Lattice Operators Research Articles

Quantum phases and transitions in spin chains with non-invertible symmetries

Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3S3 symmetry and the other with non-invertible \mathsf{Rep}(S_3)𝖱𝖾𝗉(S3) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4JK4⊠JK¯4. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3S3-symmetric and \mathsf{Rep}(S_3)𝖱𝖾𝗉(S3)-symmetric models.

Impact of gauge fixing precision on the continuum limit of nonlocal quark-bilinear lattice operators

We analyze the gauge fixing precision dependence of some nonlocal quark-bilinear lattice operators interesting in computing parton physics for several measurements, using five lattice spacings ranging from 0.032 to 0.121 fm. Our results show that gauge-dependent nonlocal measurements are significantly more sensitive to the precision of gauge fixing than anticipated. The impact of imprecise gauge fixing is significant for fine lattices and long distances. For instance, even with the typically defined precision of Landau gauge fixing of 10−8, the deviation caused by imprecise gauge fixing can reach 12%, when calculating the trace of Wilson lines at 1.2 fm with a lattice spacing of approximately 0.03 fm. Similar behavior has been observed in ξ gauge and Coulomb gauge as well. For both quasi-parton-distribution functions (quasi-PDFs) and quasi-transverse-momentum-dependent PDFs operators renormalized using the regularization independent momentum subtraction (RI/MOM) scheme, convergence for different lattice spacings at long distance is only observed when the precision of Landau gauge fixing is sufficiently high. To describe these findings quantitatively, we propose an empirical formula to estimate the required precision. Published by the American Physical Society 2024

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry g∈O(L) such that gi is fixed point free on L for any 1≤i≤|g|−1. We show that when L(2)=∅ and gi is fixed point free on L for any 1≤i≤|g|−1, VLgˆ has extra automorphisms implies either (1) the order of g is a prime or (2) L is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

Kaon mixing beyond the standard model with physical masses

We present nonperturbative results for beyond the standard model kaon mixing matrix elements in the isospin symmetric limit (mu=md) of QCD, including a complete estimate of all dominant sources of systematic error. Our results are obtained from numerical simulations of lattice QCD with Nf=2+1 flavors of dynamical domain wall fermions. For the first time, these quantities are simulated directly at the physical pion mass mπ∼139 MeV for two different lattice spacings. We include data at three lattice spacings in the range a=0.11–0.07 fm and with pion masses ranging from the physical value up to 450 MeV. Compared to our earlier work, we have added both direct calculations at physical quark masses and a third lattice spacing making the removal of discretization effects significantly more precise and eliminating the need for any significant mass extrapolation beyond the range of simulated data. We renormalize the lattice operators nonperturbatively using RI-SMOM off-shell schemes. These schemes eliminate the need to model and subtract nonperturbative pion poles that arises in the RI-MOM scheme and, since the calculations are performed with domain wall fermions, the unphysical mixing between chirality sectors is suppressed. Our results for the bag parameters in the MS¯ scheme at 3 GeV are BK≡B1=0.5240(17)(54), B2=0.4794(25)(35), B3=0.746(13)(17), B4=0.897(02)(10) and B5=0.6882(78)(94), where the first error is from lattice uncertainties and the second is the uncertainty due to the perturbative matching to MS¯. Published by the American Physical Society 2024

The classification of semi-conformal subalgebras of an even lattice vertex operator algebras with rank two

This work is to understand structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors. In this paper, we shall describe the varieties of semi-conformal vectors of a class of even lattice vertex operator algebras with rank two to characterize these vertex operator algebras. Moreover, we shall give the G-orbit decomposition of the fixed vertex operator subalgebra VL+ of a rank-two lattice vertex operator algebra VL, where G is the automorphism group preserving the conformal vector of VL+. Based on this decomposition, we classify all maximal semi-conformal subalgebras of VL+.

Higher categorical symmetries and gauging in two-dimensional spin systems

We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fusion 2-category with respect to the corresponding module 2-category. We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.

Embedding justification theory in approximation fixpoint theory

Approximation Fixpoint Theory (AFT) and Justification Theory (JT) are two frameworks to unify logical formalisms. AFT studies semantics in terms of fixpoints of lattice operators, and JT in terms of so-called justifications, which are explanations of why certain facts do or do not hold in a model. While the approaches differ, the frameworks were designed with similar goals in mind, namely to study the different semantics that arise in (mainly) non-monotonic logics. The first contribution of our current paper is to provide a formal link between the two frameworks. To be precise, we show that every justification frame induces an approximator and that this mapping from JT to AFT preserves all major semantics. The second contribution exploits this correspondence to extend JT with a novel class of semantics, namely ultimate semantics: we formally show that ultimate semantics can be obtained in JT by a syntactic transformation on the justification frame, essentially performing a sort of resolution on the rules.

Tarsis: An effective automata‐based abstract domain for string analysis

AbstractIn this paper, we introduce Tarsis, a new abstract domain based on the abstract interpretation theory that approximates string values through finite state automata. The main novelty of Tarsis is that it works over an alphabet of strings instead of single characters. On the one hand, such an approach requires a more complex and refined definition of the lattice operators and of the abstract semantics of string operators. On the other hand, it is in position to obtain strictly more precise results than state‐of‐the‐art approaches. We compare Tarsis both with simpler domains and with the standard automata model, targeting case studies containing standard yet challenging string manipulations. The performance gain w.r.t. the standard automata model is also assessed, measuring the speed‐up gained by Tarsis. Experiments confirm that Tarsis can obtain precise results without incurring in excessive computational costs.

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

Localization of polynomial long-range hopping lattice operator with uniform electric fields

Localization of polynomial long-range hopping lattice operator with uniform electric fields

Statistical convergence in vector lattices

The statistical convergence is defined for sequences with the asymptotic density on the natural numbers, in general. In this paper, we introduce the statistical convergence in vector lattices by using the finite additive measures on directed sets. Moreover, we give some relations between the statistical convergence and the lattice properties such as the order convergence and lattice operators.

Detecting Emergent Continuous Symmetries at Quantum Criticality.

New or enlarged symmetries can emerge at the low-energy spectrum of a Hamiltonian that does not possess the symmetries, if the symmetry breaking terms in the Hamiltonian are irrelevant under the renormalization group flow. We propose a tensor network based algorithm to numerically extract lattice operator approximation of the emergent conserved currents from the ground state of any quantum spin chains, without the necessity to have prior knowledge about its low-energy effective field theory. Our results for the spin-1/2 J-Q Heisenberg chain and a one-dimensional version of the deconfined quantum critical points demonstrate the power of our method to obtain the emergent lattice Kac-Moody generators. It can also be viewed as a way to find the local integrals of motion of an integrable model and the local parent Hamiltonian of a critical gapless ground state.

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

In this article, we study automorphisms of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order p. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over Zp or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order p.

Universal Scaling of Klein Bottle Entropy near Conformal Critical Points.

We show that the Klein bottle entropy [H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.261603] for conformal field theories perturbed by a relevant operator is a universal function of the dimensionless coupling constant. The universal scaling of the Klein bottle entropy near criticality provides an efficient approach to extract the scaling dimension of lattice operators via data collapse. As paradigmatic examples, we validate the universal scaling of the Klein bottle entropy for Ising and Z_{3} parafermion conformal field theories with various perturbations using numerical simulation with continuous matrix product operator approach.

Diagonalization in a quantum kicked rotor model with non-analytic potential

In this paper we study the lattice quasi-periodic operators with power-law long-range hopping and meromorphic monotone potentials, and diagonalize the operators via a Nash-Moser iteration scheme. As applications, we obtain uniform power-law localization, uniform dynamical localization and Lipschitz continuity of the integrated density of states (IDS) for such operators. Our main motivation comes from investigating quantum suppression of chaos in a quantum kicked rotor model with non-analytical potential.

Vertex algebraic construction of modules for twisted affine Lie algebras of type [formula omitted

Let g˜ be the affine Lie algebra of type A2l(2). The integrable highest weight g˜-module L(kΛ0) called the standard g˜-module is realized by a tensor product of the twisted module VLT for the lattice vertex operator algebra VL. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in [15].

Three-way preconcept and two forms of approximation operators

Three-way decisions, as a better way than two-way decisions, has played an important role in many fields. As an extension of semiconcept, preconcept constitutes a new approach for data analysis. In contrast to preconcept, formal concept or semiconcept is too conservative about dealing with data. Hence, we want to further apply three-way decisions to preconcept. In this work, we introduce three-way preconcept by an example. This new notion combines preconcept with the assistant of three-way decisions. After that, we attain a generalized double Boolean algebra consisting of three-way preconcept. Furthermore, we give two form operators, approximation operators from lattice and set equivalence relation approximation operators, respectively. Finally, we present a conclusion with some summary and future issues that need to be addressed.

Vertex operator algebras with positive central charges whose dimensions of weight one spaces are 8 and 16

We classify rational simple self-dual vertex operator algebras V of CFT and of finite type with positive central charges, where dV=dim⁡V1 is either 8 or 16, under the condition that the space spanned by characters of V-modules is equal to the solution space of some monic modular linear differential equation of the third order. It is shown that central charges are c=8 for dV=8, and c=4,16 for dim⁡dV=16, respectively. In fact, under the condition that V1 equipped with the 0th product of V are abelian Lie algebras for dV=8,c=8 and dV=16,c=16, the vertex operator algebras V are isomorphic to V2E8 and VΛ16, respectively. For dV=16 and c=4, V is isomorphic to the lattice vertex operator algebra VA2⊕A2.

Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras

For a semisimple Lie algebra s l n \frak {sl}_n , the basic representation L s l n ^ ( 1 , 0 ) L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra s l n ^ \widehat {\frak {sl}_{n}} is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L s l n ^ ( l , 0 ) L_{\widehat {\frak {sl}_{n}}}(l,0) in the l l -fold tensor product L s l n ^ ( 1 , 0 ) ⊗ l L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K ( s l l , n ) K(\frak {sl}_{l},n) , which is the commutant of the Heisenberg vertex operator algebra L h ^ ( n , 0 ) L_{\widehat {\frak {h}}}(n,0) in L s l l ^ ( n , 0 ) L_{\widehat {\frak {sl}_l}}(n,0) . The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L s l n ^ ( l 1 + ⋯ + l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L s l n ^ ( l 1 , 0 ) ⊗ ⋯ ⊗ L s l n ^ ( l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of s l l 1 + ⋯ + l s \frak {sl}_{l_1+\cdots +l_s} corresponding to the composition ( l 1 , ⋯ , l s ) (l_1, \cdots , l_s) in the rational vertex operator algebra L s l ^ l 1 + ⋯ + l s ( n , 0 ) L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0) . This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.

Existence and asymptotic stability for lattice stochastic integrodifferential equations with infinite delays

We prove a new result of resolvent operators generated by the lattice operator, and investigate the existence and asymptotic stability of mild solutions for a class of lattice stochastic integrodifferential equations with infinite delays driven by fractional Brownian motion. With the help of the Banach fixed point theorem and some inequality techniques, the existence of mild solutions are obtained. We give some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment.