Form Factor Expansion Research Articles

Enlarged Eye-Box Accommodation-Capable Augmented Reality with Hologram Replicas.

Augmented reality (AR) technology has been widely applied across a variety of fields, with head-up displays (HUDs) being one of its prominent uses, offering immersive three-dimensional (3D) experiences and interaction with digital content and the real world. AR-HUDs face challenges such as limited field of view (FOV), small eye-box, bulky form factor, and absence of accommodation cue, often compromising trade-offs between these factors. Recently, optical waveguide based on pupil replication process has attracted increasing attention as an optical element for its compact form factor and exit-pupil expansion. Despite these advantages, current waveguide displays struggle to integrate visual information with real scenes because they do not produce accommodation-capable virtual content. In this paper, we introduce a lensless accommodation-capable holographic system based on a waveguide. Our system aims to expand the eye-box at the optimal viewing distance that provides the maximum FOV. We devised a formalized CGH algorithm based on bold assumption and two constraints and successfully performed numerical observation simulation. In optical experiments, accommodation-capable images with a maximum horizontal FOV of 7.0 degrees were successfully observed within an expanded eye-box of 9.18 mm at an optimal observation distance of 112 mm.

Inelastic Decay from Integrability

A hallmark of integrable systems is the purely elastic scattering of their excitations. Such systems possess an extensive number of locally conserved charges, leading to the conservation of the number of scattered excitations, as well as their set of individual momenta. In this work, we show that inelastic decay can nevertheless be observed in circuit-QED realizations of integrable boundary models. We consider the scattering of microwave photons off impurities in superconducting circuits implementing the boundary sine-Gordon and Kondo models, which are both integrable. We show that not only is inelastic decay possible for the microwave photons, in spite of integrability, and due to a nonlinear relation between them and the elastically scattered excitations, but also that integrability in fact provides powerful analytical tools allowing us to obtain exact expressions for response functions describing the inelastic decay. Using the framework of form factors, we calculate the total inelastic decay rate and elastic phase shift of the microwave photons, extracted from a two-point response function. We then go beyond linear response and obtain the exact energy-resolved inelastic decay spectrum, using a novel method to evaluate form-factor expansions of three-point response functions, which could prove useful in other applications of integrable quantum field theories. Our results could be relevant to several recent photon-splitting experiments and, in particular, to recent experimental works that provide evidence for the elusive Schmid-Bulgadaev dissipative quantum phase transition. Published by the American Physical Society 2024

Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover

We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.

Divergent Nonlinear Response from Quasiparticle Interactions.

We demonstrate that nonlinear response functions in many-body systems carry a sharp signature of interactions between gapped low-energy quasiparticles. Such interactions are challenging to deduce from linear response measurements. The signature takes the form of a divergent-in-time contribution to the response-linear in time in the case when quasiparticles propagate ballistically-that is absent for free bosonic excitations. We give a physically transparent semiclassical picture of this singular behavior. While the semiclassical picture applies to a broad class of systems we benchmark it in two simple models: in the Ising chain using a form-factor expansion, and in a nonintegrable model-the spin-1 Affleck-Kennedy-Lieb-Tasaki chain-using time-dependent density matrix renormalization group simulations. We comment on extensions of these results to finite temperatures.

Accommodation-capable holographic waveguide head-up display with extended field of view

Augmented reality (AR) displays are enhancing user experiences by offering immersive three- dimensional (3D) content, with head-up display (HUD) being a prime application for driving safety. To enable AR-HUDs to provide sufficient information to driver, it is essential to ensure a wide field of view (FOV). Traditional methods like the magnifier principle have limitations stemming from optical aberrations and bulky form factor. Recently, optical waveguide has been used as pupil expander, which has attracted increasing attention as an optical element for compact form factor and exit-pupil expansion. However, waveguide display as exit-pupil expander only offer 2D images with infinite depth, causing mismatch issue between virtual content and real scenes. In this paper, we propose a lensless holographic waveguide display consisting of a laser light source, a spatial light modulator (SLM), a waveguide to generate accommodation-capable images, effectively extending FOV. The key distinction of this research lies in the optical design of the waveguide, which defines multiple shifted copies of the modulated wavefield as replicated virtual SLMs. A formalized algorithm is devised based on a bold assumption that numerous virtual SLMs replicated by a waveguide are considered as a single ultra-high resolution SLM. The effectiveness of the algorithm has been demonstrated through extensive validation through numerical observation simulations and optical experiments. Notably, optical experiments convincingly demonstrate the system’s ability to produce accommodation-capable true 3D content, with a four-fold extension of FOV compared to a single SLM at an observation distance of 150 mm. Additional optical experiments highlight the successful integration of AR technology, an important component for automotive HUD.

Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains

Evaluating a lattice path integral in terms of spectral data and matrix elements pertaining to a suitably defined quantum transfer matrix, we derive form-factor series expansions for the dynamical two-point functions of arbitrary local operators in fundamental Yang–Baxter integrable lattice models at finite temperature. The summands in the series are parameterised by solutions of the Bethe Ansatz equations associated with the eigenvalue problem of the quantum transfer matrix. We elaborate on the example of the XXZ chain for which the solutions of the Bethe Ansatz equations are sufficiently well understood in certain limiting cases. We work out in detail the case of the spin-zero operators in the antiferromagnetic massive regime at zero temperature. In this case the thermal form-factor series turn into series of multiple integrals with fully explicit integrands. These integrands factorize into an operator-dependent part, determined by the so-called Fermionic basis, and a part which we call the universal weight as it is the same for all spin-zero operators. The universal weight can be inferred from our previous work. The operator-dependent part is rather simple for the most interesting short-range operators. It is determined by two functions ρ and ω for which we obtain explicit expressions in the considered case. As an application we rederive the known explicit form-factor series for the two-point function of the magnetization operator and obtain analogous expressions for the magnetic current and the energy operators.

The power of Lorentzian wormholes

As shown by Louko and Sorkin in 1995, topology change in Lorentzian signature involves spacetimes with singular points, which they called crotches. We modify their construction to obtain Lorentzian semiclassical wormholes in asymptotically AdS. These solutions are obtained by inserting crotches on known saddles, like the double-cone or multiple copies of the Lorentzian black hole. The crotches implement swap-identifications, and are classically located near an extremal surface. The resulting Lorentzian wormholes have an instanton action equal to their area, which is responsible for topological suppression in any number of dimensions.We conjecture that including such Lorentzian wormhole spacetimes is equivalent to path integrating over all mostly Euclidean smooth spacetimes. We present evidence for this by reproducing semiclassical features of the genus expansion of the spectral form factor, and of a late-time two point function, by summing over the moduli space of Lorentzian wormholes. As a final piece of evidence, we discuss the Lorentzian version of West-Coast replica wormholes.

Multi-point Correlation Functions in the Boundary XXZ Chain at Finite Temperature

We consider multi-point correlation functions in the open XXZ chain with longitudinal boundary fields and in a uniform external magnetic field. We show that, at finite temperature, these correlation functions can be written in the quantum transfer matrix framework as sums over thermal form factors. More precisely, and quite remarkably, each term of the sum is given by a simple product of usual matrix elements of the quantum transfer matrix multiplied by a unique factor containing the whole information about the boundary fields. As an example, we provide a detailed expression for the longitudinal spin one-point functions at distance m from the boundary. This work thus solves the long-standing problem of setting up form factor expansions in integrable models subject to open boundary conditions.

On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions

This paper develops a technique allowing one to prove the convergence of a class of series of multiple integrals which corresponds to the form factor expansion of space-like separated two-point functions in the 1+1 dimensional massive integrable Sinh-Gordon quantum field theory.

Branch Point Twist Field Form Factors in the sine-Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement

In this paper we continue the program initiated in Part I, that is the study of entanglement measures in the sine-Gordon model. In both parts, we have focussed on one specific technique, that is the well-known connection between branch point twist field correlators and measures of entanglement in 1+1D integrable quantum field theory. Our papers apply this technique for the first time to a non-diagonal theory with an involved particle spectrum, the sine-Gordon model. In this Part II we focus on a different entanglement measure, the symmetry resolved entanglement, and develop its associated twist field description, exploiting the underlying U(1) symmetry of the theory. In this context, conventional branch point twist fields are no longer the fields required, but instead we must work with one of their composite generalisations, which can be understood as the field resulting from the fusion of a standard branch point twist field and the sine-Gordon exponential field associated with U(1) symmetry. The resulting composite twist field has correlators which as usual admit a form factor expansion. In this paper we write the associated form factor equations and solve them for various examples in the breather sector by using the method of angular quantisation. We show that, in the attractive regime, this is the sector which provides the leading contribution to the symmetry resolved entropies, both R\'enyi and von Neumann. We compute the latter in the limit of a large region size and show that they satisfy the property of equipartition, that is the leading contribution to the symmetry resolved entanglement is independent of the symmetry sector.

Out-of-equilibrium dynamics of the XY spin chain from form factor expansion

We consider the XY spin chain with arbitrary time-dependent magnetic field and anisotropy. We argue that a certain subclass of Gaussian states, called Coherent Ensemble (CE) following [1], provides a natural and unified framework for out-of-equilibrium physics in this model. We show that all correlation functions in the CE can be computed using form factor expansion and expressed in terms of Fredholm determinants. In particular, we present exact out-of-equilibrium expressions in the thermodynamic limit for the previously unknown order parameter 1-point function, dynamical 2-point function and equal-time 3-point function.

Correlation functions and transport coefficients in generalised hydrodynamics

We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

Form factors and generalized hydrodynamics for integrable systems

Our review covers microscopic foundations of generalized hydrodynamics (GHD). As one generic approach we develop form factor expansions, for ground states and generalized Gibbs ensembles (GGE). In the latter case the so obtained results are compared with predictions from GHD. One cornerstone of GHD are the GGE averaged microscopic currents, which can also be obtained through employing form factors. Discussed is a second, completely orthogonal approach based on the availability of a self-conserved current.

On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

This work extracts, by means of an exact analysis, the singular behavior of the dynamical response functions—the Fourier transforms of dynamical two-point functions—in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes that describe the local behavior of the response function near a threshold. The singular behavior is derived starting from first principles considerations: the method of analysis does not rely, at any stage, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behavior and of the associated edge exponents that arise in the vicinity of the dispersion relation of one massive excitation (hole, particle, or bound state). In addition, the present analysis shows that due to the lack of strict convexity of the particle dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These originate from multi-particle/hole/bound state excitations maximizing the energy at fixed momentum.

Exact Real-Time Longitudinal Correlation Functions of the Massive XXZ Chain.

We apply a recently developed thermal form factor expansion method to evaluate the real-time longitudinal spin-spin correlation functions of the spin-1/2 XXZ chain in the antiferromagnetically ordered regime at zero temperature. An analytical result incorporating all types of excitations in the model is obtained, without any approximations. This allows for the accurate calculation of the real-time correlations in this strongly interacting quantum system for arbitrary distances and times.

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

We introduce a framework for calculating dynamical correlations in the Lieb-Liniger model in arbitrary energy eigenstates and for all space and time, that combines a Lehmann representation with a 1/c1/c expansion. The n^\mathrm{th}nth term of the expansion is of order 1/c^n1/cn and takes into account all \lfloor \tfrac{n}{2}\rfloor+1⌊n2⌋+1 particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a "bare" 1/c1/c expansion it is uniform in space and time. The framework is based on a method for taking the thermodynamic limit of sums of form factors that exhibit non integrable singularities. We expect our framework to be applicable to any local operator. We determine the first three terms of this expansion and obtain an explicit expression for the density-density dynamical correlations and the dynamical structure factor at order 1/c^21/c2. We apply these to finite-temperature equilibrium states and non-equilibrium steady states after quantum quenches. We recover predictions of (nonlinear) Luttinger liquid theory and generalized hydrodynamics in the appropriate limits, and are able to compute sub-leading corrections to these.

Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions

We consider the problems of calculating the dynamical order parameter two-point function at finite temperatures and the one-point function after a quantum quench in the transverse field Ising chain. Both of these can be expressed in terms of form factor sums in the basis of physical excitations of the model. We develop a general framework for carrying out these sums based on a decomposition of form factors into partial fractions, which leads to a factorization of the multiple sums and permits them to be evaluated asymptotically. This naturally leads to systematic low density expansions. At late times these expansions can be summed to all orders by means of a determinant representation. Our method has a natural generalization to semi-local operators in interacting integrable models.

Low-temperature asymptotic of the transverse dynamical structure factor for a magnetically polarized XX chain

The Dyson equation for the real two-time commutator retarded one-magnon Green function of the ferromagnetically polarized XX chain is suggested following the Plakida–Tserkovnikov algorithm. Starting from this result a low-temperature integral representation for the corresponding magnon self-energy is obtained by the truncated form factor expansion, however, without any resummations. Within the suggested approach, the low-temperature asymptotics of the transverse dynamical structure factor may be readily studied. Some obtained line shapes are presented.

Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence

Generalized Hydrodynamics is a recent theory that describes large scale transport properties of one dimensional integrable models. It is built on the (typically infinitely many) local conservation laws present in these systems, and leads to a generalized Euler type hydrodynamic equation. Despite the successes of the theory, one of its cornerstones, namely a conjectured expression for the currents of the conserved charges in local equilibrium has not yet been proven for interacting lattice models. Here we fill this gap, and compute an exact result for the mean values of current operators in Bethe Ansatz solvable systems, valid in arbitrary finite volume. Our exact formula has a simple semi-classical interpretation: the currents can be computed by summing over the charge eigenvalues carried by the individual bare particles, multiplied with an effective velocity describing their propagation in the presence of the other particles. Remarkably, the semi-classical formula remains exact in the interacting quantum theory, for any finite number of particles and also in the thermodynamic limit. Our proof is built on a form factor expansion and it is applicable to a large class of quantum integrable models.

Correlated charge form factors and densities of the sd-shell nuclei

The expression of the two body term in the fac­ tor cluster expansion of the charge form factor of 40Ca is derived.lt contains the harmonic oscillator (HO) parameter bx and the parameter λ which originates from the Jastrow correlation function. This expression together with the corresponding one of 16O nucleus helps to find a mass de­pendence of λ and an approximate and fairly simple expres­ sion of the two body term of open shell nuclei in the region 16<A<40 which contains one free parameter, the HO parameter bi. The fitting to the corresponding experimen­tal charge form factor is quite improved in comparison to the HO one without correlations.