Asymptotic Bethe Ansatz Research Articles

Quasi-one-dimensional dipolar quantum gases

In this letter we consider dipolar quantum gases in a quasi-one-dimensional tube with dipole moment perpendicular to the tube direction. We deduce the effective one-dimensional interaction potential and show that this potential is not purely repulsive, but rather has an attractive part due to high-order scattering processes through transverse excited states. The attractive part can induce bound state and cause scattering resonances. This represents the dipole induced resonance in low-dimension. We work out an unconventional behavior of low-energy phase shift for this effective potential and show how it evolves across a resonance. Based on the phase shift, the interaction energy of spinless bosons is obtained using asymptotic Bethe ansatz. Despite of long-range nature of dipolar interaction, we find that a behavior similar as short-range Lieb-Linger gas emerges at the resonance regime.

On the scattering over the GKP vacuum

By converting the asymptotic Bethe Ansatz (ABA) of N=4 SYM into non-linear integral equations, we find 2D scattering amplitudes of excitations on top of the GKP vacuum. We prove that this is a suitable and powerful set-up for the understanding and computation of the whole S-matrix. We show that all the amplitudes depend on the fundamental scalar–scalar one.

Quantum corrections to spinning superstrings in AdS3 × S3 × M 4: determining the dressing phase

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS_3 x S^3 x T^4 supported by RR flux. We find that the phase should be different from the BES phase appearing in the AdS_5 x S^5 case. We use the simplest example of a rigid circular string with two equal spins in S^3 and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS_3 x S^3 x S^3 x S^1 theory and find the dependence of the 1-loop correction to the effective string tension function h(\lambda) (expected to enter the magnon dispersion relation) on the parameters alpha related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS_3 x S^3 x T^4 case.

Many-body properties of quasi–one-dimensional boson gas across a narrow CIR

We study strong interaction effects in a one-dimensional (1D) boson gas across a narrow confinement-induced resonance (CIR). In contrast to the zero-range potential, the 1D two-body interaction in the narrow CIR can be written as a polynomial of derivative δ-function interaction on many-body level. Using the asymptotic Bethe ansatz, we find that the low-energy physics of this many-body problem is described by the Tomonaga-Luttinger liquid where the Luttinger parameters are essentially modified by an effective finite-range parameter v. This parameter drastically alters quantum criticality and universal thermodynamics of the gas. In particular, it drives the Tonks-Girardeau (TG) gas from non-mutual Fermi statistics to mutual statistics or to a more exclusive super-TG gas. This novel feature is further discussed in terms of the breathing mode which is experimentally measurable.

TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT

We consider high spin, s, long twist, L, operators in the planar $ \mathcal{N}=4 $ SYM at strong coupling. Precisely, we compute the minimal anomalous dimensions (from asymptotic Bethe Ansatz) for large ’t Hooft coupling λ to the lowest order of the (string) scaling variable $ \ell \sim L/\left( {\ln\,S\sqrt{\lambda }} \right) $ with GKP string size ~ $ \ln\,S\equiv 2\ln \left( {s/\sqrt{\lambda }} \right) $ . At the leading order (ln S) · ℓ2, we can confirm the O(6) non-linear sigma model description for this bulk term, without boundary term (ln S)0. Going further, we derive, extending the O(6) regime, the exact effect of the size finiteness. In particular, we compute, at all loops, the first Casimir correction ℓ0 / ln S (in terms of the infinite size O(6) NLSM), which reveals only one massless mode (out of five), as predictable once the O(6) description has been extended. Consequently, upon comparing with string theory expansion, at one loop our findings agree for large twist, while reveal for negligible twist, already at this order, the appearance of wrapping. At two loops, as well as for next loops and orders, we can produce predictions, which may guide future string computations.

On the integrability of two-dimensional models with [formula omitted] symmetry

In this paper we study the integrability of a family of models with U(1)×SU(N) symmetry. They admit fermionic and bosonic formulations related through bosonization and subsequent T-duality. The fermionic theory is just the CPN−1 sigma model coupled to a self-interacting massless fermion, while the bosonic one defines a one-parameter deformation of the O(2N) sigma model. For N=2 the latter model is equivalent to the integrable deformation of the O(4) sigma model discovered by Wiegmann. At higher values of N we find that integrability is more sporadic and requires a fine-tuning of the parameters of the theory. A special case of our study is the N=4 model, which was found to describe the AdS4×CP3 string theory in the Alday–Maldacena decoupling limit. In this case we propose a set of asymptotic Bethe ansatz equations for the energy spectrum.

Y-system for Y = 0 brane in planar AdS/CFT

The spectrum of open strings with integrable Y=0 brane boundary conditions is analyzed in planar AdS/CFT. We give evidence that it can be described by the same Y-system that governs the spectrum of closed strings in AdS_5xS^5, except with different asymptotic and analytical properties. We determine the asymptotic solution of the Y-system that is consistent both with boundary asymptotic Bethe ansatz and boundary Luscher corrections.

Wrapping corrections beyond the 𝔰𝔩(2) sector in 𝒩 = 4 SYM

AbstractThe 𝔰𝔩(2) sector of 𝒩 = 4 SYM theory has been much studied and the anomalous dimensions of those operators are well known. Nevertheless, many interesting operators are not included in this sector. We consider a class of twist operators beyond the 𝔰𝔩(2) subsector introduced by Freyhult, Rej and Zieme. They are spin n, length‐3 operators. At one‐loop they can be identified with three gluon operators. At strong coupling, they are associated with spinning strings with two spins in AdS space and charge in S5. We exploit the Y‐system to compute the leading weak‐coupling four loop wrapping correction to their anomalous dimension. The result is written in closed form as a function of the spin n. We combine the wrapping correction with the known four loop asymptotic Bethe Ansatz contribution and analyze special limits in the spin n. In particular, at large n, we prove that a generalized Gribov‐Lipatov reciprocity holds. At negative unphysical spin, we present a simple BFKL‐like equation predicting the rightmost leading poles.

Lüscher formula for GKP string

We investigate finite-size corrections to anomalous dimensions of large-spin twist-two operators in the planar maximally supersymmetric Yang–Mills theory. We develop a framework for analysis of these corrections, that is complementary to the conventional spin-chain approach, by making use of the hole rather than the magnon picture. From the dual string theory perspective where the large-spin operator is identified with the Gubser–Klebanov–Polyakov (GKP) string, our approach is equivalent to constructing the first Lüscher correction to the energy of the GKP string by incorporating the contribution of virtual excitations propagating on it. It allows us to propose a formula that controls a particular class of large-spin corrections to the twist-two anomalous dimension and holds at any value of the coupling constant. Compared to wrapping corrections computed with magnons propagating on the spin chain, the finite-size corrections that are encoded in our formalism start at a lower-loop level. Our formalism thus calls for modification of the asymptotic contributions which are conventionally incorporated within the Asymptotic Bethe Ansatz. An educated guess allows us to remedy this pitfall and successfully confront our predictions with known results up to five-loop accuracy at weak coupling. Finally, our formula sheds light on the weak-to-strong coupling transition for the subleading large-spin corrections under study and confirms stringy expectations at strong coupling where they are found to be identical to the first Lüscher correction to the vacuum energy of the O(6) sigma model.

Asymptotic Bethe-ansatz solution for one-dimensional SU(2) spinor bosons with finite-range Gaussian interactions

We propose a one-dimensional model of spinor bosons with SU(2) symmetry and a two-body finite-range Gaussian interaction potential. We show that the model is exactly solvable when the width of the interaction potential is much smaller than the interparticle separation. This model is then solved via the asymptotic Bethe-ansatz technique. The ferromagnetic ground-state energy and chemical potential are derived analytically. We also investigate the effects of a finite-range potential on the density profiles through the local-density approximation. Finite-range potentials are more likely to lead to quasi-Bose-Einstein condensation than zero-range potentials.

Exciting the GKP string at any coupling

We analyze the spectrum of excitations around the Gubser–Klebanov–Polyakov (GKP) rotating string in the long string limit and construct a parametric representation for their dispersion relations at any value of the string tension. On the gauge theory side of the AdS/CFT correspondence, i.e., in the planar N = 4 super Yang–Mills theory, the problem is equivalent to finding the spectrum of scaling dimensions of large spin, single-trace operators. Their scaling dimensions are obtained from the analysis of the Beisert–Staudacher asymptotic Bethe ansatz equations, which are believed to solve the spectral problem of the planar gauge theory. We examine the resulting dispersion relations in various kinematical regimes, both at weak and strong coupling, and detail the matching with the Frolov–Tseytlin spectrum of transverse fluctuations of the long GKP string. At a more dynamical level, we identify the mechanism for the restoration of the SO ( 6 ) symmetry, initially broken by the choice of the Berenstein–Maldacena–Nastase vacuum in the Bethe ansatz solution to the mixing problem.

Lifting asymptotic degeneracies with the mirror TBA

We describe a qualitative feature of the AdS5 × S5 string spectrum which is not captured by the asymptotic Bethe ansatz. This is reflected by an enhanced discrete symmetry in the asymptotic limit, whereby extra energy degeneracy enters the spectrum. We discuss how finite size corrections should lift this degeneracy, through both perturbative (Lüscher) and non-perturbative approaches (the Mirror TBA), and illustrate this explicitly on two such asymptotically degenerate states.

Scattering of giant holes

We study scalar excitations of high spin operators in N=4 super Yang-Mills theory, which are dual to solitons propagating on a long folded string in AdS_3 x S^1. In the spin chain description of the gauge theory, these are associated to holes in the magnon distribution in the sl(2,R) sector. We compute the all-loop hole S-matrix from the asymptotic Bethe ansatz, and expand in leading orders at weak and strong coupling. The worldsheet S-matrix of solitonic excitations on the GKP string is calculated using semiclassical quantization. We find an exact agreement between the gauge theory and string theory results.

Superstrings in AdS2 × S2 × T6

We consider the type IIB Green–Schwarz superstring theory on AdS2 × S2 × T6 supported by homogeneous Ramond–Ramond 5-form flux and its type IIA T-duals. One motivation is to understand the solution of this theory based on integrability. This background is a limit of a 1/4 supersymmetric supergravity solution describing four intersecting D3-branes and represents a consistent embedding of AdS2 × S2 into critical superstring theory. Its AdS2 × S2 part with corresponding fermions can be described by a classically integrable PSU(1, 1|2)/SO(1, 1) × U(1) supercoset sigma model. We point out that since the RR 5-form field has non-zero components along the 6-torus directions one cannot, in general, factorize the 10D superstring theory into the supercoset part plus six bosons and six additional massless fermions. Still, we demonstrate that the full superstring model (i) is classically integrable, at least to quadratic order in fermions, and (ii) admits a consistent classical truncation to the supercoset part. Following the analogy with other integrable backgrounds and starting with the finite-gap equations of the PSU(1, 1|2)/SO(1, 1) × U(1) supercoset, we propose a set of asymptotic Bethe ansatz equations for a subset of the quantum string states.

Wrapping corrections and reciprocity beyond the $ \mathfrak{s}\mathfrak{l}(2) $ subsector in $ \mathcal{N} = 4 $ SYM

We consider N=4 SYM and a class of spin N, length-3, twist operators beyond the well studied sl(2) subsector. They can be identified at one-loop with three gluon operators. At strong coupling, they are associated with spinning strings with two spins in AdS5. We exploit the Y-system to compute the leading weak-coupling four loop wrapping correction to their anomalous dimension. The result is written in closed form as a function of the spin N. We combine the wrapping correction with the known four-loop asymptotic Bethe Ansatz contribution and analyze special limits in the spin N. In particular, at large N, we prove that a generalized Gribov-Lipatov reciprocity holds. At negative unphysical spin, we present a simple BFKL-like equation predicting the rightmost leading poles.

Review of AdS/CFT Integrability, Chapter III.4: Twist States and the Cusp Anomalous Dimension

We review the computation of the anomalous dimension of twist operators in the planar limit of N=4 SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations, this aspect of the story is emphasised.

Exact diagonalization of 1D interacting spinless Fermions

We acquire a method of constructing an infinite set of exact eigenfunctions of 1D interacting spinless Fermionic systems. Creation and annihilation operators for the interacting system are found and thereby the many-body Hamiltonian is diagonalized. The formalism is applied to several examples. One example is the theory of Jack polynomials. For the Calogero–Moser–Sutherland Hamiltonian a direct proof is given that the asymptotic Bethe ansatz is correct.

On wrapping corrections to GKP-like operators

In the recent paper arXiv:1010.5009, Maldacena et al. derive the two loop expressions for polygonal Wilson loops expectation values, or MHV amplitudes, by writing them as sums over exchanges of intermediate free particles. The spectrum of excitations of the flux tube between two null Wilson lines can be viewed as the spectrum of excitations around the infinite spin limit of finite twist operators in the sl(2) sector of N=4 SYM or the Gubser-Klebanov-Polyakov (GKP) string. This regime can be captured exploiting integrability and assuming that wrapping corrections are negligible compared to asymptotic Bethe Ansatz contributions. This assumption holds true for the N=4 SYM background GKP string, but deserves further analysis for excited states. Here, we investigate GKP cousins by considering various classes of (generalized) twist operators in beta-deformed N=4 SYM and ABJM theory. We show that the Y-system of Gromov-Kazakov-Vieira easily leads to accurate large spin expansions of the wrapping correction at lowest order in weak-coupling perturbation theory. As a byproduct, we confirm that wrapping corrections are subleading in all the considered cases.

Two-loop AdS5 × S5 superstring: testing asymptotic Bethe ansatz and finite size corrections

We continue the investigation of two-loop string corrections to the energy of a folded string with a spin S in AdS5 and an angular momentum J in S5, in the scaling limit of large J and S with = fixed. We compute the generalized scaling function at two-loop order f2(ℓ) both for small and large values of ℓ matching the predictions based on the asymptotic Bethe ansatz. In particular, in the small ℓ expansion, we derive an exact integral form for the ℓ-dependent coefficient of Catalan's constant term in f2(ℓ). Also, by resumming a certain subclass of multi-loop Feynman diagrams we obtain an exact expression for the leading ln ℓ part of which is valid to any order in the expansion. At large ℓ the string energy has a BMN-like expansion and the first few leading coefficients are expected to be protected, i.e. to be the same at weak and strong coupling. We provide a new example of this non-renormalization for the term which is generated at two loops in string theory and at one-loop in gauge theory (sub-sub-leading in 1/J). We also derive a simple algebraic formula for the term of maximal transcendentality in f2(ℓ) expanded at large ℓ. In the second part of the paper we initiate the study of 2-loop finite size corrections to the string energy by formally compactifying the spatial world-sheet direction in the string action expanded near long fast-spinning string. We observe that the leading finite-size corrections are of ‘Casimir’ type coming from terms containing at least one massless propagator. We consider in detail the one-loop order (reproducing the leading Landau–Lifshitz model prediction) and then focus on the two-loop contributions to the term (for J = 0). We find that in a certain regularization scheme used to discard power divergences the two-loop coefficient of the term appears to vanish.

Quantum dispersion relations for excitations of long folded spinning superstring in AdS 5 × S 5

We use AdS_5 x S^5 superstring sigma model perturbation theory to compute the leading one-loop corrections to the dispersion relations of the excitations near a long spinning string in AdS. This investigation is partially motivated by the OPE-based approach to the computation of the expectation value of null polygonal Wilson loops suggested in arXiv:1006.2788. Our results are in partial agreement with the recent asymptotic Bethe ansatz computation in arXiv:1010.5237. In particular, we find that the heaviest AdS mode (absent in the ABA approach) is stable and has a corrected one-loop dispersion relation similar to the other massive modes. Its stability might hold also at the next-to-leading order as we suggest using a unitarity-based argument.