices. We apply this method to a mechanical hyperbolic lattice, and characterize its band structure and zero modes of high-dimensional irreps.