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Non-Integrable QFT And Kink Confinement Giuseppe Mussardo

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Plan of the seminar Non-integrable models Form Factor Perturbation Theory (i) Confinement phenomena (ii) Decay processes Tricritical Ising Field Theory Susy non-integrable models (i) Absence of kink confinement (ii) Meta-stable vacua and susy breaking (ii) SUSY (i) E7 particles and kinks (iii) Low-high temperature duality (ii) Quantum Spin Chains (i)Quantum mechanics

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The state of the art Conformal Integrable Field Theory Thanks to Conformal and Integrable Field Theory, we have reached the control of a huge numbers of 2-d statistical models at criticality and along particular lines Partition functions Correlators of order parameters Universal Ratios Spectrum of the excitations e.g. Ising model,Potts,O(n) models, SAW, percolation, etc. etc

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However, to reach the full control of the Universality Classes, we need to face the challenge features of Non-integrable dynamics

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Hubbard model

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Level spacing distribution Poisson distribution

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Gaussian Orthogonal EnsembleGaussian Unitary Ensemble

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Riemann function

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Quantum Spin Chains Low-energy effective action: O(3) sigma model with topological term (Haldane-Affleck)

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Particular (integrable) points Massive theory having 3 particles of equal mass Massless theory having 2 doublets of right and left mover particles O(3) non-linear sigma model with (b) O(3) non-linear sigma model with topological coupling topological coupling (Zam-Zam)

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Instanton solutions (Belavin-Polyakov) i.e. in each topological sector there is the inequality

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The bound is satured when Stereographical mapping Eqs. of motion = Cauchy –Riemann condition

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General n-instanton solution The value of the action is given by

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Partition function Symmetry

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Renormalization Group Flows g θ π 2π2π 0 There is a fixed point at Θ =π,identified with the conformal field theory SU(2)_1

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M ΘπΘ c (D.Controzzi, GM, Phys. Rev. Lett. 92 (2004) 21601) triplet singlet

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Neutron scattering experiment

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R. Coldea, et al, PRL (2001) Correlation function Spin-quasiparticles in Haldane chains in CsNiCl 3

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What does it mean to say that a quantum system is integrable? The only consistent answer is in terms of the scattering of the quasi-particles

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Integrable vs NonIntegrable QFT S Elastic process S Production processes

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Analytic structure of the S-matrix s … Integrable QFT Non-Integrable QFT

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Methods to study non-integrable theories Semi-classical method Truncated Conformal Space Approach Form Factor Perturbation Theory

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Truncated Conformal Space Approach (Yurov, Al. Zam) Matrix elements on the conformal states

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Role of the boundary conditions Kink state exist only with anti-periodic or twisted b.c. Only kink-antikink states are present with periodic b.c.

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Form Factor Perturbation Theory In integrable theories we can compute the exact matrix elements on its asymptotic states of any operator (G.Delfino,GM,P. Simonetti) ( Smirnov, Karowski-Weisz)

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Perturbation theory by using the matrix elements of φ Born series

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Decay process (Fermi Golden Rule)

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Mass correction confinement of the particle adiabatic shift of its mass (i)non-local field (ii)local field

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Physical explanation Kink excitations Absent in the perturbed theory K

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++ ++ … = + + Propagator = ifHence ≈

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CFT Ising Field Theory labels the RG trajectories

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Particular (integrable) points h=0 Massive free Majorana fermion T=T c Massive theory with 8 particles CFT

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Mass Spectrum h=0Free neutral fermion ordinary particle kink anti-kink

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Confining potential Kinks are confined. Spectrum given by kink-antikink bound states of mass given by r r V(r)

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0 2 M 4 M stable unstable continuum densely filled at η→ -∞ The number of stable particles decreases as η increases, until only is stable at η→+∞

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Namely,for which The particleis then unstable for ∞ Numerical data give # particles (Fonseca-Zam)

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Decay widths of higher particles Alias, the particle with higher mass lives 4 time longer than the one with lower mass! Branching ratios of :47% into53% into (G.Delfino, P.Grinza, GM)

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Tricritical Ising Model

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TIM LTHT Phase diagram Ising Model 1 order 2 order

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By varying the chemical potential of the vacancies

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Conformal Field Theory Second unitary minimal model. The anomalous dimensions are given by the Kac table The central charge is c=7/10 (BPZ, Friedan, Qiu, Shenker))

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Fusion Rules and Structure constants of TIM

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Symmetry properties Kramers-Wannier dualitySpin symmetry

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Hidden symmetries SUSY E 7 symmetrySU(2) symmetry TIM

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Landau-Ginzburg description (Zamolodchikov)

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QFT’s of the relevant deformations of TIM Non-Integrable Integrable E 7 High T (particles) Low T (kinks and bound states thereof) Asymmetrical kinks Integrable SUSY Massless particles Kinks Integrable Particles

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CFT Tricritical Ising Field Theory in the even sector labels the RG trajectories (Lepori, GM, Toth)

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SUSY Duality

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E7: Thermal deformation (High temp.) The conservation laws are compatible with non-zero 3-particle couplings with mass ratios (Christe, GM; Fateev, Zam)

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For the S-matrix of the fundamental particle A we have 1 The remaining S are obtained by bootstrap ab

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Exact mass spectrum Mass value Parity(ht)Low tem odd kink even particle odd kink even particle even particle odd kink even particle

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Mass spectrum of the Tricritical Ising Model along the thermal axis Unstable as soon as one leaves the thermal axes

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Structure of bound states

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E7 Numerical spectrum

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SUSY 7 particles 4 particles

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Mass spectrum in the vicinity of the Thermal Axis with a non-zero chemical potential of the vacancies unstable -

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Low-temperature: Absence of kink confinement By duality and the latter matrix element does not have a pole!

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c a b Decay channels of higher mass particles Selection Rule

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SUSY Duality

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Spectrum along the (exact) negative SUSY axis

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Energy lines of degenerate vacua in finite volume 0+- Even sector

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Exact S-matrix (Aliosha) = = = =

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Form Factors of energy operator (Delfino)

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Vacuum expectation values (Fateev)

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TCSA data

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Evolution of the spectrum in the high-temperature phase

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Stable Particles and Thresolds

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Evolution of the spectrum in the low-temperature phase

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Fragility of the kink in purely bosonic theories

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N=1 Supersymmetric theories Susy transformation ;

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TIM Ising Conformal minimal models and deformations Witten’s index m even Tricritical Ising Model (m=3)Gaussian model (m=4)

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Kinks in SUSY (Jackiw, Rebbi)

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Structure of the vacua and kink excitations (BDPTW)

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Multi-frequency Super Sine-Gordon For all values ofthe origin is always a zero, i.e. SUSY exact At lowest order, the vacua of the original potential continue to remain degenerate. Form Factor Perturbation Theory (GM, JHEP 2007) stable The kinks are stable at weak coupling!

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What happens at strong coupling? Number of zeros in Since the kinks own their existence to the zeros, a variation disappear of their number implies that some of them should disappear For their topological nature, the disappearance of a kink phase transition signals a phase transition

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(q-p) = even When (q-p) = even, there will be a sequence of phase transitions that will recall the one of Tricritical Ising -> Ising. Local SUSY breaking, with meta-stable vacua

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(q-p) = odd When (q-p) = odd, there will be, in addition to a sequence of phase transitions TIM ->Ising, also a phase transition that will gaussian model will recall the one of the gaussian model. SUSY is exact In this case, there is a vacuum where SUSY is exact before and after the phase transition

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Conclusions and Open Problems Confinement of particles, decay rate of unstable particles Stability of the kinks under special circumstances such as, spin reversal symmetry or supersymmetry Finite size effects (boundary) and dynamical correlators N=2 SUSY non-integrable models SUSY breaking and metastable vacua ( Ising field theory, multi-frequency Sine-Gordon, etc) (low-temperature TIM, multi-frequency Super Sine-Gordon)

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