Truncated Conformal Space with Applications

Transcription

1 Truncated Conformal Space with Applications Máté Lencsés International Institue of Physics 10th March 2016 Boundary Degrees of Freedom and Thermodynamics of Integrable Models In collaboration with Gábor Takács M. Lencsés, G. Takács, JHEP 1509 (2015) 146 arxiv: M. Lencsés, G. Takács, JHEP 1409 (2014) 052 arxiv: M. Lencsés, G Takács, Nucl. Phys. B852 (2011) 615 arxiv:

2 The Workshop Integrable models (lattice, spin chain, QFT) Boundary (defect) degrees of freedom YBE, BA, TBA Scaling limit Critical point CFT Correlation functions, form factors Time evolution... How can one test these ideas?

3 This talk: 1+1D QFT Renormalization Group flows in the space of theories g 1 Fixed point CFT g n g 2 Fixed point CFT Fixed point CFT Central charge decreases

4 This talk: 1+1D QFT Renormalization Group flows in the space of theories g 1 Fixed point CFT g n g 2 Fixed point CFT Fixed point CFT Central charge decreases

5 This talk: 1+1D QFT Renormalization Group flows in the space of theories g 1 g n g 2 Fixed point CFT? Fixed point CFT Fixed point CFT?? Central charge decreases

6 Outline Truncated Conformal Space with Applications Truncated Conformal Space Application: q-state Potts model Other applications Conclusion

7 Harmonic oscillator Hamiltonian (dimensionless) Eigenstates: This will be our basis h HO = a a a 0 = 0 ( a ) n n = 0 n! a n = n n 1 a n = n + 1 n + 1 ( h HO n = n + 1 ) n 2

8 Hamiltonian truncation Goal: study the spectrum of some perturbation h = h HO + λh pert Exercise: calculate the matrix elements of h pert on the HO basis! Truncating the basis at a given energy n cut leads to a finite dimensional matrix, h (λ, n cut ) Diagonalize it numerically to get spectrum (λ, n cut )

9 Example, quartic perturbation h = a a λ ( a + a ) 4 ; ncut = 3 h (λ, 3) = 3λ λ 0 15λ λ 0 39λ + 5 2

10 Example, quartic perturbation h (λ, 4) = h = a a λ ( a + a ) 4 ; ncut = 4 3λ λ λ λ 6 2λ 0 39λ λ 0 75λ + 7 2

11 Example, quartic perturbation h = a a λ ( a + a ) 4 ; ncut = 5 h (λ, 5) = 3λ λ 0 2 6λ 0 15λ λ 0 6 2λ 0 39λ λ λ 0 75λ λ λ 0 123λ + 9 2

12 Example, quartic perturbation h = a a λ ( a + a ) 4 ; ncut = 6 h (λ, 6) = 3λ λ 0 2 6λ λ λ λ 6 2λ 0 39λ λ λ 0 75λ λ 2 6λ λ 0 123λ λ λ 0 183λ

13 Example, quartic perturbation h = a a λ ( a + a ) 4 ; ncut = 7 h (λ, 7) = 3λ λ 0 2 6λ λ λ λ 6 2λ 0 39λ λ λ 0 75λ λ 2 6λ λ 0 123λ λ λ 0 183λ λ λ 0

14 Example, quartic perturbation, the Result

15 Outline Truncated Conformal Space Application: q-state Potts model Other applications Conclusion

16 Conformal Space First Yurov and Zamolodchikov 1990 for the scaling Lee Yang model The HO is going to be some CFT in 1+1 dimensions, the perturbation is a sum of primaries: S = S CFT + j ˆ R ˆ µ j dx In finite volume R the spectrum is discrete The space of states depends on the CFT It can be a representation of bosonic/fermionic algebra Virasoro algebra Affine algebra (WZNW)... 0 dtφ i (x, t)

17 Truncated Conformal Space Suppose we have a minimal model Verma modules V h = span {L n1 L n2... L nk h } \nullvectors The full Hilbert space is H CFT = h, h N h, h V h V h In order to have a modular invariant partition function the above sum is finite Truncation: energy or descendant level cut-off e e Λ -c/12 Id Φ1 Φ2 -c/12 Id Φ1 Φ2

18 Truncated Conformal Space Dimensionless Hamiltonian matrix to be diagonalized h = 2π R H h nm = ( N n + N n + h n + h n c ) δ nm 12 + (mr) 2h j 2 ( κ j (2π) 2h j 1 G 1 ) B j nm where N, N are the descendant levels, h, h are the conformal weights of the state µ j = κ j m 2h j 2, m is some mass scale (the lowest mass in the off-critical theory) G mn = n m and (B j ) nm = n Φ j (1, 1) m pl These can be calculated using conformal Ward identities (=commutators with Virasoro generators)

19 Need for renormalization Let s denote the cut-off with Λ Diagonalizing the truncated Hamiltonian away from the critical point gives a cut-off dependent eigensystem One can extract cut-off dependent quantities Q (Λ) (energy levels, form factors, correlation functions etc.) The nature of the cut-off dependence depends on the conformal weight of the perturbation The more relevant the perturbation is, the faster the convergence is, for marginal and irrelevant operators, the method diverges Eg. scaling Lee Yang: 17 states are enough for surprisingly good results. In many other cases: the convergence can be very slow Renormalization: speed up the convergence, cancel divergences

20 Two ways of renormalization Feverati et al 2006; Giokas, Watts 2011 The goal is to get infinite cut-off results diagonalizing a finite dimensional Hamiltonian (at least in the low-energy sector) Counterterms One has the raw cut-off dependent quantity Q (Λ) from the diagonalization Introduce a cut-off dependent counterterm δq (Λ) in order to get infinite cut-off results Renormalization group Q ( ) = Q (Λ) + δq (Λ) Introduce cut-off dependent couplings to get infinite cut-off quantities µ i µ i (Λ)

21 Modeling cut-off dependence The basic step is to study the contribution of the shell (Λ, Λ + Λ) above the cut-off One can split up the Hilbert space into a low energy subspace below, and a high energy subspace above the cut-off One can treat the high energy subspace perturbatively (eg. by means of Schrieffer Wolff transformation) It leads to an effective Hamiltonian acting on the low-energy subspace which contains the contributions of the high energy part H eff (Λ, Λ + Λ) Λ = H (Λ) Λ + δh (Λ, Λ + Λ) Λ

22 Energy counterterms Consider now only the spectrum of low energy states Consider a state of the perturbed theory which corresponds to the unperturbed state i, this state is in the low energy subspace The contribution of the shell to the energy has the form δe i (Λ, Λ + Λ) = i δh (Λ, Λ + Λ) i The full counterterm is the following δe i (Λ) = S shells above Λ δe i (S) (with an appropriate regularization scheme)

23 RG equations One can introduce running couplings satisfying the following renormalization prescription H eff ({µ i (Λ)} Λ, Λ + Λ) Λ H ({µ i (Λ + Λ)} Λ + Λ) Λ+ Λ where means that the spectra of low energy states should be equivalent This leads to flow equations for the couplings dµ i (Λ) dλ = K i ({µ j (Λ)}, Λ) which have to be solved with initial conditions fixed at infinite cut-off

24 Outline Truncated Conformal Space Application: q-state Potts model Other applications Conclusion

25 Classical q-state Potts model Generalization of the Ising model with q different values of the site variables Potts 1952 Classical statistical physical model, defined on a two dimensional square lattice H = 1 δ T s(x),s(y) H x,y x where s (x) = 1, 2,..., q. δ s(x),q Well defined critical point (H = 0, T = T c ) for q 4 For H = 0 and T > T c : paramagnetic phase, unique ground state, T < T c : ferromagnetic phase q times degenerate ground state In absence of magnetic field the Hamiltonian is S q permutation invariant. In presence of magnetic field this symmetry is broken to S q S q 1

26 q-state Potts model: scaling field theory The critical point can be described by conformal field theory (CFT) with central charge c (q) = 1 6 t(t+1) Dotsenko, Fateev 1984 where π (t 1) q = 2 sin 2 (t + 1) The action of the scaling field theory can be written as ˆ ˆ S = S (q) CFT + τ d 2 xε (x) + h d 2 xσ (x) The operators are identified with the primaries ε = Φ 2,1 and σ = Φ (t 1)/2,(t+1)/2 Dotsenko, Fateev 1984, Nienhuis 1984 with dimensions 2h ε (q) = 1 ( ) 2 t 2h (q) σ = (t 1) (t + 3) 8t (t + 1) The couplings are related to lattice parameters τ T T c and h H

27 Zero magnetic field At zero magnetic field (h = 0) the only perturbation is Φ 2,1, this preserves integrability Zamolodchikov 1989 ˆ S = S (q) CFT + τ d 2 xφ 2,1 (x) τ > 0: paramagnetic phase; the elementary excitations are massive quasi-particles above the unique ground state τ < 0: ferromagnetic phase; there are domain wall excitations (kinks) interpolating between the degenerate ground states In both cases the complete infinite volume scattering theory is known (and makes sense for non-integer qs as well) Koberle, Swieca 1972, Chim, Zamolodchikov 1992

28 q = 3 excited state TBA Al. B. Zamolodchikov 1990; Dorey, Tateo 1996 Notations ( ) ˆ L i (θ) = log 1 + e ɛ i (θ) dλ A B (θ) = A (θ λ) B (λ) 2π The excited state TBA system is the following ML, Takacs 2014 ɛ 1,2(θ) = ±iω + mr cosh θ + log S1,2(θ θ+ k ) S 2,1(θ θ + k ) + + N l=1 N + k=1 log S2,1(θ θ l ) S 1,2(θ θ φ1 L1,2(θ) φ2 L2,1(θ) l ) e ɛ 1(θ + k ) = e ɛ 1( θ k ) = 1 e ɛ 2(θ k ) = e ɛ 2( θ + k ) = 1 E(R) = im ( sinh θ + k sinh θ + ) ( k im sinh θ l k l ˆ dθ m cosh θ (L1(θ) + L2(θ)) 2π sinh θ l ) ω = 0, ± 2π 3 is the twist in the ferromagnetic phase Fendley 1992

29 UV limit For the counterterm construction one has to identify the limiting conformal states The UV limit gives the effective central charge of the limiting CFT state c eff = c 24 ( R + L ) where c is the central charge of the fixed point CFT (in this case c = 4/5), R,L is the right/left conformal weight of the operator which creates the state Some examples State Operator State Operator GS (PM) I AĀ(PM) Φ 2/5,2/5 GS (FM) I AA(PM)/tw.AĀ(FM) Φ 2/3,2/3 tw. GS (FM) Φ 1/15,1/15 AAA(PM) Φ 7/5,7/5 /L 1 L 1 Φ 2/5,2/5 A/Ā(PM) Φ 1/15,1/15 AĀ(PM) Φ 7/5,7/5/L 1 L 1 Φ 2/5,2/5

30 TCSA energy counterterms We used descendant level cut-off The detailed calculation yields for the contribution of the nth descendant level to the energy of the state Ψ ˆ R ˆ E Ψ,n (R) E Ψ,n 1 (R) = τ 2 R dx dτ Ψ Φ(τ, x)p nφ(0, 0) Ψ CFT +O(τ 3 ) 0 0 where P n projects to the nth level. The integral can be calculated by using CFT techniques, and expansion in powers of 1/n Using zeta function regularization, one can sum up the level contributions to get the full counterterm

31 Ground state energy Counterterm method used, ML, Takacs 2014

32 Spectrum of low energy states Counterterm method used, ML, Takacs 2014

33 Non-zero magnetic field, confinement phenomenology Consider the Ising model in the ferromagnetic phase McCoy Wu scenario : Poles detach from the branch-cut of the two point function in the presence of magnetic field McCoy, Wu 1978 Interpretation: kinks form bound state particles ( mesons ) in presence of magnetic field In general: ferromagnetic phase + magnetic field pointing to a given colour which becomes energetically preferred Symmetry is broken: S q S q 1 (the non preferred colours remain equivalent) There are one true and q 1 degenerate false ground states

34 False vacua Recall the action: S = S (q) CFT + τ ˆ ˆ d 2 xε (x) + h d 2 xσ (x) We consider the case with h < 0 (preferring one ground state configuration) Using form factor perturbation theory the relative energy density of a false vacuum can be calculated easily Delfino, Grinza 2008 ) ε = δε α δε q h ( σ q α σ q q = v (q) q q 1 h α q where the values of v (q) can be determined using the formulas for VEV s in integrable perturbations of CFTs Fateev, Lukyanov, Zamolodchikov, Zamolodchikov 1998 The false vacua are absent in infinite volume, but they exist in finite volume and have linearly increasing relative energy with respect to the volume

35 Meson, baryon formation Two kink state: in magnetic field the kinks feel linear potential Three kink state (q 3): same The potential is confining The two kink states are confined to mesons, the three kink states to baryons

36 Example: ground states, kinks, mesons and baryons in the 3-state case

37 Meson masses in the Ising model McCoy Wu mass prediction m n = m(2 + λ 2/3 z n ) where m is the kink mass, λ = β(2) h and z m 2 n are the zeroes of the Airy function This can be calculated as a quantum mechanical problem of two particles in linear potential More precise results can be obtained using Bethe Salpeter equation, WKB and its improvements Fonseca, Zamolodchikov 2003, 2006; Rutkevich 2005, 2009) We found that for larger magnetic fields the improved WKB method works well Note on QM and WKB: due to the fermionic nature of the Ising kinks arise only in parity odd combinations

38 Meson masses in the three state Potts model Difference: the scattering of the two types of kinks has bosonic nature: parity even and odd combinations exist Here the kinks has charge (C) due to the remaining S 2 symmetry Mesons: linear potential QM, WKB Rutkevich 2010 Even/odd parity mesons has even/odd C parity It turned out that the WKB method works better

39 Meson and baryon masses in the three state Potts model Baryons: mass formula recently proposed based on the three body QM problem Rutkevich 2014 ( M n ± = m(3 + β (3) h /m 2) 2/3 ɛ ± n ) + O ( h 4/3) where the first ɛ n -s are calculated numerically ɛ + 1 = ɛ+ 2 = ɛ+ 3 = ɛ 1 = ɛ 2 = ɛ 3 = where + stands for parity even and for parity odd states Parity does not coincide with C parity for the baryons

40 TSCA-RG The counterterm method is very effective, but it has the disadvantage that the counterterms have to be constructed state-by-state For higher energy states, the calculation becomes more and more involved The renormalization group method can handle all the states in one time The run of the couplings governed by differential equations dµ i (Λ) dλ = K i ({µ j (Λ)}, Λ) The functions K i can be determined as a power series of the couplings and the inverse cut-off Remaining cut-off dependence can be treated using extrapolation

41 RG equations in the Ising and 3-state Potts case Ising (q = 2): dλ σ (n) dn dλ ɛ (n) dn = 1 2n e 0 (r) = Cɛσ σ 2 λɛ (n) λσ (n) n 1 Γ (1/2) 1 Cσσ ɛ 2n e 0 (r) Γ ( 3/8) 2 λ2 σ (n) n 11 4 E i (n, r) = E i (, r) + A i (r) n 3-state Potts (q = 3) dλ σ (n) dn dλ ɛ (n) dn = = + B i (r) n 2 + O(n 11/4 ) 1 Cσσ σ 2n e 0 (r) Γ (1/15) 2 λ2 σ (n) n Cɛσ σ + 2n e 0 (r) Γ (2/5) 2 λɛ (n) λσ (n) 6 n 5 1 Cσσ ɛ 2n e 0 (r) Γ ( 4/15) 2 λ2 σ (n) n E i (n, r) = E i (, r) + A i(r) n 7/5 + B i n 11/5 + O(n 12/5 )

42 TCSA-RG: false vacua in the 3-state Potts in magnetic field, h = EE0m 0.4 ΒmR TCSA level 10 even TCSA level 11 even TCSA level 9 odd 0.3 TCSA level 10 odd Extrapolated TCSA even Extrapolated TCSA odd mr RG+extrapolation method used, ML, Takacs 2015

43 TCSA-RG: extrapolation and meson identification in 3-state Potts in magnetic field h = TCSA level 9 even TCSA level 10 even TCSA level 11 even Extr. TCSA even TCSA level 8 odd TCSA level 9 odd TCSA level 10 odd Extr. TCSA odd EE0m mr RG+extrapolation method used, ML, Takacs 2015

44 TCSA-RG: Meson masses vs. magnetic field, TCSA and WKB theory 3.5 mmesonm WKB, parity even TCSA measured, charge even WKB, parity odd TCSA measured, charge odd h RG+extrapolation method used, ML, Takacs 2015

45 TCSA-RG: Baryon masses vs. magnetic field, TCSA and 3pt QM mbaryonm Theory, parity even TCSA measured, charge even Theory, parity odd TCSA measured, charge odd h RG+extrapolation method used, ML, Takacs 2015

46 Outline Truncated Conformal Space Application: q-state Potts model Other applications Conclusion

47 Boundary theories, form factors Breather form factors of boundary operator in boundary sg, ML, Takacs 2011

48 Recent interesting applications Higher dimensions φ 4 in d = 2.5: Hogervorst, Rychkov, van Rees, PRD 91, (2015) Entropies (2nd Rényi) Excited states in critical theories: Palmai PRB 90, (R) (2014); Taddia, Ortolani, Pálmai: arxiv: Off-critical Ising: Palmai Physics Letters B (2016) pp Massive version Perturbations of massive boson: Bajnok, Lájer arxiv: Ising field theory quenches: Rakovszky, Mestyán, Cullura, Kormos, Takács arxiv: WZNW CFT Beria, Brandino, Lepori, Konik, Sierra, Nucl.Phys. B877 (2013) ; Konik, Palmai, Takacs, Tsvelik Nucl. Phys. B899 (2015) D QCD: Azaria, Konik, Lecheminant, Palmai, Takacs, Tsvelik PRD 94, (2016)

49 Conclusion, outlook Conclusion Outlook TCSA is a useful testing ground for integrability results Non-integrable models can be studied Variety of quantities can be extracted Higher dimensions; intensive activity on higher dimensional CFT-s (see Poland, Simmons-Duffin Nature Physics (2016)) Other massive extensions: finite volume form factors? Better renormalization, exact RG? Marginal, irrelevant perturbations? Thank you for your attention!