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, 1976**Boundary** **CFT**

**Boundary** conditions in **CFT****CFT**_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 **to**classify boundary statesthe states C that can propagate in HopenABare given by the fusion rules NAB C of the**CFT**in string theory, this is the consistencybetween the open and closed stringsec**to**rs: the boundary states tell us howopen strings can end, eg on D-branes**Boundary** **CFT**

Changes in boundary conditions**CFT**CABthe fusion rules NAB C tell us which opera**to**rs can sit at apoint where the boundary conditions change (boundarycondition changing opera**to**rs) and which states canpropagate **from** there in**to** the bulk**Boundary** **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.

**Percolation**each 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 therectangle**Boundary** **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** **Quenches**PRL 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 Hamil**to**nian and then evolves withoutdissipation according **to** some other Hamil**to**nian, 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 dimensions**to** be carried out over a time scale much less than m0 1.**Boundary** The results**CFT**we 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 Hamil**to**nian 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 Hamil**to**nianH, which may be related **to** H 0 by varying a parameter suchas an external field. This variation, or quench, is supposeddescribed asymp**to**tically by a boundary conformal fieldtheory (B**CFT**) [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 hamil**to**nian 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 state**Boundary** **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 **CFT**where β = 4τ 0quench **from** a more general state leads **to** a generalisedGibbs ensemblethis suggests a more generally applicable physical picture**Boundary** **CFT**

A Plea for Curiosity-Driven Theoretical Physicsnone of the above results were ever part of a formalresearch proposalno ‘miles**to**nes’no ‘beneficiaries’no ‘post-doc**to**ral 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 ‘miles**to**nes’no ‘beneficiaries’no ‘post-doc**to**ral 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**