the density matrix renormalization group - max-planck-institut
TRANSCRIPT
IRI-Lille Outline
The Density Matrix Renormalization Groupand its application to non-equilibrium systems
Enrico Carlon
Institut de Recherches Interdisciplinaires(∗), LILLE
Master equation approach to NES
Introduction to the DMRG
Example: Polymer Reptation (Magnetophoresis)
Example: Pair Contact Process with Diffusion
In collaboration with:
A.Drzewinski, M.Henkel, J.Hooyberghs, J.M.J.van Leeuwen, U.Schollwock, C.Vanderzande
(∗) http://www.lifl.fr/∼iri-bn/Enrico Carlon, NESPHY03-
IRI-Lille Outline
The Density Matrix Renormalization Groupand its application to non-equilibrium systems
Enrico Carlon
Institut de Recherches Interdisciplinaires(∗), LILLE
Master equation approach to NES
Introduction to the DMRG
Example: Polymer Reptation (Magnetophoresis)
Example: Pair Contact Process with Diffusion
In collaboration with:
A.Drzewinski, M.Henkel, J.Hooyberghs, J.M.J.van Leeuwen, U.Schollwock, C.Vanderzande
(∗) http://www.lifl.fr/∼iri-bn/Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Master equation
Master Equation for a stochastic system
∂
∂t|P (t)〉 = −H|P (t)〉
Stationary state H|P0〉 = 0 Relaxation time H|P1〉 = τ−1|P1〉
Formal analogy Schrodinger equation ↔ Master equation however in the latter . . .
The elements of the vector |P (t)〉 are probabilities
Eigenstates of H decay in time as∼ exp(−Ekt) with Re(Ek) > 0
Conservation of the probability implies∑iHij = 0 (and not hermiticity as in QM)
Numerical diagonalization of H restricted to small systems!
Density Matrix Renormalization Group (DMRG): Diagonalization of an approximate Master
operator H which reproduces very well the lowest lying eigenstates of the spectrum of H .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
Quantum mechanical ground state
|ψ0〉 =∑
ij
cij |i〉|j〉 environmentsystem
with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne
Optimal truncation of the basis set of the system (|α〉, α = 1, 2 . . .m with m < Ns)?
We seek
|ψ0〉 =∑
αj
γαj |α〉|j〉
so that S = ‖|ψ0〉 − |ψ0〉‖2 is minimal . . .
Solution Find the reduced density matrix
ρ =∑
j
cijc∗i′j |i〉〈i′| = tr |ψ0〉〈ψ0|
The optimal basis is given by the ”highest”m eigenstates
ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
system environment system environment
Ns Ne Nem
Starting from exact ground state |ψ0〉 we constructed an approximate one |ψ0〉 projecting
onto the m dominant eigenstates of the reduced density matrix
Projection operator (with dim. Ns ×m)
O =
Ω1(1) Ω2(1) . . . Ωm(1)
Ω1(2) Ω2(2) . . . Ωm(2)
. . . . . . . . . . . .
Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)
Renormalization of an observable A→ A = O†AO
A matrix element 〈ψ0|A|ψ0〉 → 〈ψ0|A|ψ0〉 where |ψ0〉 = O†|ψ0〉
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
system environment system environment
Ns Ne Nem
Starting from exact ground state |ψ0〉 we constructed an approximate one |ψ0〉 projecting
onto the m dominant eigenstates of the reduced density matrix
Projection operator (with dim. Ns ×m)
O =
Ω1(1) Ω2(1) . . . Ωm(1)
Ω1(2) Ω2(2) . . . Ωm(2)
. . . . . . . . . . . .
Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)
Renormalization of an observable A→ A = O†AO
A matrix element 〈ψ0|A|ψ0〉 → 〈ψ0|A|ψ0〉 where |ψ0〉 = O†|ψ0〉
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
system environment system environment
Ns Ne Nem
Starting from exact ground state |ψ0〉 we constructed an approximate one |ψ0〉 projecting
onto the m dominant eigenstates of the reduced density matrix
Projection operator (with dim. Ns ×m)
O =
Ω1(1) Ω2(1) . . . Ωm(1)
Ω1(2) Ω2(2) . . . Ωm(2)
. . . . . . . . . . . .
Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)
Renormalization of an observable A→ A = O†AO
A matrix element 〈ψ0|A|ψ0〉 → 〈ψ0|A|ψ0〉 where |ψ0〉 = O†|ψ0〉
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
system environment system environment
Ns Ne Nem
Starting from exact ground state |ψ0〉 we constructed an approximate one |ψ0〉 projecting
onto the m dominant eigenstates of the reduced density matrix
Projection operator (with dim. Ns ×m)
O =
Ω1(1) Ω2(1) . . . Ωm(1)
Ω1(2) Ω2(2) . . . Ωm(2)
. . . . . . . . . . . .
Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)
Renormalization of an observable A→ A = O†AO
A matrix element 〈ψ0|A|ψ0〉 → 〈ψ0|A|ψ0〉 where |ψ0〉 = O†|ψ0〉
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Introduction to DMRG
environmentsystem
L/2 L/2
L = L + 2
Iterative algorithm used
to generate longer chains
DMRG works best with open boundary conditions
DMRG works extermely well for Hermitian operators L ∼ 102 − 103 sites
In non-hermitean DMRG typically one reaches L ∼ 50− 100
In non-Hermitean DMRG the density matrix is constructed by left and right eigenstates:
ρ =1
2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|
)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Example: Reptation in the Rubinstein-Duke model
Charged polymer reptating in an external field
ε
1 1
NN
(a) (b)
• = charged monomer
= neutral monomer
(a) Electrophoresis
(b) Magnetophoresis
Only the projected motion along the field is of interest → 1d stochastic system
Ex. configuration (b) +1,+1,+1, 0,−1,+1,+1,+1, 0,+1, 0
Bulk move 0,±1↔ ±1, 0 Edge move 0↔ ±1
In Magnetophoresis all reactions have rate one except for the pulled edge
−1→ 0 & 0→ +1 rate W = exp(ε)
0→ −1 & +1→ 0 rate W−1 = exp(−ε)
Enrico Carlon, NESPHY03-
IRI-Lille Relaxation time
Reptation is a very slow process.
At zero field the relaxation time is
τ ∼ N3
+1 +1 +1 +1−1 −1 0 0 0 0 −1
+1 +1 +1 +1−1 0 0 0 −10 −1
+1 +1 +1−1 0 0 −1
+1 +1 +1−1 0 0+10
0 +1
0
time
0
−1
−1
0
To calculate the ε = 0 relaxation time
with DMRG it is convenient to use
H ′ = H + ∆|ψ0〉〈ψ0|
with (∆ > Γ)∆
sp H’sp H
0Γ
Efficient trick to calculate the polymer relaxation time:
Carlon, Drzewinski and van Leeuwen PRE 2001; JCP 2002.
Enrico Carlon, NESPHY03-
IRI-Lille Relaxation time
Reptation is a very slow process.
At zero field the relaxation time is
τ ∼ N3
+1 +1 +1 +1−1 −1 0 0 0 0 −1
+1 +1 +1 +1−1 0 0 0 −10 −1
+1 +1 +1−1 0 0 −1
+1 +1 +1−1 0 0+10
0 +1
0
time
0
−1
−1
0
To calculate the ε = 0 relaxation time
with DMRG it is convenient to use
H ′ = H + ∆|ψ0〉〈ψ0|
with (∆ > Γ)∆
sp H’sp H
0Γ
Efficient trick to calculate the polymer relaxation time:
Carlon, Drzewinski and van Leeuwen PRE 2001; JCP 2002.
Enrico Carlon, NESPHY03-
IRI-Lille Relaxation time
Reptation is a very slow process.
At zero field the relaxation time is
τ ∼ N3
+1 +1 +1 +1−1 −1 0 0 0 0 −1
+1 +1 +1 +1−1 0 0 0 −10 −1
+1 +1 +1−1 0 0 −1
+1 +1 +1−1 0 0+10
0 +1
0
time
0
−1
−1
0
To calculate the ε = 0 relaxation time
with DMRG it is convenient to use
H ′ = H + ∆|ψ0〉〈ψ0|
with (∆ > Γ)∆
sp H’sp H
0Γ
Efficient trick to calculate the polymer relaxation time:
Carlon, Drzewinski and van Leeuwen PRE 2001; JCP 2002.
Enrico Carlon, NESPHY03-
IRI-Lille Relaxation time
Reptation is a very slow process.
At zero field the relaxation time is
τ ∼ N3
+1 +1 +1 +1−1 −1 0 0 0 0 −1
+1 +1 +1 +1−1 0 0 0 −10 −1
+1 +1 +1−1 0 0 −1
+1 +1 +1−1 0 0+10
0 +1
0
time
0
−1
−1
0
To calculate the ε = 0 relaxation time
with DMRG it is convenient to use
H ′ = H + ∆|ψ0〉〈ψ0|
with (∆ > Γ)∆
sp H’sp H
0Γ
Efficient trick to calculate the polymer relaxation time:
Carlon, Drzewinski and van Leeuwen PRE 2001; JCP 2002.
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Weak fields
Shape of the reptating polymer 〈yi〉 = 〈+1i〉 − 〈−1i〉
(Drzewinski, Carlon, van Leeuwen PRE 2003)
Weak fields ε ≈ 10−3 − 10−5
〈y1〉 = 0 〈yN 〉 = 2ε/3
〈yi〉+ 〈yN−i〉 = const.
Nice collapse!0 0.2 0.4 0.6 0.8 1
(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
1
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
But wrong! The middle slope
approaches 2/3 as N−1/2
Slow approach to a linear profile!0 0.1 0.2
N−1/2
0.6
0.8
1<
y’N
/2>
/ε
2/3
N=100
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Weak fields
Shape of the reptating polymer 〈yi〉 = 〈+1i〉 − 〈−1i〉
(Drzewinski, Carlon, van Leeuwen PRE 2003)
Weak fields ε ≈ 10−3 − 10−5
〈y1〉 = 0 〈yN 〉 = 2ε/3
〈yi〉+ 〈yN−i〉 = const.
Nice collapse!0 0.2 0.4 0.6 0.8 1
(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
1
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
But wrong! The middle slope
approaches 2/3 as N−1/2
Slow approach to a linear profile!0 0.1 0.2
N−1/2
0.6
0.8
1<
y’N
/2>
/ε
2/3
N=100
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Weak fields
Shape of the reptating polymer 〈yi〉 = 〈+1i〉 − 〈−1i〉
(Drzewinski, Carlon, van Leeuwen PRE 2003)
Weak fields ε ≈ 10−3 − 10−5
〈y1〉 = 0 〈yN 〉 = 2ε/3
〈yi〉+ 〈yN−i〉 = const.
Nice collapse!0 0.2 0.4 0.6 0.8 1
(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
1
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
But wrong! The middle slope
approaches 2/3 as N−1/2
Slow approach to a linear profile!0 0.1 0.2
N−1/2
0.6
0.8
1<
y’N
/2>
/ε
2/3
N=100
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Weak fields
Shape of the reptating polymer 〈yi〉 = 〈+1i〉 − 〈−1i〉
(Drzewinski, Carlon, van Leeuwen PRE 2003)
Weak fields ε ≈ 10−3 − 10−5
〈y1〉 = 0 〈yN 〉 = 2ε/3
〈yi〉+ 〈yN−i〉 = const.
Nice collapse!0 0.2 0.4 0.6 0.8 1
(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
1
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
But wrong! The middle slope
approaches 2/3 as N−1/2
Slow approach to a linear profile!0 0.1 0.2
N−1/2
0.6
0.8
1<
y’N
/2>
/ε
2/3
N=100
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Strong fields
Profiles at stronger fields (ε = 1)
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
<y i>
/ε
N = 10N = 20N = 30N = 40N = 50
0 0.2 0.4 0.6 0.8 1(i−1)/(N−1)
0
0.2
0.4
0.6
0.8
+1
−1
0
0.2 0.6 10.08
0.09
0.1
−1
〈+1i〉 increases monotonically along the chain.
Zero’s have a linear profile 〈0i+1〉 = 〈0i〉 − J (see Barkema and Schutz, EPL 1996)
〈−1i〉 is non-monotonic!
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Magnetophoresis: Interface dynamics
Interface between tail and head regions
+1 0 −1 0 −1 +1 0 0 +1 +1 +10
HEADTAIL
−1
Hypothesis Constant ratios rh,t ≡ 〈+1〉/〈−1〉 in the two regions.
In the unpulled tail rt = 1, while rh > 1 is a parameter.
fi fraction of non-zero particles at position i which are in the head zone
The number of−1’s
〈−1i〉 =
[fi
1
rh + 1+ (1− fi)
1
2
](1− 〈0i〉)
At strong pulling the interface is pushed towards the tail and fi ≈ 1 for i > i0 where:
〈−1i〉 ≈1
rh + 1(1− 〈0i〉)
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion
1+1 Reaction-diffusion models
with absorbing state transitions
are classified into distinct
universality classes
ρ
cp p
Active Inactive
Critical
Directed Percolation (DP) Ex. Contact process A→ 2A, A→ 0
Parity conserving (PC) transition Ex. BARWe A→ 3A, 2A→ 0
The pair contact process
with diffusion (PCPD)
2A→ 3A 2A→ 0
A0↔ 0A (Rate 0 ≤ d ≤ 1)
space
time
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion
1+1 Reaction-diffusion models
with absorbing state transitions
are classified into distinct
universality classes
ρ
cp p
Active Inactive
Critical
Directed Percolation (DP) Ex. Contact process A→ 2A, A→ 0
Parity conserving (PC) transition Ex. BARWe A→ 3A, 2A→ 0
The pair contact process
with diffusion (PCPD)
2A→ 3A 2A→ 0
A0↔ 0A (Rate 0 ≤ d ≤ 1)
space
time
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion
1+1 Reaction-diffusion models
with absorbing state transitions
are classified into distinct
universality classes
ρ
cp p
Active Inactive
Critical
Directed Percolation (DP) Ex. Contact process A→ 2A, A→ 0
Parity conserving (PC) transition Ex. BARWe A→ 3A, 2A→ 0
The pair contact process
with diffusion (PCPD)
2A→ 3A 2A→ 0
A0↔ 0A (Rate 0 ≤ d ≤ 1)
space
time
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion
1+1 Reaction-diffusion models
with absorbing state transitions
are classified into distinct
universality classes
ρ
cp p
Active Inactive
Critical
Directed Percolation (DP) Ex. Contact process A→ 2A, A→ 0
Parity conserving (PC) transition Ex. BARWe A→ 3A, 2A→ 0
The pair contact process
with diffusion (PCPD)
2A→ 3A 2A→ 0
A0↔ 0A (Rate 0 ≤ d ≤ 1)
space
time
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (MC)
At the critical point the density decays as (δ′ correction-to-scaling exponent):
ρ(t) ∼ t−δ(
1 +A
tδ′. . .
)
(Barkema and Carlon, PRE 2003)
Monte Carlo effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
δeff = −∂ ln ρ
∂ ln t= δ + Ct−δ
′+ . . .
0 0.1 0.2 0.3 0.4ρ, ρ∗
0
0.1
0.2
0.3
0.4
δ eff
PC
DP
pairs
particles
0 5 10 15ln t−4
−3
−2
−1
ln ρ
δeff = 0.219
Slow convergence to DP δ = 0.17(1)
Correction-to-scaling exponent δ′ ≈ δ probably due to the effect of isolated particles
δeff = δ +Dρ+ . . .
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (DMRG)
Boundary reactions 0→ A
At the central site
ρL(L/2) ∼ L−γ0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
(Barkema and Carlon, PRE 2003)
DMRG effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
γeff = − ∂ ln ρ
∂ lnL= γ + CL−γ
′+ . . .
0 0.2 0.4 0.6ρ, ρ∗0.2
0.3
0.4
0.5
0.6
γ eff
Quad. fitCubic fit4th deg. fit
PC
DPparticles
pairs
0 0.05 0.1 0.151/L
0.2
0.3
0.4
0.5
ρ(L
)
d = 0.10, p = 0.111 d = 0.15, p = 0.116 d = 0.20, p = 0.121 d = 0.50, p = 0.154 d = 0.80, p = 0.204
PC
DP
From: Carlon, Henkel, Schollwock, PRE 2001
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (DMRG)
Boundary reactions 0→ A
At the central site
ρL(L/2) ∼ L−γ0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
(Barkema and Carlon, PRE 2003)
DMRG effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
γeff = − ∂ ln ρ
∂ lnL= γ + CL−γ
′+ . . .
0 0.2 0.4 0.6ρ, ρ∗0.2
0.3
0.4
0.5
0.6
γ eff
Quad. fitCubic fit4th deg. fit
PC
DPparticles
pairs
0 0.05 0.1 0.151/L
0.2
0.3
0.4
0.5
ρ(L
)
d = 0.10, p = 0.111 d = 0.15, p = 0.116 d = 0.20, p = 0.121 d = 0.50, p = 0.154 d = 0.80, p = 0.204
PC
DP
From: Carlon, Henkel, Schollwock, PRE 2001
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (DMRG)
Boundary reactions 0→ A
At the central site
ρL(L/2) ∼ L−γ0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
(Barkema and Carlon, PRE 2003)
DMRG effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
γeff = − ∂ ln ρ
∂ lnL= γ + CL−γ
′+ . . .
0 0.2 0.4 0.6ρ, ρ∗0.2
0.3
0.4
0.5
0.6
γ eff
Quad. fitCubic fit4th deg. fit
PC
DPparticles
pairs
0 0.05 0.1 0.151/L
0.2
0.3
0.4
0.5
ρ(L
)
d = 0.10, p = 0.111 d = 0.15, p = 0.116 d = 0.20, p = 0.121 d = 0.50, p = 0.154 d = 0.80, p = 0.204
PC
DP
From: Carlon, Henkel, Schollwock, PRE 2001
Enrico Carlon, NESPHY03-
IRI-Lille Pair contact process with diffusion (DMRG)
Boundary reactions 0→ A
At the central site
ρL(L/2) ∼ L−γ0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
(Barkema and Carlon, PRE 2003)
DMRG effective exponent for
particles ρ ≡ 〈A〉 and pairs ρ∗ ≡ 〈AA〉
γeff = − ∂ ln ρ
∂ lnL= γ + CL−γ
′+ . . .
0 0.2 0.4 0.6ρ, ρ∗0.2
0.3
0.4
0.5
0.6
γ eff
Quad. fitCubic fit4th deg. fit
PC
DPparticles
pairs
0 0.05 0.1 0.151/L
0.2
0.3
0.4
0.5
ρ(L
)
d = 0.10, p = 0.111 d = 0.15, p = 0.116 d = 0.20, p = 0.121 d = 0.50, p = 0.154 d = 0.80, p = 0.204
PC
DP
From: Carlon, Henkel, Schollwock, PRE 2001
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (BARWe)
Left boundary reactions 0→ A
At the right edge
ρL(L) ∼ L−γs0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
BARWe A→ 3A 2A→ 0
γseff ≡
∂ ln ρs(L)
∂ lnL
(a) Reflecting BC
(b) Absorbing BC (A→ 0)
(Frojdh, Howard and Lauritsen, 1998)0 0.02 0.04 0.06 0.08 0.1
1/L0.7
0.74
0.78
0.82
γs
eff0 0.05 0.1
1.08
1.12
1.16
particles
pairs
pairs
particles
MC
MC
(a)
(b)
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (BARWe)
Left boundary reactions 0→ A
At the right edge
ρL(L) ∼ L−γs0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
BARWe A→ 3A 2A→ 0
γseff ≡
∂ ln ρs(L)
∂ lnL
(a) Reflecting BC
(b) Absorbing BC (A→ 0)
(Frojdh, Howard and Lauritsen, 1998)0 0.02 0.04 0.06 0.08 0.1
1/L0.7
0.74
0.78
0.82
γs
eff0 0.05 0.1
1.08
1.12
1.16
particles
pairs
pairs
particles
MC
MC
(a)
(b)
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (BARWe)
Left boundary reactions 0→ A
At the right edge
ρL(L) ∼ L−γs0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
BARWe A→ 3A 2A→ 0
γseff ≡
∂ ln ρs(L)
∂ lnL
(a) Reflecting BC
(b) Absorbing BC (A→ 0)
(Frojdh, Howard and Lauritsen, 1998)
0 0.02 0.04 0.06 0.08 0.11/L
0.7
0.74
0.78
0.82
γs
eff0 0.05 0.1
1.08
1.12
1.16
particles
pairs
pairs
particles
MC
MC
(a)
(b)
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (BARWe)
Left boundary reactions 0→ A
At the right edge
ρL(L) ∼ L−γs0 10 20 30 40
i
0
0.2
0.4
0.6
0.8
1
ρ L(i
)
L = 8,10,12,...,38
BARWe A→ 3A 2A→ 0
γseff ≡
∂ ln ρs(L)
∂ lnL
(a) Reflecting BC
(b) Absorbing BC (A→ 0)
(Frojdh, Howard and Lauritsen, 1998)0 0.02 0.04 0.06 0.08 0.1
1/L0.7
0.74
0.78
0.82
γs
eff0 0.05 0.1
1.08
1.12
1.16
particles
pairs
pairs
particles
MC
MC
(a)
(b)
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Surface critical behavior (PCPD)
Surface PCPD: (a) Reflecting BC (b) Absorbing BC
0 0.2 0.4ρs
0.5
0.6
0.7
0.8
0.9
γs
eff
d = 0.2d = 0.5d = 0.9cubic fit
PCsurf.
DPsurf.
(a)
0 0.05 0.1 0.15 0.2ρs
0.6
0.8
1
1.2
1.4
1.6
γs
eff
d = 0.2d = 0.5d = 0.9Cubic fit
PCsurf.
DPsurf.
(b)
Effective exponents extrapolate close to those of the PC class in the range 0 ≤ d ≤ 0.6.
Data for the pairs not very clear!
PC appears as a transient regime for PCPD!
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP (. . . could also be called PCDP)
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP (. . . could also be called PCDP)
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP (. . . could also be called PCDP)
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP (. . . could also be called PCDP)
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP
(. . . could also be called PCDP)
Enrico Carlon, NESPHY03-
IRI-Lille Conclusion
DMRG: useful technique to investigate 1d non equilibrium systems (even more useful if
combined with Monte Carlo simulations).
In the non-Hermitean case DMRG is not as powerful as for Hermitean problems (e.g.
quantum spin chains, fermionic models . . . ). Typical lattice sizes are 50− 100.
DMRG suitable to study: stationary profiles, effects of boundaries, or defects . . . and also
relaxation times
PCPD: From our DMRG and MC results we believe that the most plausible scenario is
that of a crossover from PC to DP (. . . could also be called PCDP)
Enrico Carlon, NESPHY03-