Physique statistique hors ´ equilibre – Examen

Universit´ es Paris 6, Paris 7 & Paris-Sud, ´ Ecole Normale Sup´ erieure, ´ Ecole Polytechnique Master 2 iCFP – Parcours de Physique Quantique

Physique statistique hors ´ equilibre – Examen

Lundi 9 janvier 2017

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Subject : Non-intersecting fluctuating lines and 1D fermions

Introduction : We study a model of non-intersecting Brownian lines in dimension 1 + 1, using the mapping on the 1D fermion gas. We consider lines going only forward in one of the two spatial directions (axis τ in Fig. 1), which can thus be considered as a fictitious “time”. Each line is described by a displacement field x n (τ ) ∈ R for τ ∈ [0, L]. Such models appear in various contexts : directed polymers, fluctuating interface models (for atomic terraces, wetting problems,..), etc.

Single line : The basic ingredient of the model is the functional for the elastic energy E[x(τ )] = c

2 Z p

dτ ² + dx ² − dτ

' c 4

Z L 0

dτ (∂ τ x(τ )) ² ,

where it is assumed that distorsions are small, |∂ _τ x| 1. The positive coefficient c is the elastic coefficient (the stress). The Gibbs measure exp

− E[x(τ )]/T is identified with the Wiener measure for the Brownian motion with diffusion constant D = T /c. Such a Brownian trajectory can be conveniently described by the Fokker-Planck equation

∂ τ P τ (x|y) = D ∂ _x ² P τ (x|y) , (1) where P _τ (x|y) is the conditional probability for the Wiener process (free Brownian motion).

Part C is almost independent of parts A and B.

τ

1

y

y

N

y

y

y

... x

x

x

(τ) x (τ)

1

(τ)

...

...

0 L

y

Figure 1 : N non-intersecting elastic lines (directed polymers, fluctuating interfaces,...).

A. Properties of a single line.– We first study the fluctuations of a single line.

1/ Eq. (1) describes the single line. Recall its solution on R .

Hint : The initial condition is lim

P

(x|y) = δ(x − y).

2/ Fluctuations.– We consider the situation where the line issues and ends at the same point, say x(0) = x(L) = 0. Justify

hx(τ ) ² i = A L

Z

R

dx P L−τ (0|x) x ² P τ (x|0) (2)

What is the normalisation constant A L ? Compute the variance hx(τ ) ² i and plot it neatly as a function of τ ∈ [0, L]. Where are the fluctuations maximum ?

B. N non-intersecting lines.– We now consider N lines (x 1 (τ ), · · · , x _N (τ ))

= ~ x(τ ), still described by the free diffusion, with however some constraints for non-intersection

∂ _τ P _τ (~ x|~ y) = D ~ ∇ ² P _τ (~ x|~ y) with x ₁ < x ₂ < · · · < x _N , (3) where ∇ ≡ ~ (· · · , ∂/∂x i , · · · ).

1/ Recall the general form for the decomposition of the propagator over the spectrum of the operator H N = −D ~ ∇ ² , denoted {E _n , Ψ n (~ x)} _{n=0, 1,···,∞} (with R

d ^N ~ x Ψ ^∗ _n (~ x)Ψ m (~ x) = δ n,m ).

Hint : Since the Fokker-Planck operator is Hermitian here, we can simply use the mapping of (3) onto the imaginary time Schr¨ odinger equation with Hamiltonian H

.

2/ We assume that the N lines start from ~ x(0) = ~ y and reach the same set of values at final

“time”, ~ x(L) = ~ y (like in Fig. 1). Express formally the average of an observable hA(~ x(τ ))i at

“time” τ in terms of the propagator P _τ (~ x|~ y). Argue that for τ and L − τ → ∞, this average can simply be written in terms of the ground state Ψ 0 (~ x) of the operator H N .

C. Compressibility and 1D fermionic gas.– We study the response of N 1 lines to an external perturbation. In the limit L → ∞, their statistical properties are controlled by the ground state Ψ 0 (~ x) of the operator H _N . Thus, we can use the mapping between (3) and the Schr¨ odinger equation for a gas of N free spinless fermions of “mass” m = 1/(2D) = c/(2T ) (we set ~ = 1), where T is the original temperature of the lines and c their elastic constant.

We now consider the fermions, described by the Schr¨ odinger equation i∂ t Ψ(~ x, t) = H _N Ψ(~ x, t) : fermions occupy 1D plane waves ψ _k (x) = ^{√ 1}

L e ^ikx of energy ε _k = _2m ^k

. In a box of finite size L , the wave vector k is quantised (it is sufficient to remember that P

k → L /(2π) R dk).

1/ We consider a single fermion : the density operator is ˆ n(x) = δ(x − x) and its Fourier ˆ component ˆ n _q = e ^{−iqˆ x} , where ˆ x is the position operator. Give the matrix element h ψ _k |ˆ n _q |ψ _k

i.

2/ We now consider the fermion gas. A perturbation is introduced under the form of a scalar external potential V (x, t) = (1/L ) P

q V q (t) e ^iqx . The corresponding Hamiltonian is H ˆ _pert (t) =

N

X

i=1

V (ˆ x _i , t) = Z

dx V (x, t) ˆ n(x) = 1 L

X

q

V _q (t) ˆ n −q ; (4) from now on ˆ n q = P

i e ^{−iqˆ x}

is the density operator of the fermion gas. Its response is h n(x, t)i ˆ _V = hˆ n(x)i +

Z

dt ⁰ dx ⁰ χ(x − x ⁰ , t − t ⁰ ) V (x ⁰ , t ⁰ ) + O(V ² ) (5) i.e. hˆ n _q (t)i _V = n L δ _q,0 + R

R dt ⁰ χ e _q (t − t ⁰ ) V _q (t ⁰ ) + O(V ² ), where n = hˆ n(x)i is the mean density.

a) Express χ e q (t) as an equilibrium correlation function.

b) Write its Fourier transform χ(q, ω) = e R

R dt χ e q (t) e ^iωt as a sum over contributions of plane

waves. Analyse its analytical structure. Interpret the position of the poles of the integrand.

3/ Static compressibility.– The connection to the problem of fluctuating lines is made more transparent by considering the static response χ(q)

= χ(q, e 0) at zero temperature.

a) We denote by f (ε _k ) the Fermi-Dirac distribution. Show that χ(q) = −

Z dk π

f (ε k )

ε _k − ε _k+q (6)

where the integral is a Cauchy principal part integral (see appendix).

b) The zero temperature fermion gas involves the Fermi energy ε F = _2m ¹ k _F ² controlling the number of fermions N. Compute explicitely χ(q).

c) Kohn anomaly.– Show that lim q→0 χ(q) is related to the density of states at Fermi level ρ ₀ = _πk ^m

F

. Analyse χ(q) for q → 2k _F and q → ∞. Plot neatly −χ(q)/ρ ₀ as a function of q/(2k _F ).

Express k _F and ρ ₀ as a function of the parameters for the fluctuating lines (n, T and c).

d) Bonus.— Argue that the length of the lines L coincides with the inverse temperature of the fermions β ferm = 1/T ferm (go back to question B.2). Discuss the validity of the T ferm = 0 approximation, and then that of the small distorsion approximation made in the very begining (express the two conditions in terms of the parameters of the lines). For finite T _ferm , what is the value of χ(2k F ) (in terms of T, n and c) ?

Appendix :

• Convention for Fourier transform in dimension d : ϕ _q =

Z

Vol

d ^d x e ^−iqx ϕ(x) & ϕ(x) = 1 Vol

X

q

ϕ _q e ^iqx −→

Vol→∞

Z d ^d q

(2π) ^d ϕ _q e ^iqx (7)

• In pratice, Cauchy principal part integral ⁻ R

dx _x−x ^ϕ(x)

0

= Re R

dx _x−x ^ϕ(x)

0

±i0

can be calculated with

− Z b

a

dx ϕ(x) x − x 0

= lim

η→0

Z x

−η a

+ Z b

x

+η

dx ϕ(x) x − x 0

, (8)

where ϕ(x) is a regular function and x 0 ∈ [a, b] (if x 0 ∈ / [a, b], this is the usual integral).

• We recall that the grand canonical average of a commutator of many body operators, sums of one particle operators, of the form ˆ A = P N

i=1 a ˆ ⁽ⁱ⁾ and ˆ B = P N

i=1 ˆ b ⁽ⁱ⁾ , is h[ ˆ A , B]i ˆ = X

α

f α h ϕ α | ˆ a, ˆ b

| ϕ α i = X

α, β

(f α − f _β ) a _αβ b _βα , (9) where the sum runs over one particle stationary states. a _αβ = h ϕ _α |ˆ a| ϕ _β i is a matrix element of the one particle operator and f _α = f (ε _α ) denotes the occupancy of the individual eigenstate.

The problem is inspired by the article :

P.-G. de Gennes, Model for fibrous structures with steric constraints, J. Chem. Phys. 48(5), 2257–2259 (1968).

Solutions sur la page du cours : cf. http://lptms.u-psud.fr/christophe_texier/

Christophe Texier January 2017 Non-intersecting fluctuating lines and 1D fermions – Solutions

A. Single line.

1/ Solution is P _τ (x|y) = ^{√ 1}

4πDτ exp

− ^(x−y) _4Dτ

. Note that here, D has dimension [D] = [x] ² /[τ ] = [length]

2/ Averaging takes the form

hx(τ ) ² i = R

R dx P L−τ (0|x) x ² P τ (x|0) P _L (0|0)

(if one removes the x ² in the numerator, one gets obviously h1i = 1, as required). The Gaussian integral gives the fluctuations of the Brownian bridge, hx(τ ) ² i = 2D τ 1 − _L ^τ

. Fluctuations are maximum at the middle τ = L/2, where the line is less constrained hx(L/2) ² i = DL/2 = T L/(2c). Discussion : fluctuations increase with temperature and decrease with the stress. Sca- ling with the length is δx ∼ √

L as expected.

Dimensional analysis : [T ] = [energy] and [c] = [force] = [energy]/[length]. Ok.

B. N lines.

1/ The decomposition of the propagator over the spectrum of H _N = −D ~ ∇ ² is P _τ (~ x|~ y) = h ~ x | e ^{−τ H}

| ~ y i =

∞

X

n=0

Ψ _n (~ x) Ψ ^∗ _n (~ y) e ^{−τ E}

.

2/ Averaging of a function of coordinates at “time” τ has the same structure as for one line.

Assuming a discrete spectrum, for long lines we write P _τ (~ x|~ y) ' Ψ ₀ (~ x) Ψ ^∗ ₀ (~ y) e ^{−τ E}

, hence hA(~ x(τ ))i =

N ! R ∞

−∞ dx ₁ R ∞

x

dx ₂ · · · R ∞

x

dx _N P L−τ (~ y|~ x) A(~ x) P _τ (~ x|~ y)

P L (~ y|~ y) (10)

' N ! R ∞

−∞ dx 1

R ∞

x

dx 2 · · · R ∞

x

dx N Ψ 0 (~ y) Ψ ^∗ ₀ (~ x) e ^−(L−τ)E

A(~ x) Ψ 0 (~ x) Ψ ^∗ ₀ (~ y) e ^{−τ E}

Ψ ₀ (~ y) Ψ ^∗ ₀ (~ y) e ^−LE

^.

Finally

hA(~ x(τ ))i '

τ & L−τ→∞ N ! Z ∞

−∞

dx 1

Z ∞ x

dx 2 · · · Z ∞

x

dx N |Ψ ₀ (~ x)| ² A(~ x) . (11) which corresponds to average in the ground state of H _N . The factor N ! is required if we choose to normalise the eigenstates as R

R

d~ x |Ψ _n (~ x)| ² = 1.

From non-intersecting lines to 1D fermions : The constraints for non-intersection can be des- cribed by wave functions which vanish at coinciding points Ψ(· · · , x i , x i+1 = x i , · · · ) = 0.

Moreover, being restricted to the sector x 1 < x 2 < · · · < x N , we can construct a basis of wave functions antisymmetric under exchange of particles, i.e. fermionic eigenstates.

Remark : from the point of view of the diffusion, the Dirichlet boundary condition implies that normalisation is not conserved, i.e. R

d~ x P τ (~ x|~ y) ∼ e ^−E

^τ decays as time grows. However, dividing

by P _L (~ y|~ y) in (10) implies that averaging is taken only over non-intersecting trajectories which

have survived.

C. Compressibility and 1D fermionic gas.– We now study the 1D fermion gas (the begining of the part C reproduces a result of the lectures).

1/ Matrix elements for the one-body density operator : h ψ _k |ˆ n _q | ψ _k

i = R _dx

L e −ikx−iqx+ik

x = δ k

,k+q .

2/ Compressibility.– We apply one of the main results of the lectures : χ e _q (t) = − i

L θ _H (t) h[ˆ n _q (t), n ˆ −q ]i (12) in terms of many-body density operator. The correlator can be expressed in terms of one-body matrix elements (relation of the appendix) :

χ e _q (t) = − i

L θ _H (t) X

k, k

(f _k − f _k

) |h ψ _k |ˆ n _q |ψ _k

i| ² e ^i(ε

^−ε

^)t where f _k ≡ f (ε _k ) = 1/ e ^β(ε

^−µ) + 1

. A Fourier transform over time finally gives the formula obtained in the exercice session (TD 8) :

χ(q, ω) = e 1 L

X

k

f k − f k+q

ω + ε _k − ε _k+q + i0 ⁺ (13)

where the 0 ⁺ in the denominator is a regulator. It ensures that singularities of the response function are in the lower complex plane of the frequency (causality). Besides we recognise the energy of a particle-hole excitation ε k+q − ε k .

3/ Static compressibility.– For zero frequency, we could drop the regulator which is not needed χ(q) = _L ¹ P

k

f

−f

ε

−ε

. We prefer to use the symmetry of the spectrum ε _k = ε −k and split the sum (which requires to keep the regulator)

χ(q) = 1 L

X

k

f k − f k+q

ε _k − ε _k+q + i0 ⁺ = 1 L

X

k

f _k

ε _k − ε _k+q + i0 ⁺ − 1 L

X

k

f _k ε k−q − ε _k + i0 ⁺

= 1 L

X

k

f _k

ε _k − ε _k+q + i0 ⁺ + 1 L

X

k

f _k ε _k − ε _k+q − i0 ⁺ finally, using _Ω±i0 ¹

= PP _Ω ¹ ∓ iπ δ(Ω),

χ(q) = 2 L

X

k

PP f _k

ε _k − ε _k+q = − Z dk

π

f _k ε _k − ε _k+q

The Fermi-Dirac distribution at T ferm = 0 is f k = θ H (k F − |k|). Writing ε k+q − ε k = _m ^q (k + q/2) we find

χ(q) = − m q π −

Z +k

−k

dk

k + q/2 = m q π ln

q − 2k _F q + 2k F

(14) We obtain easily the limiting behaviours :

χ(q) ' −ρ ₀ ×



 



 



1 for q k _F

1

2 ln _|q−2k ^4k

F

| for q ∼ 2k _F 2k

q

2

for q k _F

(15)

where ρ ₀ = _πk ^m

F

is the DoS per unit volume at Fermi level. The logarithmic singularity is called

the Kohn anomaly. It is due to a particle-hole excitation where the particle and the hole are

close to the two Fermi points.

0.0 0.5 1.0 1.5 2.0 2.5 0.0

0.5 1.0 1.5 2.0 2.5

q  2k F

- Χ 0 H q L Ρ 0

Figure 2 : Zero temperature compressibility with logarithmic divergence at q = 2k _F . The mapping on the problem of fluctuating lines with density n is



 

 

k F = π n m −→ _2T ^c ρ ₀ → _2π

^c T n

Bonus :

• If we come back to the formula for the averaging (10), we see that the length L of the lines play the same role as the inverse temperature for the fermions. This is more clear is one traces over the initial/final positions

hA(~ x(τ ))i

R d~ y

−→ Tr

e ^{−L H}

A(~ x) Tr

e ^{−L H}

⁽¹⁶⁾

• The temperature of the fermions is mapped onto the inverse of the length T ferm = 1/L and the Fermi energy ε F = k ² _F /(2m) = π ² n ² T /c. The limit of the degenerate gas (T _ferm ε F ) corresponds to a high temperature regime for the lines T n ² L/c.

• In the very begining, we have assumed small distorsions ∂x/∂τ 1. Velocity is of the order of the Fermi velocity v _F = k _F /m = 2πnT /c. Small velocity v _F 1 requires small temperature for the lines T c/n.

We conclude that the validity of the T ferm = 0 calculation is n ² L/c T c/n.

• For finite temperature, we expect that the Kohn anomaly is smothed by the fermion tempe- rature as χ(2k F ) ∼ −ρ ₀ ln |ε _F /T ferm |. Using the mapping onto the parameters of the lines we get

χ(q = 2πn) ∼ − c

T n ln n ² T L/c

(17)