Free Boson Research Articles

Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fiber bundles

The discrete phase space and continuous time representation of relativistic quantum mechanics are further investigated here as a continuation of paper I.1 The main mathematical construct used here will be that of an area filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as [Formula: see text], a circle of radius [Formula: see text] where [Formula: see text]. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell [Formula: see text]. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a [Formula: see text]-dimensional phase space hyper-tori [Formula: see text] as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fiber bundles. We also study free scalar Bosons in the background [Formula: see text]-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein–Gordon equation. The second quantized field quanta of this system can cohabit with the tiny Planck oscillators inside the [Formula: see text] phase cells for eternity. Finally, a generalized free second quantized Klein–Gordon equation in a higher [Formula: see text]-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite sqrt{Toverline{T}} deformations

The conformal symmetry algebra in 2D (Diff(S1)⊕Diff(S1)) is shown to be related to its ultra/non-relativistic version (BMS3≈GCA2) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT2, the BMS3 generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, T and overline{T} , closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS3 becomes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a sqrt{Toverline{T}} term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra. The deformation can also be described through the original CFT2 on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to T and overline{T} . BMS3 symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of N free bosons, which coincides with ultra-relativistic limits only for N = 1. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS3 (or flat) versions.

Deformation of the Ising model and its ultraviolet completion

Pure deformations of conformal field theories are generally asymptotically incomplete in the ultra-violet (UV) due to square-root singularities in the ground state energy on a cylinder of circumference R, such that the theory is ill-defined for distances shorter than some critical R *. In this article we show how a theory can be completed if one includes an infinite number of additional irrelevant perturbations. This is fully demonstrated in the case of the Ising model at c IR = 1/2 in the infra-red (IR), where we find two completions with central charges c UV = 3/2 and c UV = 7/10, the latter being the tri-critical Ising model. Both of these UV completions have supersymmetry which is broken in the renormalization group flow to low energies. We also consider multiple deformations of a free massless boson, where we cannot find a UV completion that is consistent with the c-theorem. For negative coupling g, which violates the c-theorem, in both cases we find c UV = −c IR as g → −∞. Finally we also study pure deformations of the off-critical Ising model.

Truly two-dimensional black holes under dimensional transitions of spacetime

A sufficiently massive star in the end of its life will inevitably collapse into a black hole as more deconfined degrees of freedom make the core ever softer. One possible way to avoid the singularity in the end is by dimensional phase transition of spacetime. Indeed, the black hole interior, two-dimensional (2D) in nature, can be described well as a perfect fluid of free massless Majorana fermions and gauge bosons under a 2D supersymmetric mirror model with new understanding of emergent gravity from dimensional evolution of spacetime. In particular, the 2D conformal invariance of the black hole gives rise to desired consistent results for the interior microphysics and structures including its temperature, density and entropy.

Quantum simulation of second-quantized Hamiltonians in compact encoding

We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the $\phi^4$-theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.

On factorizable S-matrices, generalized TTbar, and the Hagedorn transition

We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable S-matrix of an integrable QFT deformed by CDD factors. Such S-matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators. The TBA equations, of course, determine the ground state energy E(R) of the finite-size system, with the spatial coordinate compactified on a circle of circumference R. We limit attention to theories involving just one kind of stable particles, and consider deformations of the trivial (free fermion or boson) S-matrix by CDD factors with two elementary poles and regular high energy asymptotics — the “2CDD model”. We find that for all values of the parameters (positions of the CDD poles) the TBA equations exhibit two real solutions at R greater than a certain parameter-dependent value R*, which we refer to as the primary and secondary branches. The primary branch is identified with the standard iterative solution, while the secondary one is unstable against iterations and needs to be accessed through an alternative numerical method known as pseudo-arc-length continuation. The two branches merge at the “turning point” R* (a square-root branching point). The singularity signals a Hagedorn behavior of the density of high energy states of the deformed theories, a feature incompatible with the Wilsonian notion of a local QFT originating from a UV fixed point, but typical for string theories. This behavior of E(R) is qualitatively the same as the one for standard TTbar deformations of local QFT.

Lorentzian dynamics and factorization beyond rationality

We investigate the emergence of topological defect lines in the conformal Regge limit of two-dimensional conformal field theory. We explain how a local operator can be factorized into a holomorphic and an anti-holomorphic defect operator connected through a topological defect line, and discuss implications on analyticity and Lorentzian dynamics including aspects of chaos. We derive a formula relating the infinite boost limit, which holographically encodes the “opacity” of bulk scattering, to the action of topological defect lines on local operators. Leveraging the unitary bound on the opacity and the positivity of fusion coefficients, we show that the spectral radii of a large class of topological defect lines are given by their loop expectation values. Factorization also gives a formula relating the local and defect operator algebras and fusion categorical data. We then review factorization in rational conformal field theory from a defect perspective, and examine irrational theories. On the orbifold branch of the c = 1 free boson theory, we find a unified description for the topological defect lines through which the twist fields are factorized; at irrational points, the twist fields factorize through “non-compact” topological defect lines which exhibit continuous defect operator spectra. Along the way, we initiate the development of a formalism to characterize non-compact topological defect lines.

Entanglement renormalization for quantum fields with boundaries and defects

The continuous multiscale entanglement renormalization ansatz (cMERA) [J. Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)] gives a variational wave functional for ground states of quantum field-theoretic Hamiltonians. A cMERA is defined as the result of applying to a reference unentangled state a unitary evolution generated by a quasilocal operator, the entangler. This makes the extension of the formalism to the case where boundaries and defects are present nontrivial. Here we show how this generalization works, using the ($1+1$)-dimensional free boson cMERA as a proof-of-principle example, and restricting ourselves to conformal boundaries and defects. In our prescription, the presence of a boundary or defect induces a modification of the entangler localized only to its vicinity, in analogy with the so-called principle of minimal updates for the lattice tensor network MERA.

Narain to Narnia

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern-Simons theory in three dimensions and an ensemble average of Narain's family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi-Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.

Harmonic analysis of 2d CFT partition functions

We apply the theory of harmonic analysis on the fundamental domain of SL(2, ℤ) to partition functions of two-dimensional conformal field theories. We decompose the partition function of c free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space ℍ/SL(2, ℤ), and of target space moduli space O(c, c; ℤ)\\O(c, c; ℝ)/O(c)×O(c). This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS3 gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

T overline{T} -flow effects on torus partition functions

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the T overline{T} deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature.

Entanglement Renormalization of Thermofield Double States.

Entanglement renormalization is a method for "coarse graining" a quantum state in real space, with the multiscale entanglement renormalization ansatz as a notable example. We obtain an entanglement renormalization scheme for finite-temperature (Gibbs) states by applying the multiscale entanglement renormalization ansatz to their canonical purification, the thermofield double state. As an example, we find an analytically exact renormalization circuit for a finite-temperature two-dimensional toric code that maps it to a coarse-grained system with a renormalized higher temperature, thus explicitly demonstrating its lack of topological order. Furthermore, we apply this scheme to one-dimensional free boson models at a finite temperature and find that the thermofield double corresponding to the critical thermal state is described by a Lifshitz theory. We numerically demonstrate the relevance and irrelevance of various perturbations under real space renormalization.

Underlying SUSY in a generalized Jaynes\u2013Cummings model

We present a general qubit-boson interaction Hamiltonian that describes the Jaynes–Cummings model and its extensions as a single Hamiltonian class. Our model includes non-linear processes for both the free qubit and boson field as well as non-linear, multi-boson excitation exchange between them. It shows an underlying algebra with supersymmetric quantum mechanics features allowing an operator based diagonalization that simplifies the calculations of observables. As a practical example, we show the evolution of the population inversion and the boson quadratures for an initial state consisting of the qubit in the ground state interacting with a coherent field for a selection of cases covering the standard Jaynes–Cummings model and some of its extensions including Stark shift, Kerr-like, intensity dependent coupling, multi-boson exchange and algebraic deformations.

CKP hierarchy and free bosons

In this paper, free bosons are used to study some integrable properties of Kadomtsev-Petviashvili (KP) hierarchy of C type (CKP hierarchy), from the aspects of tau functions. First, the modified CKP hierarchy is constructed by using free bosons, and the corresponding Lax structure is given. Then, the constrained CKP hierarchy is found to be related to the modified CKP hierarchy, and the corresponding solutions are derived by using free bosons. Next, by using the relations between the Darboux transformations and the squared eigenfunction symmetries, we express the Darboux transformations of the CKP hierarchy in terms of free bosons, by which one can better understand the essential properties of the CKP Darboux transformations. In particular, the additional symmetries of the CKP hierarchy can be viewed as the infinitesimal generator of the CKP Darboux transformations. Based on these results, we finally obtain the actions of the CKP additional symmetries on the CKP tau functions constructed by free bosons.

Entanglement branes and factorization in conformal field theory

In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the CFT edge modes. Finally, we relate our factorization map to the co-product formula for the CFT symmetry algebra, which we show is equivalent to a Boguliubov transformation in the case of a free boson.

Symmetry decomposition of relative entropies in conformal field theory

We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal U(1) symmetry. We calculate various symmetry resolved Rényi relative entropies between one interval reduced density matrices of CFT primary states using the replica method. By taking the replica limit, the symmetry resolved relative entropy can be obtained. We also take the XX spin chain model as a concrete lattice realization of this CFT to perform numerical computation. The CFT predictions are tested against exact numerical calculations finding perfect agreement.

Non-Wilson-Fisher kinks of $O(N)$ numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs

It is well established that the O(N)O(N) Wilson-Fisher (WF) CFT sits at a kink of the numerical bounds from bootstrapping four point function of O(N)O(N) vector. Moving away from the WF kinks, there indeed exists another family of kinks (dubbed non-WF kinks) on the curve of O(N)O(N) numerical bounds. Different from the O(N)O(N) WF kinks that exist for arbitary NN in 2<d<42<d<4 dimensions, the non-WF kinks exist in arbitrary dimensions but only for a large enough N>N_c(d)N>Nc(d) in a given dimension dd. In this paper we have achieved a thorough understanding for few special cases of these non-WF kinks, which already hints interesting physics. The first case is the O(4)O(4) bootstrap in 2d, where the non-WF kink turns out to be the SU(2)_1SU(2)1 Wess-Zumino-Witten (WZW) model, and all the SU(2)_{k>2}SU(2)k>2 WZW models saturate the numerical bound on the left side of the kink. This is a mirror version of the Z_2Z2 bootstrap, where the 2d Ising CFT sits at a kink while all the other minimal models saturating the bound on the right. We further carry out dimensional continuation of the 2d SU(2)_1SU(2)1 kink towards the 3d SO(5)SO(5) deconfined phase transition. We find the kink disappears at around d=2.7d=2.7 dimensions indicating the SO(5)SO(5) deconfined phase transition is weakly first order. The second interesting observation is, the O(2)O(2) bootstrap bound does not show any kink in 2d (N_c=2Nc=2), but is surprisingly saturated by the 2d free boson CFT (also called Luttinger liquid) all the way on the numerical curve. The last case is the N=\inftyN=∞ limit, where the non-WF kink sits at (\Delta_\phi, \Delta_T)=(d-1, 2d)(Δϕ,ΔT)=(d−1,2d) in dd dimensions. We manage to write down its analytical four point function in arbitrary dimensions, which equals to the subtraction of correlation functions of a free fermion theory and generalized free theory. An important feature of this solution is the existence of a full tower of conserved higher spin current. We speculate that a new family of CFTs will emerge at non-WF kinks for finite NN, in a similar fashion as O(N)O(N) WF CFTs originating from free boson at N=\inftyN=∞.

Fermionic and bosonic quantum field theories from quantum cellular automata in three spatial dimensions

Quantum walks on lattices can give rise to relativistic wave equations in the long-wavelength limit, but going beyond the single-particle case has proven challenging, especially in more than one spatial dimension. We construct quantum cellular automata for distinguishable particles based on two different quantum walks, and show that by restricting to the antisymmetric and symmetric subspaces, respectively, a multiparticle theory for free fermions and bosons in three spatial dimensions can be produced. This construction evades a no-go theorem that prohibits the usual fermionization constructions in more than one spatial dimension. In the long-wavelength limit, these recover Dirac field theory and Maxwell field theory, i.e., free QED.

Non-abelian infrared divergences on the celestial sphere

We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the ‘celestial sphere’ intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theory, we then show that the infrared operator can be computed as a correlator of vertex operators in a conformal field theory of Lie-algebra-valued free bosons on the celestial sphere. We verify by means of the OPE that the theory correctly predicts the all-order structure of collinear limits, and the tree-level factorisation of soft real radiation.