Quantum sine-Gordon thermodynamics: The Bethe ansatz method

Fowler, Michael; Zotos, X.

doi:10.1103/physrevb.24.2634

Cited by 79 publications

(33 citation statements)

References 19 publications

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“…The constraints on the length of strings in the XXZ model were first postulated in [82] and later shown to be equivalent to normalizability of the Bethe wave function of the associated string configuration [207,208]. We could of course have applied that result here, but we prefer this more intuitive 'derivation'.…”

Section: Point Of Reference: the Xxz Spin Chainmentioning

confidence: 98%

Integrability of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring and its deformations

Tongeren

Stijn²

2014

J. Phys. A: Math. Theor.

103

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This article reviews the application of integrability to the spectral problem of strings on AdS 5 × S 5 and its deformations. We begin with a pedagogical introduction to integrable field theories culminating in the description of their finite-volume spectra through the thermodynamic Bethe ansatz. Next, we apply these ideas to the AdS 5 × S 5 string and in later chapters discuss how to account for particular integrable deformations. Through the AdS/CFT correspondence this gives an exact description of anomalous scaling dimensions of single trace operators in planar N = 4 supersymmetry Yang-Mills theory, its 'orbifolds', and β and γ-deformed supersymmetric Yang-Mills theory. We also touch upon some subtleties arising in these deformed theories. Furthermore, we consider complex excited states (bound states) in the su(2) sector and give their thermodynamic Bethe ansatz description. Finally we discuss the thermodynamic Bethe ansatz for a quantum deformation of the AdS 5 × S 5 superstring S-matrix, with close relations to among others Pohlmeyer reduced string theory, and briefly indicate more recent developments in this area.

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Section: Point Of Reference: the Xxz Spin Chainmentioning

confidence: 98%

Integrability of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring and its deformations

Tongeren

Stijn²

2014

J. Phys. A: Math. Theor.

103

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“…Both models are related to the spin-1/2 Heisenberg chain [140,141], and it is not surprising that they can be solved by Bethe Ansatz resp. the quantum-inverse-scattering method [27,28,87,189,190]. Haldane constructed a renormalized Bethe Ansatz solution and could determine some correlation functions of these models [191].…”

Section: Spin Gapsmentioning

confidence: 99%

One-dimensional Fermi liquids

1995

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We review the progress in the theory of one-dimensional (1D) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particlehole bubble producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless one-dimensional quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t − J-or the extended Hubbard model. Also 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.We discuss in detail various solutions of the Luttinger model which emphasize different aspects of the physics of 1D Fermi liquids. Correlation functions are calculated in detail using bosonization, and the relation of this method to other approaches is discussed. The correlation functions decay as non-universal power-laws, and scaling relations between their exponents are parameterized by the effective coupling constant. Charge-spin separation only shows up in dynamical correlations. The Luttinger liquid concept is developed from perturbations of the Luttinger model. Mainly specializing to the 1D Hubbard model, we review a variety of mappings for complicated models of interacting electrons onto Luttinger models, and thereby obtain their correlation functions. We also discuss the generic behaviour of systems not falling into the Luttinger liquid universality class because of gaps in their low-energy spectrum. The Mott transition provides an example for the transition from Luttinger to non-Luttinger behaviour, and recent results on this problem are summarized. Coupling chains by interactions or tunneling allows transverse coherence to establish in the single-or two-particle dynamics, and drives the systems away from a Luttinger liquid. We discuss the influence of charge-spin separation and of the anomalous dimensions on the transverse dynamics of the electrons. The edge states in the quani tum Hall effect provide a realization of a modified, chiral Luttinger liquid whose detailed properties differ...

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“…The thermodynamics of the Sine-Gordon model is most efficiently studied [32] via the recently developed Thermal Bethe Ansatz approach [33], which circumvents problems associated with solving the infinite number of coupled nonlinear integral equations that emerge in the standard approach based on the string hypothesis [34] (note that the coupling constant β in our problem is a continously varying quantity and no truncation to a finite number of coupled equations is possible). It was shown in [32] that the free energy of the Sine-Gordon model can be expressed in terms of the solution of a single nonlinear integral equation for the complex quantity ε(θ) (we set the spin velocity to 1 for simplicity) (14) where β = (k B T ) −1 , M is the soliton mass and…”

Section: Sine-gordon Thermodynamicsmentioning

confidence: 99%