Boundary CFT from Strings to Percolation to Quantum Quenches

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Boundary CFT from Strings to Percolation to Quantum Quenches

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E. Brézin et al. in Phase Transitions and Critical Phenomena,Vol. 6, edited by C. Domb and M.S. Green, 1976Boundary CFT


Boundary conditions in CFTCFT_T(z), T(z)-T xy = 0 at the boundary ⇒ T(z) = T(¯z)T(¯z) is the analytic continuation of T(z)boundary CFT is simpler than bulk CFT!Boundary CFT


Annulus partition functionHclosedZ annulus−L ĤAB= Tr eopen= 〈A|e −W Ĥ closed|B〉AH openBfor a rational CFT, this allows us toclassify boundary statesthe states C that can propagate in HopenABare given by the fusion rules NAB C of theCFTin string theory, this is the consistencybetween the open and closed stringsectors: the boundary states tell us howopen strings can end, eg on D-branesBoundary CFT


Changes in boundary conditionsCFTCABthe fusion rules NAB C tell us which operators can sit at apoint where the boundary conditions change (boundarycondition changing operators) and which states canpropagate from there into the bulkBoundary CFT


1. Phys. A Math. Gen. 25 (1992) LZOl-L206. Rinted in the UKLETTER TO THE EDITORCritical percolation in finite geometriesJohn L CardyDepartment of Physics, University of California, Santa Barbara, CA 93106, USAReceived ZS November I991Abstract. The methods of conformal field theory arc used to compute the crossing probabilitiesbetween segments of the boundary of a compact two-dimensional region at thepercolation threshold. There probabilities are shown to be invariant not only under changesof scale, but also under mappings of the region which are conformal in the interior andcontinuous on the boundary. This is a larger invariance than that expected for genericcritical systems. Specific predictions are presented for the crossing probability betweenopposite sides of a rectangle, and are compared with recent numerical work. The agreementis excellent.Percolation crossing formula, 1991Conformal field theory has been very successful in determining universal quantitiesassociated with two-dimensional isotropic systems at their critical points [l, 21. Therange of predictions which can be made appears to be bounded by the enthusiasmand industriousness of the theorist rather than Boundary by any intrinsic CFT limitations of the theory.


Percolationeach hexagon independently coloured black or white withprobability p or 1 − pwhat is the probability P of a left-right crossing on onlyblack hexagons as the lattice spacing → 0?for p > p c = 1 2 , P → 1; for p < p c, P → 0at p = p c it is a non-trivial function of the shape of therectangleBoundary CFT


The answer iswhereP = Γ(2 3 )Γ( 1 3 )2 η1/3 2F 1 ( 1 3 , 2 3 , 4 3 ; η)η = ((1 − k)/(1 + k)) 2 with width/height = K(1 − k 2 )/2K(k 2 )this formula led to the mathematical development ofSchramm-Loewner Evolution (SLE) [Schramm, 2000] andwas finally proven rigorously by Smirnov [2001]Boundary CFT


Quantum QuenchesPRL 96, 136801 (2006)PHYSICAL REVIEW LETTERS week ending7 APRIL 2006Time Dependence of Correlation Functions Following a Quantum QuenchPasquale Calabrese 1 and John Cardy 21 Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands2 Oxford University, Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, United Kingdomand All Souls College, Oxford, United Kingdom(Received 13 January 2006; published 3 April 2006)We show that the time dependence of correlation functions in an extended quantum system in ddimensions, which is prepared in the ground state of some Hamiltonian and then evolves withoutdissipation according to some other Hamiltonian, may be extracted using methods of boundary criticalphenomena in d 1 dimensions. For d 1 particularly powerful results are available using conformalfield theory. These are checked against those available from solvable models. They may be explained interms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of thecorrelation length in the initial state, then propagate classically through the system.DOI: 10.1103/PhysRevLett.96.136801P. Calabrese + JC, 2006PACS numbers: 73.43.Nq, 11.25.Hf, 64.60.Htthe quantum critical point has dynamic exponent z 1Suppose that an extended quantum system in d dimensionsto be carried out over a time scale much less than m0 1.Boundary The resultsCFTwe find from CFT suggest a rather simple(for example, a quantum spin system) is prepared at (or, equivalently, a linear quasiparticle dispersion relationtime t 0 in a pure state j 0 i which is the ground state of ! vjkj) because then the 1 1-dimensional problem issome Hamiltonian H 0 (or, more generally, in a thermalstate at a temperature less than the gap m 0 to the firstexcited state.) For times t>0 the system evolves unitarilyaccording to the dynamics given by a different HamiltonianH, which may be related to H 0 by varying a parameter suchas an external field. This variation, or quench, is supposeddescribed asymptotically by a boundary conformal fieldtheory (BCFT) [7,8]. Some of these methods have recentlybeen applied [9] to studying the time evolution of theentanglement entropy, but, as we shall argue, they aremore generally applicable. Further details of these calculationswill appear elsewhere [10].


prepare an extended quantum system in a pure state |ψ 0 〉for times t > 0, evolve the state unitarily with hamiltonian Hhow do correlation functions of local observables〈Φ 1 (x 1 , t)Φ 2 (x 2 , t) · · · 〉 evolve?do they become stationary? If so what is the stationarystate?we studied this in the case whenH = H CFT and |ψ 0 〉 = e −τ 0H |B〉where |B〉 is a conformally invariant boundary stateBoundary CFT


ΦΦ2τ0in imaginary time:〈B|e −τ 0H e itH Φ 1 (x 1 )Φ 2 (x 2 ) · · · e −itH e −τ 0H |B〉= 〈Φ 1 (x 1 , τ)Φ 2 (x 2 , τ) · · · 〉 slab with τ → τ 0 + itthis can be computed by conformal mapping to the upperhalf plane, with the results:correlations in region of length l become stationary after atime t = l/2vthermalisation: reduced density matrix is Gibbs ensembleρ l ∝ e −βH CFTwhere β = 4τ 0quench from a more general state leads to a generalisedGibbs ensemblethis suggests a more generally applicable physical pictureBoundary CFT


A Plea for Curiosity-Driven Theoretical Physicsnone of the above results were ever part of a formalresearch proposalno ‘milestones’no ‘beneficiaries’no ‘post-doctoral training programme’no box-ticking on some research assessmentthere was no scramble to post them on the arXivthey never appeared on the front of Nature or Sciencethey grew out of pure scientific curiosity (in a way that Ihope Dirac would have approved)current funding, research assessment and academicpromotion procedures are stifling this central aspect ofour subject!Boundary CFT


A Plea for Curiosity-Driven Theoretical Physicsnone of the above results were ever part of a formalresearch proposalno ‘milestones’no ‘beneficiaries’no ‘post-doctoral training programme’no box-ticking on some research assessmentthere was no scramble to post them on the arXivthey never appeared on the front of Nature or Sciencethey grew out of pure scientific curiosity (in a way that Ihope Dirac would have approved)current funding, research assessment and academicpromotion procedures are stifling this central aspect ofour subject!Boundary CFT

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