Integrability out of equilibrium: current, fluctuations, correlations Benjamin Doyon King s College London

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1 Integrability out of equilibrium: current, fluctuations, correlations Benjamin Doyon King s College London Main collaborators: Denis Bernard (École Normale Supérieure de Paris), and recently Olalla A. Castro Alvaredo (City University London) Students: Yixiong Chen and Marianne Hoogeveen Dijon September 2013

2 My works on this subject J. Phys. A 45 (2012) with Denis Bernard J. Phys. A 46 (2013) with Denis Bernard arxiv: , to appear in Les Houches Lecture Notes arxiv: with Denis Bernard, accepted in Ann. Inst. H. Poincaré arxiv: with Yixiong Chen arxiv: with A. De Luca, J. Viti and D. Bernard arxiv: with Marianne Hoogeveen and Denis Bernard work in progress with O. A. Castro Alvaredo, Y. Chen, M. Hoogeveen

3 Physical situation Independent quantum chains in different thermal states t<0: β l β r R/2 R/2 t =0: R Unitary evolution... t>0:

4 Steady-state limit R tv F observation lengths observation lengths Asymptotic reservoir Infinite steady state region (no gradient) Asymptotic reservoir tv F R Transient regions

5 Anything happens? Generic non-integrable models: Fourier s law says that no gradient => no current; thermal equilibrium in steady state region. Nothing. Integrable models: conservation laws allow a flow of energy to exist; non-equilibrium steady state. Something interesting! Any critical point: collective behaviours allow a flow of energy; non-equilibrium steady state. Something interesting! [D. Bernard & BD, 2012] Infinite steady state region (no gradient)

6 Initially: Evolution Hamiltonian: Steady state limit: How / what to calculate ness = lim t 0 ρ 0 = e β lh l β r H r H = H l + H r + H contact lim R Observables supported on a finite region Tr e iht 0 ρ 0 e iht0 Tr (ρ 0 ) Current: J(β l, β r )=J ness, J = d dt H r H l 2 = i[h, Hr H l 2 ] Correlations: O(x 1,t 1 )O(x 2,t 2 ) ness

7 Fluctuations of energy transfer: H r H l H l Q = 1 2 H r Q = q 0 t H l H r Q = q 0 + q Ω t (q) = probability of q at time t = Tr P q0 +qe iht P q0 ρ 0 P q0 e iht P q0 +q q 0 F (z) = generating function of scaled cumulants = n z n n! lim t 1 t qn cumulant Ω t = n z n n! C n

8 Fluctuations of energy transfer: It turns out that: 1 F (z) = lim t t log e zq(t) e zq ness, Q(t) =e iht Qe iht charge transfer in free fermion models; indirect measurement protocol: [Levitov & Lesovik 1993; Klich 2002; Schonhamer 2007; D. Bernard & BD 2012] energy transfer in CFT: [D. Bernard & BD 2012] Surprisingly, for energy transfer in integrable models: F (z) = z 0 dz J(β l z, β r + z ) holds whenever there is «pure transmission» SQ = -QS where S is the scattering matrix. [D. Bernard & BD 2013] [D. Bernard & BD 2013] implies the standard fluctuation relations of Jarzynski, Wojcik based on Gallavotti; see Esposito, Harbola, Mukamel (RMP 2009). F (z) =F (β l β r z)

9 Scaling limit: (relativistic) QFT Temperatures «Microscopic» energy scale CFT Critical point 0 Massive QFT Gap

10 Formal description of steady state in massive QFT [D. Bernard & BD 2012] [BD 2012] ness =Tr e W / Tr e W W = β l 0 dθe θ n θ + β r 0 dθe θ n θ Total energy of right-moving asymptotic particles Total energy of left-moving asymptotic particles *In massive QFT: states are described by asymptotically free particles characterized by their rapidities

11 Formal description of steady state in massive QFT [D. Bernard & BD 2012] [BD 2012] back in time... State agreement with free fermion calculations (XY model) [W. H. Aschbacher & C.-A. Pillet, 2003]

12 Formal description of steady state in CFT [D. Bernard & BD 2012, 2013] ness =Tr e W / Tr e W W = β l L 0 2πR + β r L 0 2πR (R ) Total energy of right-movers Total energy of left-movers *In CFT: densities separate into right-movers and left-movers energy density: momentum density: h(x) =h + (x)+h (x) p(x) =h + (x) h (x)

13 Current / fluctuations in CFT [D. Bernard & BD 2012, 2013] J(β l, β r )= πc 12 (β 2 l βr 2 )= πck2 B 12 (T l 2 Tr 2 ) central charge of Virasoro algebra h + (x) c 24 + n Z - confirmed by numerics [C Karrasch, R. Ilan & J. E. Moore, 2012] - agreement with Luttinger liquid results [M. Mintchev & P. Sorba, 2012] [D. B. Gutman, Yu. Gefen & A. D. Mirlin, 2010] L n e 2πinx R Virasoro

14 Current / fluctuations in CFT F (z) =f(z; β l )+f( z; β r ) [D. Bernard & BD 2012, 2013] [BD, M. Hoogeveen & D. Bernard 2013] f(z; β) = cπ 12 1 β z 1 β Recall: F (z) = generating function of scaled cumulants Independent Poisson process for every energy value: independent energy packets, travelling in both directions F (z) = dq ω(q)(e zq 1), ω(q) = cπ 12 e β l q (q >0) e β rq (q <0) intensity cumulant generating function for a single Poisson process

15 Poisson process interpretation current through low-energy rightmoving this surface particles high-energy rightmoving particles low-energy leftmoving particles high-energy leftmoving particles

16 Current / fluctuations Ising model W = dθ w(θ) a (θ)a(θ), w(θ) =m cosh θ [W. H. Aschbacher & C.-A. Pillet, 2003] [A. De Luca, J. Viti, D. Bernard & BD 2013] βl (θ > 0) β r (θ < 0) The current J is quadratic in the creation and annihilation operators, so a rather simple free-fermion calculation gives J = Tr e W J Tr (e W ) = m2 4π Fluctuations: Independent Poisson processes, dθ sinh 2θ 1+e w(θ) => Landauer form: p dp times thermal distributions on left / right ω(q) = n=1 ( 1) n 1 2πn 2 F (z) = z 0 e β l q e β rq using = dz J(β l z, β r + z ) (q >nm) (q < nm) dq ω(q)(e zq 1)

17 Current in massive integrable QFT [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] We extend the thermodybamic Bethe ansatz [Al. B. Zamolodchikov, 1990] to the non-equilibrium steady state density matrix: J = Tr e W p(0) Tr (e W ) In order to calculate the trace, we make the system finite and periodic, replacing the operator W with an approximate finite-length version W L. Following Zamolodchikov, a state v is described by Bethe roots with energies and momenta that approximate the relativistic ones. So we set: W L v = k w k v, w k = e k v : energies e k, momenta p k current operator = momentum density βl (p k > 0) β r (p k < 0)

18 Current in massive integrable QFT J = lim [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] Using translation invariance and taking the large length limit, we get: L L 1 Tr L e W L P Tr L e W L total momentum operator We define a «free energy» and calculate the current from it: f a := lim L L 1 log Tr L e W L ap, J = d da f a. a=0 We can then use the methods of the thermodynamic Bethe ansatz... [Al. B. Zamolodchikov, 1990]

19 Current in massive integrable QFT J = dθ 2π x(θ) =m sinh θ + (θ) =w(θ) where we recall that m cosh θ x(θ) w(θ) =m cosh θ [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] 1+e (θ) dγ dγ 2π ϕ(θ γ) x(γ) 1+e (γ) 2π ϕ(θ γ) log(1 + e (γ) ) βl (θ > 0) β r (θ < 0) dressed momentum non-equil. pseudo energy driving term with a jump ϕ(θ) = i d dθ S(θ) differential scattering phase

20 Current in massive integrable QFT [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] L-function has a jump and an asymmetry L(θ) = log(1 + e (θ) ) Sine-Gordon soliton and breather L-functions at high and medium temperatures Large temperature / small gap limit: we recover the exact CFT result J(β l, β r ) πc 12 (T 2 c = 3 π 2 (0) d l T 2 r ) log(1 + e )+ 1+e An expression for the central charge [Al. B. Zamolodchikov, 1990] Temperatures Critical point 0 CFT Massive QFT «Microscopic» energy scale Gap

21 Non-equilibrium c-functions [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] Dividing current by CFT form, get something that equates the central charge at fixed points. In fact, it is monotonic between fixed points: a c-function [A. B. Zamolodchikov 1986, Al. B. Zamolodchikov 1990], which measures the number of degrees of freedom along an RG trajectory. c 1 (T l,t r )= J(β l, β r ) (π/12)(t 2 l T 2 r ) Non-equilibrium c-function in Al. B. Zamolodchikov s roaming trajectories model high-t: UV fixed point (c=1) low-t: IR fixed point (c=0)

22 Non-equilibrium c-functions [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] It turns out that all cumulants give rise to c-functions: c n (T l,t r )= 12C n (T l,t r ) πn!(t n+1 l +( 1) n T n+1 r ) Explanation: by lowering the mass, keeping the temperatures the same, we allow more excitations to be present, hence there is more current, and more fluctuations d dm C n 0.

23 Poisson process interpretation [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] We can calculate the Poisson process intensity by Fourier transform of the current; it is an intensity if and only if it is positive: ω(q) = 1 q dλ 2π J(β l iλ, β r + iλ) e iλq. F (z) = z using = 0 dz J(β l z, β r + z ) dq ω(q)(e zq 1) Poisson processes iff ω(q) 0

24 Poisson process interpretation [O. A. Castro Alvaredo, Y. Chen, BD & M. Hoogeveen in preparation] It turns out that indeed, this Fourier transform is positive! Asymptotically CFT c=1 π 12 eβ rq Asymptotically CFT c=1 π 12 e β lq Exactly Ising (one-particle) Exactly 0 Exactly Ising (one-particle)

25 Correlation functions in steady state (Ising model) [Y. Chen & BD 2013] extending: [BD 2005] We use a «spectral decomposition» in the space of operators («Liouville space») in order to obtain a convergent expansion valid at large distances, following an old idea in C* algebra. ρ A B ρ = Tr ρ A B Tr (ρ) [Gelfand & Naimark 1943, Segal 1947] Inner product on the space of operators with respect to a density matrix: Choice of basis states: vac ρ := 1 ρ Q ρ 1,..., N (θ 1,...,θ N ):= A, B End(H) creation / annihilation operators θ 1,...,θ N ρ 1,..., N := Q ρ 1,..., N (θ 1,...,θ N ) a 1 (θ 1 ) a N (θ N ) ρ convenient normalization: N i=1 1+e iw (θ i ).

26 Correlation functions in steady state The spectral decomposition is: (Ising model) [Y. Chen & BD 2013] Tr [ρσ(x)σ(0)] Tr[ρ] = ρ vac σ(x)σ(0) ρ left-action: A B ρ = AB ρ = ρ vac σ (x)σ (0) vac ρ = ρ vac σ (x) state ρρ state σ (0) vac ρ states Form factor expansion! We use ρ = e W with W = dθ w(θ) a (θ)a(θ), and we take σ(x) to be the Ising spin field (say in ordered phase).

27 Correlation functions in steady state (Ising model) Mixed-state form factors are defined as: [Y. Chen & BD 2013] Results: f ρ 1,..., N (θ 1,...,θ N ):= ρ vac σ (0) θ 1,...,θ N ρ 1,..., N. f ρ;± 1, 2 (θ 1, θ 2 )=h ± 1 (θ 1 )h ± 2 (θ 2 ) h + (θ) = direction of branch of twist field (right or left) i 2π g(θ) =exp e iπ «leg factors» tanh θ 2 θ 1 ± i( 2 1 )0 2 4 g(θ + i0), h (θ) = ih + (θ) dθ 2πi log tanh θ θ + i vacuum form factor d dθ log expectation value σ ± ρ tanh w(θ ) 2

28 Correlation functions in steady state (Ising model) Analytic structure of leg factors determines leading asymptotic behaviour of two-point function. This reproduces the correct asymptotic in the GGE occurring after a magnetic-field quench as calculated by Calabrese, Essler and Faggoti ( ). In the non-equilibrium steady state: there are branch points: Im(θ) [Y. Chen & BD 2013] branch points π Re(θ)

29 Correlation functions in steady state (Ising model) [Y. Chen & BD 2013] We find an oscillatory behaviour at large distances, e.g. in the disordered phase: e xe ness mx cos(ν log(mx)+b) with a frequency determined by the temperatures: ν = 1 π log coth m 2T r tanh m 2T l

30 Conclusion and perspectives Generalization to presence of impurities Current and fluctuations in integrable spin chains Non-equilibrium correlation functions / mixed-state form factors in integrable models Higher dimensions (in progress - AdS/CFT...)