Integrable deformations of the O(3) sigma model. The sausage model

Fateev, V.A.; Onofri, E.; Zamolodchikov, A. B.

doi:10.1016/0550-3213(93)90001-6

Cited by 169 publications

(293 citation statements)

References 43 publications

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“…The point of using this loop-cluster definition for the importance sampling of classical spin systems, as was treated rigorously in [17] [18], is that the sum over discrete steps in the probabilistic Markov chain, M, can then essentially be interchanged with the sum over discrete loop-clusters in the definition of the lattice partition function, which leads to more efficient numerical sampling. As was noticed in [15] and [16], however, when this loop-cluster definition of the new lattice configurations is applied to the Trotter-Suzuki formalism in (4), this definition has the advantage that closed loops on the D +1 dimensional lattice in (4) automatically preserve the definition of the trace. Therefore, as is used explicitly in [15], using this approach it is possible to interchange, t, with, M, and to define the lattice partition via the with respect to a trace operation defined over the length of the Markov chain, rather than with respect to Euclidean-time.…”

Section: B Continuous-timementioning

confidence: 95%

“…Firstly, it was found that the infrared (IR) fixed point of the quantum spin chain system at θ = π is directly related to the unstable renormalization group flow of the SU(2) Wess-Zumino-Witten model, such that the central charge of the quantum spin chain system at θ = π is c = 1 [1]. Secondly, it was identified in [3] that the quantum spin chain system is asymptotically free and, thirdly, it was established in [4] that the action of the quantum spin chain can be described effectively by the Sine-Gordon model in the vicinity of the IR fixed point at θ = π. These three results indicate that conformal symmetries are important for classifying the quantum spin chain system, and also, that a nonperturbative description of the system would be needed in order to treat the quantum fluctuations of quantum spin chain systems at arbitrary values of θ.…”

Section: Introductionmentioning

confidence: 99%

“…However, no procedure has yet been defined to renormalize the fluctuations in θ of the quantum spin chain at arbitrary θ. Although it was established in [4] that perturbative renormalization is a well defined program for the effective action given by the Sine-Gordon model at the IR fixed point, it is not clear the extent to which this renormalization program can be generalised to arbitrary values of θ.…”

Section: Introductionmentioning

confidence: 99%

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The partition function zeroes of quantum critical points

Crompton¹

2009

Nuclear Physics B

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The Lee-Yang theorem for the zeroes of the partition function is not strictly applicable to quantum systems because the zeroes are defined in units of the fugacity e h∆τ , and the Euclidean-time lattice spacing ∆τ can be divergent in the infrared (IR). We recently presented analytic arguments describing how a new space-Euclidean time zeroes expansion can be defined, which reproduces Lee and Yang's scaling but avoids the unresolved branch points associated with the breaking of nonlocal symmetries such as parity. We now present a first numerical analysis for this new zeros approach for a quantum spin chain system. We use our scheme to quantify the renormalization group flow of the physical lattice couplings to the IR fixed point of this system. We argue that the generic Finite-Size Scaling (FSS) function of our scheme is identically the entanglement entropy of the lattice partition function and, therefore, that we are able to directly extract the central charge, c, of the quantum spin chain system using conformal predictions for the scaling of the entanglement entropy.

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Section: B Continuous-timementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The partition function zeroes of quantum critical points

Crompton¹

2009

Nuclear Physics B

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“…It is well-known that S n shrinks under Ricci flow [26], and when its curvature is of order 1/α ′ , sigma model perturbation theory is no longer reliable. For the case of S 2 , it is known (both using techniques from integrable models and numerical simulations) what the limiting world-sheet quantum field theory is -it is a free field theory of massive fields [27], [28].…”

Section: Rg Flows In Closed String Theory Linear and Geometric Stmentioning

confidence: 99%

A proposal for studying off-shell stability of vacuum geometries in string theoryThis paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.

Suneeta

2009

Can. J. Phys.

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We briefly review studies of off-shell stability of vacuum geometries in semiclassical gravity. We propose a study of off-shell stability of vacua in string theory by a distinct, though somewhat related approach -by studying their stability under suitable world-sheet sigma model renormalization group (RG) flows. Stability under RG flow is a mathematically well-posed and tractable problem in many cases, as we illustrate through examples. The advantage is that we can make definite predictions about late time behaviour and endpoints of off-shell processes in string theory. This is a contribution to the proceedings of Theory Canada 4, CRM Montreal.

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“…In dimension 2, there exists the well-known example due to Fateev-Onofri-Zamolodchikov [19], King [28] and Rosenau [38], which is often called sausage model. In this paper we present examples of ancient solutions on spheres (as well as some other compact manifolds).…”

Section: Introductionmentioning

confidence: 99%

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

Bakas

Kong

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kähler manifold. Di¤ering from Fateev's examples most of our examples are non-collapsed. The construction of this paper, di¤erent from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jenson, and Ziller in the sense that the existence problem to a backward nonlinear parabolic PDE is reduced to the study of nonlinear ODE systems. The key step of solving the reduced nonlinear ODE system is to find suitable monotonic and conserved quantities under Ricci flow with symmetry. Besides supplying new possible singularity models for Ricci flow on spheres and projective spaces, our solutions are counter-examples to some folklore conjectures on ancient solutions of Ricci flow on compact manifolds. As a by-product, we infer that some nonstandard Einstein metrics on spheres and complex projective spaces are unstable fixed points of the Ricci flow.

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Integrable deformations of the O(3) sigma model. The sausage model

Cited by 169 publications

References 43 publications

The partition function zeroes of quantum critical points

The partition function zeroes of quantum critical points

A proposal for studying off-shell stability of vacuum geometries in string theoryThis paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

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