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**Non-Integrable QFT And Kink Confinement**

Giuseppe Mussardo

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**Plan of the seminar Non-integrable models**

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

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The state of the art 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 e.g. Ising model ,Potts ,O(n) models, SAW, percolation, etc. Partition functions Correlators of order parameters Spectrum of the excitations Universal Ratios etc

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**Non-integrable dynamics**

However, to reach the full control of the Universality Classes, we need to face the challenge features of Non-integrable dynamics There are several resons for that. The integrable models have zero measure in the space of the Hamiltonians First of all, the majority of deformations are non-integrable, To control the class of universality (universal ratios) we need to consider them. Last but not least they are a theoretical challenge

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Hubbard model

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**Level spacing distribution**

Poisson distribution

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**Gaussian Unitary Ensemble Gaussian Orthogonal Ensemble**

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**Rydberg atom (n=50-100) and magnetic field B few tesla**

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Riemann function

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**Gaussian Unitary Ensemble (breaks time invariance).**

Berry suggests a classical chaotic hamiltonian, with isolated periodic orbits with time period equals to the Primes. Recent paper by Sierra and Towsend: the total numbers of zeros of Riemann function as Coincides with the energy level of lowest Landau level of a quantum particle in a magnetic and electric field.

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**Quantum Spin Chains Low-energy effective action: O(3) sigma model**

with topological term (Haldane-Affleck)

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**(b) O(3) non-linear sigma model with**

topological coupling Particular (integrable) points Massive theory having 3 particles of equal mass (Zam-Zam) Massless theory having 2 doublets of right and left mover particles (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**

θ π 2π There is a fixed point at Θ =π,identified with the conformal field theory SU(2)_1

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**(D.Controzzi, GM, Phys. Rev. Lett. 92 (2004) 21601)**

Θ π c triplet singlet RESONANCE (D.Controzzi, GM, Phys. Rev. Lett. 92 (2004) 21601)

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**Neutron scattering experiment**

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**Spin-quasiparticles in Haldane chains in CsNiCl3**

Correlation function Spin-quasiparticles in Haldane chains in CsNiCl3 R. Coldea, et al, PRL (2001)

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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**

The excitations make scattering processes, exactly like the elementary particles in the high-energy accelerators Elastic process Production processes

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**Analytic structure of the S-matrix**

Integrable QFT Non-Integrable QFT s Integrable case: the simplest analytic structure, with square-root branch cuts and poles in the physical strip Non-integrable case: the story is much more complicated, the S-matrix has a cascade of branch-cuts, corresponding to the different thresholds of the productions processes, the poles move around, there are in general also poles in the second sheet with an imaginary part, which correspond to resonances …

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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**

(G.Delfino,GM,P. Simonetti) In integrable theories we can compute the exact matrix elements on its asymptotic states of any operator (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 (i) non-local field confinement of the particle (ii)**

Gamma is a universal property of the RG trajectory, So it can be computed even at the CFT point (i) non-local field confinement of the particle (ii) local field adiabatic shift of its mass

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**Absent in the perturbed theory**

Physical explanation K Kink excitations Absent in the perturbed theory

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Propagator = ≈ + + + … + + + = if Hence

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**labels the RG trajectories**

Ising Field Theory labels the RG trajectories CFT

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**Particular (integrable) points**

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

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**Mass Spectrum h=0 Free neutral fermion ordinary particle kink**

anti-kink

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**r r V(r) Confining potential**

Kinks are confined. Spectrum given by kink-antikink bound states of mass given by

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**The number of stable particles decreases as η increases, **

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

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**Namely, for which The particle is then unstable for ∞**

# particles 3 4 2 1 ∞ Numerical data give (Fonseca-Zam)

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**Decay widths of higher particles**

(G.Delfino, P.Grinza, GM) Alias, the particle with higher mass lives 4 time longer than the one with lower mass! Branching ratios of :47% into 53% into

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**Tricritical Ising Model**

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Phase diagram 1 order 2 order TIM LT HT Ising Model

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**By varying the chemical potential of the vacancies**

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**Conformal Field Theory**

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

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**Fusion Rules and Structure constants of TIM**

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**Kramers-Wannier duality**

Symmetry properties Spin symmetry Kramers-Wannier duality

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Hidden symmetries TIM SU(2) symmetry E 7 symmetry SUSY

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**Landau-Ginzburg description**

(Zamolodchikov)

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**QFT’s of the relevant deformations of TIM**

Particles Non-Integrable High T (particles) Integrable E 7 Low T (kinks and bound states thereof) Asymmetrical kinks Integrable Massless particles Integrable SUSY Kinks

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**Tricritical Ising Field Theory in the even sector**

(Lepori, GM, Toth) labels the RG trajectories CFT

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**Duality SUSY In the low-temperature phase KINKS ARE STABLE**

No matter how we vary the couplings SUSY

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**E7: Thermal deformation (High temp.)**

(Christe, GM; Fateev, Zam) The conservation laws are compatible with non-zero 3-particle couplings with mass ratios

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**The remaining S are obtained by bootstrap**

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

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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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4 particles 7 particles SUSY

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**Mass spectrum in the vicinity of the Thermal Axis **

with a non-zero chemical potential of the vacancies - unstable The corrections to the masses are NEGATIVE

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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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**Decay channels of higher mass particles**

Selection Rule c a b

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SUSY Duality

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**Spectrum along the (exact) negative SUSY axis**

Observe that there are 3 lowest lines exponentially splitted and DOUBLET of lines

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**Energy lines of degenerate vacua in finite volume**

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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**Conformal minimal models and deformations**

Witten’s index m even Gaussian model (m=4) Tricritical Ising Model (m=3) TIM Ising

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**Kinks in SUSY (Jackiw, Rebbi)**

Fermion has localised zero-energy solution

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**Structure of the vacua and kink excitations**

(BDPTW) Effective mass of the fermion: if positive, the vacuum is not degenerate, if negative it is doubly degenerate

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**Multi-frequency Super Sine-Gordon**

(GM, JHEP 2007) For all values of the origin is always a zero, i.e. SUSY exact At lowest order, the vacua of the original potential continue to remain degenerate. The kinks are stable at weak coupling! The res vanishes when w=1 In the bosonic case, it is the first term that matters (see O(3) sigma model). In SUSY it is instead the VEV that is responsible for the vanishing of the residue! In SUSY the perturbing operator is made of two terms, and the fermionic VEV cancels the bosonic one. Form Factor Perturbation Theory

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**What happens at strong coupling?**

Number of zeros in Since the kinks own their existence to the zeros, a variation of their number implies that some of them should disappear At same critical values of the coupling the system has massless excitations that rule its long-distance behavior. At these critical values N jumps by a step of 2 Because the way the zeros disappear is by colliding and then moving to complex values. When this happens, the kinks between the colliding zeros become massless. Notice that in this evolution of the zeros, the ones at 0,q and 2q (in units of pi) never move! For their topological nature, the disappearance of a kink signals a phase transition

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**Local SUSY breaking, with meta-stable vacua**

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 Near the critical point and at the right minima, the effective dynamics is the one of the tricritical ising model. At this vacuum, there is a cascade of phase transitions: C=3/2 -> 7/10 ->1/2. The massless goldstino is the Majorana fermion of the Ising model. Increasing the value of coupling, the life-time of the false and metastable vacuum shortens and the effective theory loses, at a certain point, its validity.

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**When (q-p) = odd, there will be, in addition to a sequence of **

phase transitions TIM ->Ising, also a phase transition that will will recall the one of the gaussian model. In this case, there is a vacuum where SUSY is exact before and after the phase transition There is 3 colliding zeros, the two zeros on the left and the right of the one at q pi, will strangle the one in the middle. In this case the effective theory is the one close to the phase transition of the gaussan model. In this case there is a cascade Of RG flows: from c=3/2 -> 1 -> 0

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**Conclusions and Open Problems**

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

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