The Density Matrix Renormalization Group - Max-Planck-Institut

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-



∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉












∂

∂t|P (t)〉 = −H|P (t)〉










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 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .




|ψ0〉 =∑

ij


with i = 1, 2 . . .Ns and j = 1, 2 . . .Ne


We seek

|ψ0〉 =∑

αj

γαj |α〉|j〉



ρ =∑

j



ρ|Ωi〉 = ωi|Ωi〉 ω1 ≥ ω2 ≥ . . .Enrico Carlon, NESPHY03-


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〉




Ns Ne Nem




O =

Ω1(1) Ω2(1) . . . Ωm(1)

Ω1(2) Ω2(2) . . . Ωm(2)

. . . . . . . . . . . .

Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)






Ns Ne Nem




O =

Ω1(1) Ω2(1) . . . Ωm(1)

Ω1(2) Ω2(2) . . . Ωm(2)

. . . . . . . . . . . .

Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)






Ns Ne Nem




O =

Ω1(1) Ω2(1) . . . Ωm(1)

Ω1(2) Ω2(2) . . . Ωm(2)

. . . . . . . . . . . .

Ω1(Ns) Ω2(Ns) . . . Ωm(Ns)





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|

)



environmentsystem

L/2 L/2

L = L + 2







ρ =1

2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|

)



environmentsystem

L/2 L/2

L = L + 2







ρ =1

2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|

)



environmentsystem

L/2 L/2

L = L + 2







ρ =1

2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|

)



environmentsystem

L/2 L/2

L = L + 2







ρ =1

2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|

)



environmentsystem

L/2 L/2

L = L + 2







ρ =1

2tr(|ψl0〉〈ψl0|+ |ψr0〉〈ψr0|

)


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)




ε

1 1

NN

(a) (b)


= neutral monomer

(a) Electrophoresis

(b) Magnetophoresis





−1→ 0 & 0→ +1 rate W = exp(ε)

0→ −1 & +1→ 0 rate W−1 = exp(−ε)




ε

1 1

NN

(a) (b)


= neutral monomer

(a) Electrophoresis

(b) Magnetophoresis





−1→ 0 & 0→ +1 rate W = exp(ε)

0→ −1 & +1→ 0 rate W−1 = exp(−ε)




ε

1 1

NN

(a) (b)


= neutral monomer

(a) Electrophoresis

(b) Magnetophoresis





−1→ 0 & 0→ +1 rate W = exp(ε)

0→ −1 & +1→ 0 rate W−1 = exp(−ε)




ε

1 1

NN

(a) (b)


= neutral monomer

(a) Electrophoresis

(b) Magnetophoresis





−1→ 0 & 0→ +1 rate W = exp(ε)

0→ −1 & +1→ 0 rate W−1 = exp(−ε)




ε

1 1

NN

(a) (b)


= neutral monomer

(a) Electrophoresis

(b) Magnetophoresis





−1→ 0 & 0→ +1 rate W = exp(ε)

0→ −1 & +1→ 0 rate W−1 = exp(−ε)


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.





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



H ′ = H + ∆|ψ0〉〈ψ0|

with (∆ > Γ)∆

sp H’sp H

0Γ







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



H ′ = H + ∆|ψ0〉〈ψ0|

with (∆ > Γ)∆

sp H’sp H

0Γ







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



H ′ = H + ∆|ψ0〉〈ψ0|

with (∆ > Γ)∆

sp H’sp H

0Γ




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






〈y1〉 = 0 〈yN 〉 = 2ε/3


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




N−1/2

0.6

0.8

1<

y’N

/2>

/ε

2/3

N=100






〈y1〉 = 0 〈yN 〉 = 2ε/3


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




N−1/2

0.6

0.8

1<

y’N

/2>

/ε

2/3

N=100






〈y1〉 = 0 〈yN 〉 = 2ε/3


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




N−1/2

0.6

0.8

1<

y’N

/2>

/ε

2/3

N=100


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!




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







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







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







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







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







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





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〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)




+1 0 −1 0 −1 +1 0 0 +1 +1 +10

HEADTAIL

−1





〈−1i〉 =

[fi

1

rh + 1+ (1− fi)

1

2

](1− 〈0i〉)


〈−1i〉 ≈1

rh + 1(1− 〈0i〉)


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







ρ

cp p

Active Inactive

Critical





2A→ 3A 2A→ 0

A0↔ 0A (Rate 0 ≤ d ≤ 1)

space

time







ρ

cp p

Active Inactive

Critical





2A→ 3A 2A→ 0

A0↔ 0A (Rate 0 ≤ d ≤ 1)

space

time







ρ

cp p

Active Inactive

Critical





2A→ 3A 2A→ 0

A0↔ 0A (Rate 0 ≤ d ≤ 1)

space

time


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+ . . .




ρ(t) ∼ t−δ(

1 +A

tδ′. . .

)






′+ . . .

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



δeff = δ +Dρ+ . . .




ρ(t) ∼ t−δ(

1 +A

tδ′. . .

)






′+ . . .

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



δeff = δ +Dρ+ . . .




ρ(t) ∼ t−δ(

1 +A

tδ′. . .

)






′+ . . .

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



δeff = δ +Dρ+ . . .




ρ(t) ∼ t−δ(

1 +A

tδ′. . .

)






′+ . . .

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



δeff = δ +Dρ+ . . .




ρ(t) ∼ t−δ(

1 +A

tδ′. . .

)






′+ . . .

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



δeff = δ +Dρ+ . . .


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


DMRG effective exponent for


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




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






′+ . . .

0 0.2 0.4 0.6ρ, ρ∗0.2

0.3

0.4

0.5

0.6

γ eff


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





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






′+ . . .

0 0.2 0.4 0.6ρ, ρ∗0.2

0.3

0.4

0.5

0.6

γ eff


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





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






′+ . . .

0 0.2 0.4 0.6ρ, ρ∗0.2

0.3

0.4

0.5

0.6

γ eff


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



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)




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


γseff ≡

∂ ln ρs(L)

∂ lnL

(a) Reflecting BC



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)




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


γseff ≡

∂ ln ρs(L)

∂ lnL

(a) Reflecting BC


(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)




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


γseff ≡

∂ ln ρs(L)

∂ lnL

(a) Reflecting BC



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)


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!




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)







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)







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)







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)







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)







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)





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)








relaxation times










relaxation times










relaxation times










relaxation times


that of a crossover from PC to DP

(. . . could also be called PCDP)








relaxation times




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The Density Matrix Renormalization Group - Max-Planck-Institut