Asymptotic Bethe Ansatz Research Articles

Universal description of dissipative Tomonaga-Luttinger liquids with SU(N) spin symmetry: Exact spectrum and critical exponents

Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with SU($N$) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with SU($N$) symmetry. We uncover that the spectrum of dissipative TL liquids with SU($N$) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of $c=1$ U(1) Gaussian CFT, and $N-1$ spin modes characterized by level-$1$ SU($N$) Kac-Moody algebra with the conformal anomaly $c=N-1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the SU($N$) Calogero-Sutherland model with inverse-square long-range interactions.

Quantum Spectral Curve for AdS3/CFT2: a proposal

We conjecture the Quantum Spectral Curve equations for string theory on AdS3× S3× T4 with RR charge and its CFT2 dual. We show that in the large-length regime, under additional mild assumptions, the QSC reproduces the Asymptotic Bethe Ansatz equations for the massive sector of the theory, including the exact dressing phases found in the literature. The structure of the QSC shares many similarities with the previously known AdS5 and AdS4 cases, but contains a critical new feature — the branch cuts are no longer quadratic. Nevertheless, we show that much of the QSC analysis can be suitably generalised producing a self-consistent system of equations. While further tests are necessary, particularly outside the massive sector, the simplicity and self-consistency of our construction suggests the completeness of the QSC.

Exchange interactions, Yang-Baxter relations and transparent particles

We introduce a class of particle models in one dimension involving exchange interactions that have scattering properties satisfying the Yang-Baxter consistency condition. A subclass of these models exhibits reflectionless scattering, in which particles are “transparent” to each other, generalizing a property hitherto only known for the exchange Calogero model. The thermodynamics of these systems can be derived using the asymptotic Bethe-Ansatz method.

The holographic dual of strongly \u03b3-deformed mathcal{N} = 4 SYM theory: derivation, generalization, integrability and discrete reparametrization symmetry

Recently, we constructed the first-principle derivation of the holographic dual of planar mathcal{N} = 4 SYM in the double-scaled γ-deformed limit directly from the CFT side. The dual fishchain model is a novel integrable chain of particles in AdS5. It can be viewed as a discretized string and revives earlier string-bit approaches. The original derivation was restricted to the operators built out of one of two types of scalar fields. In this paper, we extend our results to the general operators having any number of scalars of both types, except for a very special case when their numbers are equal. Interestingly, the extended model reveals a new discrete reparametrization symmetry of the “world-sheet”, preserving all integrals of motion. We use integrability to formulate a closed system of equations, which allows us to solve for the spectrum of the model in full generality, and present non-perturbative numerical results. We show that our results are in agreement with the Asymptotic Bethe Ansatz of the fishnet model up to the wrapping order at weak coupling.

Asymptotic Bethe ansatz of ABJM open spin chain from giant graviton

In our previous work, the two-loop integrability of ABJM determinant like operator has been well established. In this paper, we push the integrability to all loop orders. The asymptotic Bethe ansatz equations for ABJM determinant like operator (open string attached on giant graviton) are obtained. In the derivation, the symmetries preserved by the bulk and the boundary played a crucial role. Taking the weak coupling limit and applying appropriate fermionic dualities, we obtained a different set of scalar sector Bethe equations with our previous results. When the “gauge” transformation on Bethe equations was introduced, the discrepancy disappeared.

Supersymmetric t-J models with long-range interactions: thermodynamics and criticality

We analyze the thermodynamics and the critical behavior of the supersymmetric t-J model with long-range interactions. Using the transfer matrix formalism, we obtain a closed-form expression for the free energy per site both for a finite number of sites and in the thermodynamic limit. Our approach, which is different from the usual ones based on the asymptotic Bethe ansatz and generalized exclusion statistics, can in fact be applied to a large class of models whose spectrum is described in terms of supersymmetric Young tableaux and their associated Haldane motifs. In the simplest and most interesting case, we identify the five ground state phases of the model and derive the complete low-temperature asymptotic series of the free energy per site, the magnetization and charge densities, and their susceptibilities. We verify the model’s characteristic spin-charge separation at low temperatures, and show that it holds to all orders in the asymptotic expansion. Using the low-temperature asymptotic expansions of the free energy, we also analyze the critical behavior of the model in each of its ground state phases. While the standard phase is described by two independent conformal field theories (CFTs) with central charge c = 1 in correspondence with the spin and charge sectors, we find that the low-energy behavior of the and phases is that of a single c = 1 CFT. We show that the model exhibits an even richer behavior on the boundary between zero-temperature phases, where it can be non-critical but gapless, critical in the spin sector but not in the charge one, or critical with central charge c = 3/2.

Positivity of hexagon perturbation theory

The hexagon-form-factor program was proposed as a way to compute three-and higher-point correlation functions in mathcal{N}=4 super-symmetric Yang-Mills theory and in the dual AdS5×S5 superstring theory, by exploiting the integrability of the theory in the ’t Hooft limit. This approach is reminiscent of the asymptotic Bethe ansatz in that it applies to a large-volume expansion. Finite-volume corrections can be incorporated through Lüscher-like formulae, though the systematics of this expansion is largely unexplored so far. Strikingly, finite-volume corrections may feature negative powers of the ’t Hooft coupling g in the small-g expansion, potentially leading to a breakdown of the formalism. In this work we show that the finite-volume perturbation theory for the hexagon is positive and thereby compatible with the weak-coupling expansion for arbitrary n-point functions.

Many-body stabilization of a resonant p -wave Fermi gas in one dimension

Using the asymptotic Bethe Ansatz, we study the stabilization problem of the one-dimensional spin-polarized Fermi gas confined in a hard-wall potential with tunable p-wave scattering length and finite effective range. We find that the interplay of two factors, i.e., the finite interaction range and the hard-wall potential, will stabilize the system near resonance. The stabilization occurs even in the positive scattering length side, where the system undergoes a many-body collapse if any of the factors is absent. At p-wave resonance, the fermion system is found to feature the "quasi-particle condensation" for any value of effective range, which is stabilized if the range is above twice the mean particle distance. Slightly away from resonance, the correction to the stability condition linearly depends on the inverse scattering length. Finally, a global picture is presented for the energetics and stability properties of fermions from weakly attractive to deep bound state regime. Our results raise the possibility for achieving stable p-wave superfluidity in quasi-1D atomic systems, and meanwhile, shed light on the intriguing s- and p-wave physics in 1D that violate the Bose-Fermi duality.

ABJM quantum spectral curve and Mellin transform

The present techniques for the perturbative solution of quantum spectral curve problems in mathcal{N}=4 SYM and ABJM models are limited to the situation when the states quantum numbers are given explicitly as some integer numbers. These techniques are sufficient to recover full analytical structure of the conserved charges provided that we know a finite basis of functions in terms of which they could be written explicitly. It is known that in the case of mathcal{N}=4 SYM both the contributions of asymptotic Bethe ansatz and wrapping or finite size corrections are expressed in terms of the harmonic sums. However, in the case of ABJM model only the asymptotic contribution can still be written in the harmonic sums basis, while the wrapping corrections part can not. Moreover, the generalization of harmonic sums basis for this problem is not known. In this paper we present a Mellin space technique for the solution of multiloop Baxter equations, which is the main ingredient for the solution of corresponding quantum spectral problems, and provide explicit results for the solution of ABJM quantum spectral curve in the case of twist 1 operators in sl(2) sector for arbitrary spin values up to four loop order with explicit account for wrapping corrections. It is shown that the result for anomalous dimensions could be expressed in terms of harmonic sums decorated by the fourth root of unity factors, so that maximum transcendentality principle holds.

Chiral limit of mathcal{N} = 4 SYM and ABJM and integrable Feynman graphs

We consider a special double scaling limit, recently introduced by two of the authors, combining weak coupling and large imaginary twist, for the γ-twisted mathcal{N} = 4 SYM theory. We also establish the analogous limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in the planar limit. In spite of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions defined by specific, integrable Feynman diagrams. We discuss the details of this diagrammatics. We construct the doubly-scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories. Each entry of the mixing matrix of local conformal operators in the simplest of these theories — the bi-scalar model in 4D and tri-scalar model in 3D — is given by a single Feynman diagram at any given loop order. The related diagrams are in principle computable, up to a few scheme dependent constants, by integrability methods (quantum spectral curve or ABA). These constants should be fixed from direct computations of a few simplest graphs. This integrability-based method is advocated to be able to provide information about some high loop order graphs which are hardly computable by other known methods. We exemplify our approach with specific five-loop graphs.

Y-system for \u03b3-deformed ABJM theory

We investigate the integrable aspects of the planar γ-deformed ABJM theory and propose the twisted asymptotic Bethe ansatz equations. A more general method through a twisted generating functional is discussed, based on which, the asymptotic large L solution of Y-system is modified in order to match the asymptotic Bethe ansatz equations. Several applications of our method in the sl(2)-like sector and some important examples in β-deformed ABJM are presented as well.

Three-point functions in N = 4 $$ \mathcal{N}=4 $$ SYM: the hexagon proposal at three loops

Basso, Komatsu and Vieira recently proposed an all-loop framework for the computation of three-point functions of single-trace operators of ${\cal N}=4$ super-Yang-Mills, the "hexagon program". This proposal results in several remarkable predictions, including the three-point function of two protected operators with an unprotected one in the $SU(2)$ and $SL(2)$ sectors. Such predictions consist of an "asymptotic" part---similar in spirit to the asymptotic Bethe Ansatz of Beisert and Staudacher for two-point functions---as well as additional finite-size "wrapping" L\"uscher-like corrections. The focus of this paper is on such wrapping corrections, which we compute at three-loops in the $SL(2)$ sector. The resulting structure constants perfectly match the ones obtained in the literature from four-point correlators of protected operators.

Worldsheet scattering for the GKP string

We initiate the perturbative study of the S-matrix for the excitations on top of the GKP vacuum at strong coupling. Using the string sigma model action expanded around the null cusp classical solution, we compute the tree-level S-matrix elements and compare them with the predictions from the asymptotic Bethe Ansatz. We also check the factorization of the three-body S-matrix for various bosonic processes, finding precise agreement with the constraints imposed by integrability.

12 loops and triple wrapping in ABJM theory from integrability

Adapting a method recently proposed by C. Marboe and D. Volin for $$ \mathcal{N}=4 $$ super-Yang-Mills, we develop an algorithm for a systematic weak coupling expansion of the spectrum of anomalous dimensions in the sl(2)-like sector of planar $$ \mathcal{N}=6 $$ super-Chern-Simons. The method relies on the Quantum Spectral Curve formulation of the problem and the expansion is written in terms of the interpolating function h(λ), with coefficients expressible as combinations of Euler-Zagier sums with alternating signs. We present explicit results up to 12 loops (six nontrivial orders) for various twist-1 and twist-2 operators, corresponding to triple and double wrapping terms, respectively, which are beyond the reach of the Asymptotic Bethe Ansatz as well as Luscher’s corrections. The algorithm works for generic states in this sector and in principle can be used to compute arbitrary orders of the weak coupling expansion. For the simplest operator with L=1 and spin S=1, the Pade extrapolation of the 12-loop result nicely agrees with the available Thermodynamic Bethe Ansatz data in a relatively wide range of values of the coupling. A Mathematica notebook with a selection of results is attached.

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

We give a derivation of quantum spectral curve (QSC) - a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed in our recent paper Phys.Rev.Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -- a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: Scattering, particles and minimal area Wilson loops

Moving from Beisert–Staudacher equations, the complete set of Asymptotic Bethe Ansatz equations and S-matrix for the excitations over the GKP vacuum is found. The resulting model on this new vacuum is an integrable spin chain of length R=2ln⁡s (s=spin) with particle rapidities as inhomogeneities, two (purely transmitting) defects and SU(4) (residual R-)symmetry. The non-trivial dynamics of N=4 SYM appears in elaborated dressing factors of the 2D two-particle scattering factors, all depending on the ‘fundamental’ one between two scalar excitations. From scattering factors we determine bound states. In particular, we study the strong coupling limit, in the non-perturbative, perturbative and giant hole regimes. Eventually, from these scattering data we construct the 4D pentagon transition amplitudes (perturbative regime). In this manner, we detail the multi-particle contributions (flux tube) to the MHV gluon scattering amplitudes/Wilson loops (OPE or BSV series) and re-sum them to the Thermodynamic Bubble Ansatz.

Six-loop anomalous dimension of twist-two operators in planar N = 4 $$ \mathcal{N}=4 $$ SYM theory

We compute the general form of the six-loop anomalous dimension of twist-two operators with arbitrary spin in planar $$ \mathcal{N}=4 $$ SYM theory. First we find the contribution from the asymptotic Bethe ansatz. Then we reconstruct the wrapping terms from the first 35 even spin values of the full six-loop anomalous dimension computed using the quantum spectral curve approach. The obtained anomalous dimension satisfies all known constraints coming from the BFKL equation, the generalised double-logarithmic equation, and the small spin expansion.

Reciprocity and Self-Tuning Relations without Wrapping

We consider scalar Wilson operators ofN= 4 SYM at high spin,s, and generic twist in the multicolor limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain “reciprocity” and functional “self-tuning” relations up to all terms1/s(ln s)n(inclusive) at any fixed ’t Hooft couplingλ. Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order ins. On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalous dimension expansions at the next order(ln s)2/s2, at fixed ’t Hooft coupling, in such a way to reestablish the aforementioned relation (which fails otherwise).

Twist-2 at five loops: wrapping corrections without wrapping computations

Using known all-loop results from the BFKL and generalized doublelogarithmic equations and large spin limit we have recomputed the five-loop planar anomalous dimension of twist-2 operators without consideration of any wrapping effects. One part of the anomalous dimension was calculated in an usual way with the help of Asymptotic Bethe Ansatz. The rest part, related with the wrapping effects, was reconstructed from a known constraints with the help of methods from the numbers theory.

The spectrum of tachyons in AdS/CFT

We analyze the spectrum of open strings stretched between a D-brane and an anti-D-brane in planar AdS/CFT using various tools. We focus on open strings ending on two giant gravitons with different orientation in $AdS_5 \times S^5$ and study the spectrum of string excitations using the following approaches: open spin-chain, boundary asymptotic Bethe ansatz and boundary thermodynamic Bethe ansatz (BTBA). We find agreement between a perturbative high order diagrammatic calculation in ${\cal N}=4$ SYM and the leading finite-size boundary Luscher correction. We study the ground state energy of the system at finite coupling by deriving and numerically solving a set of BTBA equations. While the numerics give reasonable results at small coupling, they break down at finite coupling when the total energy of the string gets close to zero, possibly indicating that the state turns tachyonic. The location of the breakdown is also predicted analytically.