Free fermionsInteracting fermions

An out of equilibrium setup

Free fermions at the edge of interacting systems
Jean-Marie Stéphan
Camille Jordan Institute, University of Lyon 1, France

New trends in integrable systems
Osaka 2019
Based mostly on[JMS, SciPost Physics 2019][JMS, J. Stat. Mech 2017]

Free fermionsInteracting fermions

Outline1

Free fermions

2Interacting fermions

3

An out of equilibrium setup

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup


∂2
2
H=
dx ψ (x) − 2 + x ψ(x)∂xRZ

{ψ(x)† , ψ(y)} = δ(x − y)

†

,



{ψ(x)† , ψ(y)† } = {ψ(x), ψ(y)} = 0

Well known model in a confining potential, relevant for cold
atomic systems.
Exactly solvable, free fermions.
Single particle states can be obtained in explicit form [Hermite].

Free fermionsInteracting fermions



∂2
H=
dx ψ (x) − 2 + x ψ(x)
∂x
R
Z

†

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup



∂2
H=
dx ψ (x) − 2 + x ψ(x)
∂x
R
Z

†

R
Introduce the modes χ† (λ) = dx u(λ, x) ψ † (x). ThenZ
H=dλ(λ)χ† (λ)χ(λ)
R

provided∂2− 2 + x u(λ, x) = (λ)u(λ, x)∂xThe “single particle” wave functions are Airy functionsZ
dq i(qx+q3 /3)u(λ, x) = Ai(x+λ) , (λ) = −λ , Ai(x) =
e
.
2π
R

Free fermionsInteracting fermions

An out of equilibrium setup

Remarks[Pokrovsky & Talapov 1979, Prähofer & Spohn 2000, Spohn 2005]

The potential does not confine to a given region of space.
The ground state is a Dirac sea, with propagatorZ ∞†hψ (x)ψ(y)i =dλAi(x + λ)Ai(y + λ)0Particle number is infinite: on the interval [−a, ∞) it diverges
2 3/2
as 3π
a .
Full counting statistics (FCS) for charge Q(x) = ψ † (x)ψ(x)R∞

Υs (α) = heα s dx Q(x) i

Free fermionsInteracting fermions

An out of equilibrium setup

Full counting statistics in the ground state
FCS can be computed exactlyΥs (α)

=
Wick

=
=

1+
1+
1+

Z
∞
X
αn
n=1
∞
X
n=1
∞
X
n=1

def

=

n!

An

dx1 . . . dxn hQ(x1 ) . . . Q(xn )i(eα − 1)nn!

Z(eα − 1)n
n!

Z

An

An

dx1 . . . dxn h: Q(x1 ) . . . Q(xn ) :i
dx1 . . . dxn det (hψ † (xi )ψ(xj )i)
1≤i,j≤ndet(I + (eα − 1)GAiry )s

The quantity E(s) = limα→−∞ Υs (α) = dets (I − GAiry ) is theprobability that the interval A = [s, ∞) is empty of fermions.
This is called the Tracy-Widom distribution [Tracy & Widom, 1993].

Free fermionsInteracting fermions

An out of equilibrium setup

Tracy-Widom distribution

0.5
p(s) = dE
ds
0.4
0.3
0.2
0.1
0

−4

−3

−2

s

−1

0

1

Free fermionsInteracting fermions

An out of equilibrium setup

Back to the harmonic trap: semiclassical solution


∂2
2
H=
dx ψ (x) − 2 + x − µ ψ(x)
∂x
R
Z

†

Assume separation of scales. Around some point x0 , the single
particle ground state looks like the projector onto−d2
< µ − x20(dδx)2−→k 2 < µ − x20 = k(x0 )This is a disk in phase (k, x0 ) space. Thinking in Fourier, theprojection acts as a filter with sine-kernel response functionhψ † (x0 + δx)ψ(x0 + δy)i =sin [k(x0 )(δx − δy)]π(δx − δy)Free fermionsInteracting fermions

An out of equilibrium setup

Connection to random matrices|ϕ(x1 , . . . , xN )|2 ∝P 2Y(xi − xj )2 e− j xj
i<jwhich is the joint eigenvalue pdf for GUE. See [Eisler 2013; Calabrese,
Majumdar & Le Doussal 2014, Dean, Le Doussal, Majumdar & Schehr 2014, . . . ]p
Density hψ † (x0 )ψ(x0 )i = k(xπ 0 ) = π1 µ − x20 [Wigner semicircle law].
√
Here µ = 2N , where N is particle number.
What about the edge scaling close to µ = 2N ? From semiclassics([Praehoffer & Spohn 2000; Spohn 2006]):√
k 2 + ( µ + x)2 ≤ µ√
2⇒ k 2 + 2 µx H
+H
x

H≤ 0

Free fermionsInteracting fermions−

An out of equilibrium setupd2√+ 2 µx ≤ 02dxWith x = (4µ)−1/6 y, we get
−

d2
+y ≤0
dy 2

Hence, Airy kernel scaling at the edge, and Tracy-Widom follows.

Semiclassically, we go from a disk k 2 + x2 ≤ µ to a parabolic
region q 2 + y ≤ 0.

Free fermionsInteracting fermions

Another example: dimers

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

Dimer coverings on the Aztec diamond

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Free fermionsInteracting fermions

Arctic circle theorem

An out of equilibrium setup[Jockusch, Propp and Shor 1998]

Free fermionsInteracting fermions

An out of equilibrium setup

N 1/3 scaling near the edge: rightmost particle follows theTracy-Widom distribution [Johansson 2005]

Free fermionsInteracting fermions

What about interactions?

An out of equilibrium setup

Free fermionsInteracting fermionsInteracting dimers (repulsive)

An out of equilibrium setup

Free fermionsInteracting fermionsInteracting dimers (attractive)

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

Can also do six vertex model

a

a

b

∆=

b

a2 + b2 − c2
= 1 − eλ2abccFree fermionsInteracting fermions

An out of equilibrium setup

There is still a limit shape. Conjecture for the arctic curve insix vertex [Colomo & Pronko ]T-W scaling is difficult to establish with interactions, but
widely believed to be true.

Proof for a different model, e. g. ASEP with step initialconditions [Tracy & Widom, 2009].

Free fermionsInteracting fermions

An out of equilibrium setup

Why should the edge be free?
Bethe equations (e.g. six vertex model/XXZ spin chain)


e−ν


N
sinh (λj + iγ/2) L Y sinh(λj − λk + iγ)=
sinh (λj − iγ/2)
sinh(λj − λk − iγ)
k6=j

For γ = π/2, the rhs simplifies tofermion point.

Q

k6=j (−1). This is the free

Away from free fermions and for fixed N/L the Bethe
equations are not well understood. The Bethe roots densify
on an unknown curve in the complex plane.
However, close to the edge N/L → 0, and the solutions
become free fermions-like again.

Free fermionsInteracting fermions

An out of equilibrium setup

Lieb Liniger in a harmonic trap (repulsive)

Z
H=



~g
~2 ∂ 22− µ + x Ψ(x) + Ψ†2 (x)Ψ(x)2dxΨ (x) −
2
2m ∂x
2
†

Look at an ~ → 0 limit (still interacting).

The trapping potential breaks integrability, but assuming LDA, thedensity profile may be computed using thermodynamic Bethe
ansatz (TBA). [Dunjko, Laurent & Olshanii 2001; Gangardt & Shlyapnikov 2003;
Brun & Dubail 2018]

Free fermionsInteracting fermions

An out of equilibrium setup

TBA for the homogeneous ground state (V (x) = 0):
Z kF

dq
1
V (k, q)ρ(q, µ) =2π
−kF 2πZ kF
dq
V (k, q)(q, µ) = k 2 − µ(k, µ) −2π−kF

ρ(k, µ) −Ask (kF , µ) = 0. The kernel V is known explicitly, and vanishes√
for free fermions, in which case we recover kF (µ) = µ.

LDA just tells us to replace µ → µ(x) = µ − x2 .

Free fermionsInteracting fermions

An out of equilibrium setup

Edge scaling
Claim: can look at the edge scaling using TBA considerations. In
that case interactions only provide extra subleading corrections,compared to free fermions.

√d2µx ≤ 0, which exactly
So TW scaling still holds, we find − dx
2 +2
the same result irrespective of interactions for the harmonic trap.

Physically, interacting particles are diluted near the edge, they justrenormalize to free fermions. In terms of Luttinger parameter,
K(x) varies with position, but K(xe ) = 1.

Free fermionsInteracting fermions

An out of equilibrium setup

XXZ in a varying magnetic fieldXyxz
Sxx Sx+1+ Sxy Sx+1
+ ∆Sxz Sx+1
+ h(x/R)Sxz

x∈Z+1/2

The edge is at xe = Rh−1 (1 + ∆). We find the edge scaling−

h0 (h−1 (1 + ∆))
d2
+
2
x<0
dx2
R

Tracy-Widom scaling occurs on a scale `∆ now, with
1/3
R
`∆ =
2h0 (h−1 (1 + ∆))Variance now depends on interactions, through ∆.

Free fermionsInteracting fermions

An out of equilibrium setup

Numerical checks

Density profile1

Sxz + 1/2

0.8

∆ = −0.4∆ = 0.4∆=1∆ = 1.7

0.6
0.4
0.2
0

−2

−1

0

x/R

1

2

Free fermionsInteracting fermions

An out of equilibrium setup

Distribution of the rightmost down spin from the emptinessQ(1+2Sjz )formation probability Ex = h ∞
j=x i
2
0.5

TW
∆ = 0.4, R = 256∆ = 1, R = 256∆ = 1.8, R = 256

0.4
0.3
0.2
0.1
0

−4

−3

−2

−1

(x − xe )/`∆

01

Free fermionsInteracting fermions

An out of equilibrium setup

An exception: Calogero-Sutherland

H=

N
X
j=1

∂2
− 2 + x2j
∂xj!
+

X β(β/2 − 1)i6=j(xi − xj )2

Luttinger parameter is known to be K = 2/β, constant,everywhere in the domain. So the edge is not free for β 6= 2.
Ground state wave function|ϕ(x1 , . . . , xN )|2 ∝Yi<j(xi − xj )β e−

2
j xj

P

which is exactly the eigenvalue pdf for β-ensemble in randommatrix theory, which leads to β-deformed Tracy-Widom.

Free fermionsInteracting fermions

Out of equilibrium setups

An out of equilibrium setup

Free fermionsInteracting fermions

The Hamiltonian (or Trotter) limit

An out of equilibrium setup

Free fermionsInteracting fermions

The Hamiltonian (or Trotter) limit

An out of equilibrium setup

Free fermionsInteracting fermions

The Hamiltonian (or Trotter) limit

An out of equilibrium setup

Free fermionsInteracting fermions

The Hamiltonian (or Trotter) limit

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

The Hamiltonian (or Trotter) limit

In six vertex language, set a = 1 first, and choose a value of ∆.

Then set b = R/N , vertical spacing ∆y = 1/N and take the limit
N → ∞.
The partition function becomes
Z(R) = hψ|eRHXXZ |ψiwhere |ψi = |. . . ↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓ . . .i

Free fermionsInteracting fermions

An out of equilibrium setup

Six vertex model with domain wall boundary conditions
Introduced by [Korepin 1982]. Determinant formula [Izergin 1987]
2[sin ]NZ = QN −1
det
2 0≤i,j≤N −1
k=0 k!

Z ∞−∞1 − e−γudu ui+j e−u
1 − e−πuThen take the Hamiltonian limit [JMS, J. Stat. Mech 2017]
1

2

Z(τ ) = heτ H i = e− 24 (τ sin γ) det(I − V )Bα (x, y) =√V (x, y) = B0 (x, y) ω(y)
√
√
√
√
√
yJα ( x)Jα0 ( y) − xJα ( y)Jα0 ( x)2(x − y)ω(y) = Θ(y) −1 − e−γy/(2τ sin γ)
1 − e−πy/(2τ sin γ)

Free fermionsInteracting fermions

An out of equilibrium setup

Asymptotics
cos γ = ∆ < 1

Easiest: use [Zinn-Justin 2000] [Bleher, Fokin 2006]π2(τ sin γ)2− log Z(τ ) =−1+ O(log τ )(π − γ)224
Interpretation: free energy of the fluctuating region.

Free fermionsInteracting fermions

An out of equilibrium setup

The Wick rotation (conjectures)
For t ∈ R, R(t) = |Z(it)|2 is a (quantum) return probability.
Root of unity, γ = πpq
− log R(t) =q2(t sin γ)2+ O(log t)−12(q − 1)
12

Coincides with analytic continuation of the asymptotic result
only when p = 1.
non root of unity− log R(t) = t sin γ + O(log t)

Free fermionsInteracting fermions|Ψ(t)i = e−itH |Ψ0 i

H=

X

,

An out of equilibrium setup|Ψ0 i = |. . . ↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓ . . .i

12
3
Sx1 Sx+1+ Sx2 Sx+1
+ ∆Sx3 Sx+1x∈ZFree fermion case (∆ = 0) [Antal, Rácz, Rákos, and Schütz, 1999, . . . ]Interactions: Numerics [Gobert, Kollath, Schollwöck & Schütz 2005,. . . ]Generalized hydrodynamics (GHD) framework (|∆| < 1, ballistic)[Castro-Alvaredo, Doyon, Yoshimura 2016] [Bertini, Collura, De Nardis, Fagotti 2016]√
This quench: light cone xe (t) = t 1 − ∆2 [JMS, 2017]
Full density profile from GHD [De Luca, Collura, Viti 2017]

Free fermionsInteracting fermions

An out of equilibrium setup

0.5
0.03
0.02

0.4
hSxz (t)i + 1/2

0.01
0

0.3

1

xf

xe

0.2

∆=0
∆=
√1/2
∆ = 2/2∆ = 0.8
GHD

0.1
0

1/2

0

1
x/t

GHD
√edge
xe = t 1 − ∆2free edge
xf = tContrary to previous situations density profile is linear near thelight cone, compared to previous root behavior.

Free fermionsInteracting fermions

An out of equilibrium setup

Tracy-Widom edge for ∆ = 0 (free fermions) [Eisler & Racz 2013]

From GHD considerations in the bulk, [De Luca, Collura & Viti 2017]
guessed a new (diffusive) kernel for the edge. In our language, thisreads free fermions k + x/t < 0.
Fastest quasi-particle goes as x ' t [Sabetta & Misguich 2013].
[Bulchandani & Karrasch 2018] observed t1/3 scaling near this other front,
which they interpreted as signature of Tracy-Widom.

Subleading corrections to GHD are generically diffusive in the bulk
[De Nardis, Bernard & Doyon 2018]

Free fermionsInteracting fermions

An out of equilibrium setup

Rescaled density profiles√

X = x−t √1−∆t

2

Rescaled density decays as ρ(X  1) ∼ 4π12 X

0.45

t = 60t = 120t = 240
toy-model

0.4√∆ = 1/ 2

0.35
0.3
E(X)

0.25
0.2

0.15
0.1
0.05
0

−2

−1

0
X

√
X = (x − xe )/ t

1

2

Free fermionsInteracting fermions

An out of equilibrium setup

Rescaled distribution of the last particle (fat tails)√∆ = 1/ 2
0.5
0.4

t = 60t = 120t = 240t = 60t = 120t = 2400.3
0.6

xe 0.8

xf

x/t

x/t

0.2
0.1
0

−4

−3

−2

−1

0
X

1

√
X = (x − xe )/ t

2

3

4

Free fermionsInteracting fermions

An out of equilibrium setup

Skewnesssk =h(x−hxi)3 i.
h(x−hxi)2 i3/2

sk ' 0.22408 for Tracy-Widom.

1.8

∆=0
∆ = 1/2∆=√
0.6∆ = 1/ 2∆ = 0.8
T-W

1.6
1.4
skewness

1.2
1
0.8
0.6
0.4
0.2
0

100

1000
t

Free fermionsInteracting fermions

An out of equilibrium setup

Not clear if it’s free fermions at the edge in that case.

√
The tail on the right of xe = t 1 − ∆2 is very long. Skewness
appears to diverge.

Hard cutoff at xLR = t, due to Lieb-Robinson type bounds.
t1/3 behavior near x = t is the right tail of the quantum
delocalization of the rightmost particle. Not Tracy-Widom.

Free fermionsInteracting fermions

Exact computations?

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermionsL-shaped partition function and Emptiness

An out of equilibrium setup

Free fermionsInteracting fermions

An out of equilibrium setup

L-shaped partition function and EmptinessEN (r, s) =ZN (r, s)
ZN (0, 0)

EN (a, b, ∆|r, s) can be computed exactly. [Colomo & Pronko 2007]

2

Thermodynamic limit: EN ∼ e−N f (r/N,s/N )The function f is not known. However it vanishes on thearctic curve, which is known exactly. [Colomo & Pronko 2009]

Free fermionsInteracting fermions

An out of equilibrium setup

Time-dependent Emptiness formation probability
y/R
y/R
R
ẼR (x, y) = lim EN (a, b = N, ∆|r = N −x+N
, s = N +x+N
)
2
2
N →∞

Problem: The Colomo Pronko formula is a multiple integral type
formula, and the number of integrals explodes in the Hamiltonian
limit.

However, it is possible to write it as [Colomo unpublished]
I
I
dwx Y
dw1
...
φj,k det(. . .) det(. . .)
C 2iπ
C 2iπ
1≤j<k≤xwhere there are x integrals. This is more suitable for Hamiltonian
limit.

Free fermionsInteracting fermions

An out of equilibrium setup

Summary
The edge of several inhomogeneous interacting systems’renormalizes’ to free fermions. The most typical example inthat case is Tracy-Widom.

Quantum quench problems provide us with new edge
universality classes, which are worth exploring.

Free fermionsInteracting fermionsThank you!

An out of equilibrium setup