Density-matrix renormalization-group approach to large deviations and dynamical phase transitions Mieke Gorissen and Carlo Vanderzande Hasselt University - Belgium Lyon - 12 June 2012 Carlo Vanderzande DMRG-approach to large deviations
Current fluctuations in exclusion processes The (a)symmetric exclusion process (ASEP) with open boundaries ( N sites) q p q p q α β γ δ Densities of reservoirs at the boundary α δ ρ a = ρ b = α + γ β + δ Each realisation of the stochastic process can be characterised by the total number of particles Q T passing through the system for T ≫ 1 . Carlo Vanderzande DMRG-approach to large deviations
Current fluctuations in exclusion processes The average and fluctuations of Q T can be determined from the cumulant generating function 1 T ln � e sQ T � µ ( s, L ) = lim T →∞ by taking derivatives at s = 0 . Thermodynamics of histories or s-ensemble - Weight histories of the process with e sQ T s = 0 : typical histories 1 | s | ≫ 1 : histories with a very large current 2 By tuning s we can study rare events Carlo Vanderzande DMRG-approach to large deviations
The symmetric exclusion process For the symmetric exclusion process ( p = q = 1 / 2 ) one has 1 µ ( s, L ) = 1 8 N 2 F ( − 4 M ( s )) + 1 − a − b 1 M ( s ) + O ( N − 3 ) N M ( s ) + N 2 M ( s ) is a known analytical function 2 1 F is a universal function with a singularity at π 2 / 2 2 The third term is non-universal ( a = 1 / ( α + γ ) , b = 1 / ( β + δ ) ) 3 The singularity of F is reached when M ( s ) < − π 2 / 8 For the SSEP the singularity is never reached - no dynamical phase transition No exact results for q � = p and open boundaries → need for a numerical approach that can reach large N -values and gives precise results 1 A. Imparato, V. Lecomte and F. van Wijland, PRE 80 , 011131 (2009) 2 B. Derrida, B. Dou¸ cot and P.-E. Roche, J. Stat. Phys. 115 , 717 (2004) Carlo Vanderzande DMRG-approach to large deviations
DMRG-approach Markov evolution P ( C, t ) � ∂ t P ( C, t ) = H ( C, C ′ ) P ( C ′ , t ) C ′ The generator H of the ASEP can be mapped onto a quantum spin chain (XXZ-model). The stationary state corresponds to the ground state of − H . The stationary state of one-dimensional stochastic many particle systems is a matrix product state (MPS). The density matrix renormalisation group (DMRG) (White, 1992) is the most precise numerical technique to determine ground state properties of quantum (spin) chains. It corresponds to a variational optimisation over MPS-states (Dukelsky et al. , 1998). First applications of DMRG to stochastic problems: Hieida (1998), Carlon et al. (1999). Carlo Vanderzande DMRG-approach to large deviations
DMRG-approach Cumulant generating function µ ( s, N ) = λ ( s, N ) where λ ( s, N ) is the largest eigenvalue of a generalised generator H ( C, C ′ ) e sα ( C,C ′ ) H s ( C, C ′ ) = C � = C ′ and α ( C, C ′ ) = +1( − 1) if a particle leaves (enters) the system on the right when C ′ → C . Expectation values like the density ρ i at site i ρ i ( s, N ) = � L 0 | ˆ n i | R 0 � with � L 0 | and | R 0 � the left and right eigenvector associated to the largest eigenvalue of H s First application of DMRG to current/activity fluctuation: M. Gorissen, J. Hooyberghs and C.V., PRE 79 , 020101 (2009). Carlo Vanderzande DMRG-approach to large deviations
DMRG-approach Problem Dimension of vector space = 2 N - puts a limit to system size that can be studied by exact diagonalisation DMRG technique RG-idea: eliminate variables → ”choose” m ( < 2 N ) vectors and 1 project H (Hamiltonian, generator) in space spanned by these vectors How to choose these m vectors : use the density-matrix 2 L = 8 (l) (r) ρ ρ L = 10 Carlo Vanderzande DMRG-approach to large deviations
DMRG-approach DMRG-algorithm Take a system with N even, ”Hamiltonian” H N : calculate ground 1 state | ψ 0 � - density matrix ρ = | ψ 0 �� ψ 0 | For stochastic systems: symmetric combination of projection on left and right eigenvectors Construct left and right reduced density matrices 2 ρ ( l ) = Tr ′ ρ ( r ) = Tr ′ r ρ l ρ Take the m eigenvectors of ρ ( l ) ( ρ ( r ) ) with largest eigenvalue: 3 | ϕ l � 1 , . . . , | ϕ l � m ( | ϕ r � 1 , . . . , | ϕ r � m ) Add two extra sites i and i + 1 in the middle of the system: project 4 H N +2 in the space spanned by {| ϕ l � 1 , . . . , | ϕ l � m , |±� N/ 2+1 , |±� N/ 2+2 , | ϕ r � 1 , . . . , | ϕ r � m } Reduction of ”number of degrees of freedom” : 2 N +2 → 4 m 2 Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process p = 1 / 2 + ν/ (2 N ) , q = 1 / 2 − ν/ (2 N ) ( ν > 0 ) : diffusive model We determined M ( s ) using the additivity principle 3 Comparison with DMRG results for N up to 120. 250 DMRG, N = 40 DMRG, N = 80 DMRG, N = 120 200 additivity principle 150 µ (s,4/7,5/18,N)N 100 50 0 -50 -50 -40 -30 -20 -10 0 10 20 30 s ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 3 Bodineau and Derrida, PRL 92 180601 (2004) Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process Is there a dynamical transition? 1 1.2 monotonous profile monotonous profile profile with minimum 0.9 profile with minimum DMRG 40 1 0.8 DMRG 100 0.7 0.8 0.6 M"(s) M"(s) 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0.1 0 0 -25 -20 -15 -10 -5 0 5 -40 -30 -20 -10 0 10 20 s s ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 No dynamical transition for parameter values investigated. Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process Is the universal function F appearing in this diffusive model ? 6 - π 2 /8 monotonous profile profile with minimum 4 2 M(s) 0 -2 -4 -6 -25 -20 -15 -10 -5 0 5 s ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 Finite size corrections are not described by the universal function F . Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process Corrections are 1 /N 2 as can be expected for a diffusive model µ ( s, N ) = 1 N M ( s ) + 1 N 2 H ( s ) + O ( N − 3 ) 30 N = 40 N = 60 N = 80 25 N = 100 20 ( µ (s,N) - 1/N M(s))N 2 15 10 5 0 -5 -25 -20 -15 -10 -5 0 5 s ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process Density profile corresponding to a large current 0.6 DMRG, N = 40 DMRG, N = 80 DMRG, N = 120 0.55 additivity principle 0.5 0.6 0.45 ρ (i) 0.56 0.4 ρ (1) 0.52 0.35 0.48 0.3 0 0.0125 0.025 1/N 0.25 0 0.2 0.4 0.6 0.8 1 i/N ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 , j = 5 . 1214 ..., s = 10 (typical current: j ⋆ = 2 . 5845 ... ) Carlo Vanderzande DMRG-approach to large deviations
Results I : The weakly asymmetric exclusion process Density profile corresponding to a small current 0.6 DMRG, N = 40 DMRG, N = 80 DMRG, N = 120 0.5 additivity principle 0.6 time integrated 0.5 0.4 late time 0.4 ρ (i) ρ (i) 0.3 0.3 0.2 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 0.1 i/N 0 0 0.2 0.4 0.6 0.8 1 i/N ν = 10 , ρ a = 4 / 7 , ρ b = 5 / 18 , j = 0 . 00041 ..., s = − 10 Reference: M. Gorissen and C.V., arxiv.org/abs/1201.6264 Carlo Vanderzande DMRG-approach to large deviations
Results II: The totally asymmetric exclusion process For the TASEP ( p = 1 , q = 0 ) numerical results indicate 4 µ ( s, N ) = s 1 N 3 / 2 G ( sN 1 / 2 , ∆ αN 1 / 2 ) 4 + with ∆ α = α − 1 / 2 , the distance to the low-density/maximal current phase transition. β LD MC 1 2 HD α 1 2 4 M. Gorissen and C.V., J. Phys. A, 44 , 115005 (2011) Carlo Vanderzande DMRG-approach to large deviations
Results II: The totally asymmetric exclusion process The current shows a dynamical phase transition 0.9 0.8 0.7 0.6 J*(s,L)/(L+1) 0.5 0.4 0.3 0.2 0.1 0 -0.1 -1 -0.5 0 0.5 1 s From the scaling form for µ ( s, N ) one finds that the k -th cumulant of the current in the MC-phase scales as � Q k T � c ∼ N k/ 2 − 3 / 2 Lazarescu and Mallick (J. Phys. A, 44 , 315001 (2011)) have conjectured a parametric representation of the current cumulant generating function for the TASEP. Check with DMRG through numerical differentiation of µ ( s, N ) Carlo Vanderzande DMRG-approach to large deviations
Results II: The totally asymmetric exclusion process -0.04 0.016 DMRG results DMRG results exact results 0.015 exact results -0.045 0.014 -0.05 0.013 -0.055 0.012 -0.06 0.011 * (L) * (L) C 4 C 3 0.01 -0.065 0.009 -0.07 0.008 -0.075 0.007 -0.08 α = 0.35, β = 0.64 0.006 � = 0.65, � = 0.65 -0.085 0.005 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 L L Carlo Vanderzande DMRG-approach to large deviations
Conclusions The DMRG is a precise numerical tool that can be used to calculate cumulant generating functions, density profiles, gaps, ... for one-dimensional non-equilibrium models with discrete variables. Allows to formulate/verify finite size scaling theories Use of tDMRG to investigate time-dependent behavior? Carlo Vanderzande DMRG-approach to large deviations
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