Free fermions and -determinantal processes Fabio Deelan Cunden - - PowerPoint PPT Presentation

Free fermions and α-determinantal processes

Fabio Deelan Cunden (University College Dublin) based on J. Phys. A: Math. Theor. 52, 165202 (2019) joint work with Satya N. Majumdar (Paris) and Neil O’Connell (Dublin) Statistical Mechanics Seminar - University of Warwick, Jan 23, 2020

1 / 23

Determinantal point processes

A point process or ‘random point configuration’ on R can be described in terms of its correlation functions (when they exist) ̺n(x1, . . . , xn) = lim

δ→0

P (exactly one particle in (xi, xi + δ), for all i = 1, . . . , n) δn

2 / 23

Determinantal point processes

A point process or ‘random point configuration’ on R can be described in terms of its correlation functions (when they exist) ̺n(x1, . . . , xn) = lim

δ→0

P (exactly one particle in (xi, xi + δ), for all i = 1, . . . , n) δn Determinantal / fermionic processes (Macchi 70’s) are point processes with ̺n(x1, . . . , xn) = det

1≤i,j≤nK(xi, xj),

n ∈ N. The function K(x, y) is called correlation kernel. Macchi-Soshnikov criterion: If K is self-adjoint, locally trace class, with 0 ≤ K ≤ 1, then the random point configuration exists and is unique (and exhibits repulsion). Examples:

• Free fermions (Pauli exclusion principle);
• Eigenvalues of random matrices (Dyson, Mehta, Gaudin - level repulsion);
• Non-intersecting paths.

2 / 23

α-determinantal point processes

Let A be a n × n matrix.and α a scalar. The α-determinant of A (Vere-Jones, 1988) is det A =

• σ∈Sn

sgn(σ)A1σ(1)A2σ(2) · · · Anσ(n) where m(σ) = # disjoint cycles in the permutation σ. We simply replace the signature sgn(σ) = (−1)n−m(σ) by αn−m(σ) in the definition

• f the ordinary determinant. It is clear that

det−1 A = det A, det1 A = per A, det0 A = A11A22 · · · Ann.

3 / 23

α-determinantal point processes

Let A be a n × n matrix.and α a scalar. The α-determinant of A (Vere-Jones, 1988) is det A =

• σ∈Sn

(−1)n−m(σ)A1σ(1)A2σ(2) · · · Anσ(n) where m(σ) = # disjoint cycles in the permutation σ. We simply replace the signature sgn(σ) = (−1)n−m(σ) by αn−m(σ) in the definition

• f the ordinary determinant. It is clear that

det−1 A = det A, det1 A = per A, det0 A = A11A22 · · · Ann.

3 / 23

α-determinantal point processes

Let A be a n × n matrix and α a scalar. The α-determinant of A (Vere-Jones, 1988) is detαA =

• σ∈Sn

αn−m(σ)A1σ(1)A2σ(2) · · · Anσ(n) where m(σ) = # disjoint cycles in the permutation σ. We simply replace the signature sgn(σ) = (−1)n−m(σ) by αn−m(σ) in the definition

• f the ordinary determinant. It is clear that

det−1 A = det A, det1 A = per A, det0 A = A11A22 · · · Ann. An α-determinantal point process (Shirai and Takahashi, 2003) with kernel K is defined, when it exists, as the point process with n-point correlation functions ̺n(x1, . . . , xn) = detα

1≤i,j≤nK(xi, xj).

α = −1: determinantal process; α = 1: permanental process; α = 0: Poisson process (with intensity K(x, x)).

3 / 23

α-determinantal point processes

An α-DPP with kernel K is defined, when it exists, as the point process with n-point correlation functions ̺n(x1, . . . , xn) = detα

1≤i,j≤nK(xi, xj).

Existence criterion when α < 0 and K is self-adjoint and locally trace class:

4 / 23

α-determinantal point processes

An α-DPP with kernel K is defined, when it exists, as the point process with n-point correlation functions ̺n(x1, . . . , xn) = detα

1≤i,j≤nK(xi, xj).

Existence criterion when α < 0 and K is self-adjoint and locally trace class:

• The point process exists (and is unique) iff − 1

α ∈ N and 0 ≤ −αK ≤ 1.

4 / 23

α-determinantal point processes

An α-DPP with kernel K is defined, when it exists, as the point process with n-point correlation functions ̺n(x1, . . . , xn) = detα

1≤i,j≤nK(xi, xj).

Existence criterion when α < 0 and K is self-adjoint and locally trace class:

• The point process exists (and is unique) iff − 1

α ∈ N and 0 ≤ −αK ≤ 1.

• The α-DPP is a superposition (union) of − 1

α i.i.d. DPPs with kernel −αK.

4 / 23

Free fermions in a harmonic potential

N spin-polarized noninteracting fermions in a one-dimensional harmonic potential. Suppose that the system is in a stationary state (an eigenstate of H).

5 / 23

Free fermions in a harmonic potential

N spin-polarized noninteracting fermions in a one-dimensional harmonic potential. Suppose that the system is in a stationary state (an eigenstate of H). Qestions:

• 1. Find the particle density of the system.
• 2. Find the pair correlation function.
• 3. Find the n-point correlation function.
• 4. Study the particle statistics asymptotics as N → ∞.

Fact: the number statistics of free fermions forms a determinantal point process.

5 / 23

Qantum harmonic oscillator

Single particle Schr¨

• dinger equation:
• − ∂2

∂x2 + x2 4

• ψ(x) = Eψ(x),

x ∈ R. Solutions: ψk(x) = hk(x)e−x2/4, Ek = k + 1/2, k ∈ Z+

6 / 23

Qantum harmonic oscillator

Single particle Schr¨

• dinger equation:
• − ∂2

∂x2 + x2 4

• ψ(x) = Eψ(x),

x ∈ R. Solutions: ψk(x) = hk(x)e−x2/4, Ek = k + 1/2, k ∈ Z+ Non-interacting fermions: antisymmetric superpositions of the ψk’s Ψk1,...,kN (x1, . . . , xN) = 1 √ N! det

1≤i,j≤N ψki(xj),

with 0 ≤ k1 < k2 < · · · < kN. They are normalised eigenfunctions of the Hamiltonian H =

• i
• − ∂2

∂x2

i

+ x2

i

4

• with eigenvalues E = k1 + · · · + kN + N/2

in the subspace of completely antisymmetric states Ψ(xσ(1), . . . , xσ(N)) = sgn(σ)Ψk1,...,kN (x1, . . . , xN).

6 / 23

Free fermions in a harmonic potential

Denote by J = {k1, . . . , kN} ⊂ Z+ the occupied levels. |ΨJ(x1, . . . , xN)|2 = 1 N! det

1≤i,j≤N ψki(xj)

det

1≤i,j≤N ψki(xj) = 1

N! det

1≤i,j≤N KJ(xi, xj),

KJ(x, y) =

• k∈J

ψk(x)ψk(y) : integral kernel of projection onto span {ψk(x): k ∈ J}

7 / 23

Free fermions in a harmonic potential

Denote by J = {k1, . . . , kN} ⊂ Z+ the occupied levels. |ΨJ(x1, . . . , xN)|2 = 1 N! det

1≤i,j≤N ψki(xj)

det

1≤i,j≤N ψki(xj) = 1

N! det

1≤i,j≤N KJ(xi, xj),

KJ(x, y) =

• k∈J

ψk(x)ψk(y) : integral kernel of projection onto span {ψk(x): k ∈ J} ΨJ defines a determinantal point process on R with correlation kernel KJ(x, y): ̺n(x1, . . . , xn) = N! (N − n)! ˆ |ΨJ(x1, . . . , xN)|2dxn+1 · · · dxN = N! (N − n)! ˆ 1 N! det

1≤i,j≤N KJ(xi, xj)dxn+1 · · · dxN

= det

1≤i,j≤n KJ(xi, xj)

(by Mehta-Gaudin integration lemma)

7 / 23

Free fermions in a harmonic potential

Particle density: ̺1(x) = KJ(x, x) Pair correlation fcn: ̺2(x1, x2) = det KJ(x1, x1) KJ(x1, x2) KJ(x2, x1) KJ(x2, x2)

• = KJ(x1, x1)KJ(x2, x2) − |KJ(x1, x2)|2

. . . ̺n(x1, . . . , xn) = det

1≤i,j≤n KJ(xi, xj)

. . . with KJ(x, y) =

• k∈J

ψk(x)ψk(y), J = {k1, . . . , kN}.

8 / 23

Free fermions in a harmonic potential

Particle density: ̺1(x) = KJ(x, x) Pair correlation fcn: ̺2(x1, x2) = det KJ(x1, x1) KJ(x1, x2) KJ(x2, x1) KJ(x2, x2)

• = KJ(x1, x1)KJ(x2, x2) − |KJ(x1, x2)|2

. . . ̺n(x1, . . . , xn) = det

1≤i,j≤n KJ(xi, xj)

. . . with KJ(x, y) =

• k∈J

ψk(x)ψk(y), J = {k1, . . . , kN}. Large-N asymptotics of ̺n(x1, . . . , xn)? Need to understand the asymptotics of the correlation kernel KJ(x, y).

8 / 23

Ground state: J = {0, 1, . . . , N − 1}

9 / 23

Ground state and the GUE eigenvalue process

Suppose that J = [0 . . N): ΨGS(x1, . . . , xN) = 1 √ N! det

1≤i,j≤N ψi−1(xj).

This is the ground state. The corresponding projection kernel KJ(x, y) =

N−1

• k=0

ψk(x)ψk(y) is the kernel of the Gaussian Unitary Ensemble of random matrix theory.

10 / 23

Ground state and the GUE eigenvalue process

Suppose that J = [0 . . N): ΨGS(x1, . . . , xN) = 1 √ N! det

1≤i,j≤N ψi−1(xj).

This is the ground state. The corresponding projection kernel KJ(x, y) =

N−1

• k=0

ψk(x)ψk(y) is the kernel of the Gaussian Unitary Ensemble of random matrix theory. Let X be a N × N random complex Hermitian matrix with independent standard Gaussian entries. Then, the eigenvalues of X have joint probability density p(x1, . . . , xN) = |ΨGS(x1, . . . , xN)|2. (from the Weyl denominator formula. Mehta, Gaudin, Dyson, ...)

10 / 23

Ground state and the GUE eigenvalue process: asymptotics

Using Christoffel-Darboux formula for orthogonal polynomials, KJ(x, y) =

N−1

• k=0

ψk(x)ψk(y) = √ N ψN(x)ψN−1(y) − ψN−1(x)ψN(y) x − y . (semi)Classical asymptotics of orthogonal polynomials ψk(x) = e−x2/4hk(x).

11 / 23

Ground state and the GUE eigenvalue process: asymptotics

Using Christoffel-Darboux formula for orthogonal polynomials, KJ(x, y) =

N−1

• k=0

ψk(x)ψk(y) = √ N ψN(x)ψN−1(y) − ψN−1(x)ψN(y) x − y . (semi)Classical asymptotics of orthogonal polynomials ψk(x) = e−x2/4hk(x). Macroscopic particle density: Wigner semicircle law ̺1(x) = e−x2/2

N−1

• k=0

hk(x)2 N→∞ ∼

11 / 23

Ground state and the GUE eigenvalue process: asymptotics

Using Christoffel-Darboux formula for orthogonal polynomials, KJ(x, y) =

N−1

• k=0

ψk(x)ψk(y) = √ N ψN(x)ψN−1(y) − ψN−1(x)ψN(y) x − y . (semi)Classical asymptotics of orthogonal polynomials ψk(x) = e−x2/4hk(x). Macroscopic particle density: Wigner semicircle law ̺1(x) = e−x2/2

N−1

• k=0

hk(x)2 N→∞ ∼ 1 2π √ 4N − x2.

=

• 10
• 5

5 10 0.2 0.4 0.6 0.8 1.0

Figure: Density for N = 10 fermions in a harmonic trap. Comparison with the semicircular distribution.

11 / 23

Ground state and the GUE eigenvalue process: asymptotics

Microscopic scaling limit: sine process (Dyson, Mehta, Constin-Lebowitz, etc.) lim

N→∞

1 ̺1(0)KJ

• x

̺1(0), y ̺1(0)

• =

12 / 23

Ground state and the GUE eigenvalue process: asymptotics

Microscopic scaling limit: sine process (Dyson, Mehta, Constin-Lebowitz, etc.) lim

N→∞

1 ̺1(0)KJ

• x

̺1(0), y ̺1(0)

• = sin π(x − y)

π(x − y) = ⇒ ˜ ̺n(x1, . . . , xn) = lim

N→∞

1 (̺1(0))n ̺n x1 ̺1(0), . . . , xn ̺1(0)

• =

det

1≤i,j≤N

sin π(xi − xj) π(xi − xj) . Ex: ˜ ̺1(x) = 1, ˜ ̺2(x1, x2) = 1 − sin π(x1 − x2) π(x1 − x2) 2 (Friedel oscillations)

12 / 23

Ground state and the GUE eigenvalue process: asymptotics

Microscopic scaling limit: sine process (Dyson, Mehta, Constin-Lebowitz, etc.) lim

N→∞

1 ̺1(0)KJ

• x

̺1(0), y ̺1(0)

• = sin π(x − y)

π(x − y) = ⇒ ˜ ̺n(x1, . . . , xn) = lim

N→∞

1 (̺1(0))n ̺n x1 ̺1(0), . . . , xn ̺1(0)

• =

det

1≤i,j≤N

sin π(xi − xj) π(xi − xj) . Ex: ˜ ̺1(x) = 1, ˜ ̺2(x1, x2) = 1 − sin π(x1 − x2) π(x1 − x2) 2 (Friedel oscillations)

Figure: Lef: two-point corr. function. Right: T. Jeltes et al. (Orsay + Amsterdam), Nature 445, 402 (2007).

12 / 23

Excited states: J = {k1, k2, . . . , kN} k1 < k2 < · · · < kN

13 / 23

Block projection process: asymptotics

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

KJ(x, y) =

(a+1)2M−1

• k=a2M

ψk(x)ψk(y)

14 / 23

Block projection process: asymptotics

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

KJ(x, y) =

(a+1)2M−1

• k=a2M

ψk(x)ψk(y) =

(a+1)2M−1

• k=0

ψk(x)ψk(y) −

a2M−1

• k=0

ψk(x)ψk(y)

14 / 23

Block projection process: asymptotics

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

KJ(x, y) =

(a+1)2M−1

• k=a2M

ψk(x)ψk(y) =

(a+1)2M−1

• k=0

ψk(x)ψk(y) −

a2M−1

• k=0

ψk(x)ψk(y) Large-M asymptotics: particle density ̺1(x) = KJ(x, x)

M→∞

∼

14 / 23

Block projection process: asymptotics

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

KJ(x, y) =

(a+1)2M−1

• k=a2M

ψk(x)ψk(y) =

(a+1)2M−1

• k=0

ψk(x)ψk(y) −

a2M−1

• k=0

ψk(x)ψk(y) Large-M asymptotics: particle density ̺1(x) = KJ(x, x)

M→∞

∼ 1 2π

• 4(a + 1)2M − x2 −

√ 4a2M − x2

• .

14 / 23

Block projection process: asymptotics

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

KJ(x, y) =

(a+1)2M−1

• k=a2M

ψk(x)ψk(y) =

(a+1)2M−1

• k=0

ψk(x)ψk(y) −

a2M−1

• k=0

ψk(x)ψk(y) Large-M asymptotics: particle density ̺1(x) = KJ(x, x)

M→∞

∼ 1 2π

• 4(a + 1)2M − x2 −

√ 4a2M − x2

• .

For large a, the one-point function approaches the arcsine law ̺1(x)

a→∞

∼ (2a + 1)M π 1 √ 4a2M − x2 ,

14 / 23

Excited states and block projection process

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

k=a2M k=(a+1)2M

ℕ

̺1(x) = KJ(x, x)

M→∞

∼ 1 2π

• 4(a + 1)2M − x2 −

√ 4a2M − x2

• .

a=0

• 10
• 5

5 10 0.2 0.4 0.6 0.8 1.0 a=1

• 15
• 10
• 5

5 10 15 0.5 1.0 1.5 a=2

• 20
• 10

10 20 0.5 1.0 1.5 2.0 a=5

• 40
• 20

20 40 0.5 1.0 1.5 2.0 2.5 3.0 3.5

15 / 23

Excited states and block projection process

Microscopic scaling limit: lim

M→∞

1 ̺1(0)KJ

• x

̺1(0), y ̺1(0)

• = k(x − y).

k(x − y) = sin (π(a + 1)(x − y)) π(x − y) − sin (πa(x − y)) π(x − y) = sin π

2 (x − y) π 2 (x − y)

cos ω(x − y), where ω = π (a + 1/2).

16 / 23

Excited states and block projection process

Microscopic scaling limit: lim

M→∞

1 ̺1(0)KJ

• x

̺1(0), y ̺1(0)

• = k(x − y).

k(x − y) = sin (π(a + 1)(x − y)) π(x − y) − sin (πa(x − y)) π(x − y) = sin π

2 (x − y) π 2 (x − y)

cos ω(x − y), where ω = π (a + 1/2). What is the large a asymptotics of the correlation functions? Examples: ˜ ̺1(x) = 1, ˜ ̺2(x1, x2) = 1 − sin π

2 (x1 − x2) π 2 (x1 − x2)

2 cos2 ω(x1 − x2)

a=0

0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 a=2

0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 a=1

0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 a=5

0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0

16 / 23

Block projection process: asymptotics

˜ ̺n(x1, . . . , xn) = det

1≤i,j≤n k(xi − xj) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

k(xσ(i) − xi) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

sin π

2 (xi − xσ(i)) π 2 (xi − xσ(i)) n

• i=1

cos ω(xσ(i) − xi)

17 / 23

Block projection process: asymptotics

˜ ̺n(x1, . . . , xn) = det

1≤i,j≤n k(xi − xj) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

k(xσ(i) − xi) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

sin π

2 (xi − xσ(i)) π 2 (xi − xσ(i)) n

• i=1

cos ω(xσ(i) − xi) For large a, the product of cosines becomes lim

ω→∞ n

• i=1

cos ω(xσ(i) − xi) = 1 2 n−m(σ) .

17 / 23

Block projection process: asymptotics

˜ ̺n(x1, . . . , xn) = det

1≤i,j≤n k(xi − xj) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

k(xσ(i) − xi) =

• σ∈Sn

(−1)n−m(σ)

n

• i=1

sin π

2 (xi − xσ(i)) π 2 (xi − xσ(i)) n

• i=1

cos ω(xσ(i) − xi) For large a, the product of cosines becomes lim

ω→∞ n

• i=1

cos ω(xσ(i) − xi) = 1 2 n−m(σ) . Therefore, as a → ∞, the process is α-determinantal with α = −1/2. ˜ ̺n(x1, . . . , xn)

a→∞

→

• σ∈Sn
• −1

2 n−m(σ)

n

• i=1

sin π

2 (xi − xσ(i)) π 2 (xi − xσ(i))

= det−1/2

1≤i,j≤n

sin π

2 (xi − xj) π 2 (xi − xj)

, in a weak sense. The union of two independent sine processes.

17 / 23

Scaling limits and α-determinantal processes

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1.0 2-point function

a0=3 a0=∞ Figure: Two-point correlation function and its weak limit.

18 / 23

Scaling limits and α-determinantal processes

Summarising: KJ(x, y) =

• k∈J

ψk(x)ψk(y). Suppose that the set of energy levels is J =

• a2M . . (a + r)2M
• , with r positive
• integer. There are two cases for the rescaled processes in the bulk:
• If a = 0, then

lim

M→∞ ̺1(0)−n

det

1≤i,j≤n KJ(̺1(0)−1xi, ̺1(0)−1xj) = det−1 1≤i,j≤n

sin π(xi − xj) π(xi − xj) ;

• If a > 0, then

lim

a→∞ lim M→∞ ̺1(0)−n

det

1≤i,j≤n KJ(̺1(0)−1xi, ̺1(0)−1xj) = det− 1

2

1≤i,j≤n

sin π

2 (xi − xj) π 2 (xi − xj)

How to find a suitable limit procedure to obtain α-determinantal processes out of KJ(x, y) where α = − 1

m, with m generic positive integer?

19 / 23

Scaling limits and α-determinantal processes

This can be generalised to set of occupied levels J with several blocks B ≥ 1.

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | a02M (a0+r)2M

1 2 B-1 . . .

a12M a22M aB-12M (a1+r)2M (a2+r)2M (aB-1+r)2M

J of even type

ℕ

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | a12M a22M aB-12M (aB-1+r)2M (a2+r)2M (a1+r)2M (r/2)2M

1 2 B-1 . . .

J of odd type

ℕ

20 / 23

Scaling limits and α-determinantal processes

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | a02M (a0+r)2M

1 2 B-1 . . .

a12M a22M aB-12M (a1+r)2M (a2+r)2M (aB-1+r)2M

J of even type

ℕ

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | a12M a22M aB-12M (aB-1+r)2M (a2+r)2M (a1+r)2M (r/2)2M

1 2 B-1 . . .

J of odd type

ℕ

Theorem (C., Majumdar, O’Connell 2018) The scaling limit of the DPP with kernel KJ is an α-determinantal point process ˜ ̺n(x1, . . . , xn)

ai→∞

→ detα

1≤i,j≤n

sin πα(xi − xj) πα(xi − xj) with α = − 1 2B

• r

α = − 1 2B − 1, depending on whether J is of even or odd type. Union of − 1

α i.i.d. sine processes.

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Scaling limits and α-determinantal processes

Var

• #
• −L

2 , L 2

• =

ˆ L/2

−L/2

KJ(x, x)dx − ¨ L/2

−L/2

KJ(x, y)2dxdy. lim

ai→∞ lim M→∞ Var

• #
• −

L 2̺1(0), L 2̺1(0)

• = L + α

¨ L/2

−L/2

sin απ(x − y) απ(x − y) 2 dxdy

α=-1/4 α=-1/3 α=-1/2 * * * * * * * * * □ □ □ □ □ □ □ □ □

○ ○ ○ ○ ○ ○ ○ ○ ○ 1 2 3 4

L

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Number variance * B=2 : a0=1 a1=4 r=1 □ B=2 : a0=0 a1=4 r=2

○ B=1 : a0=2

r=1

Figure: Number variance Var

• #
• −

L 2̺1(0) , L 2̺1(0)

• .

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Scaling limits and α-determinantal processes

x p x ρ(x)

Slits (Complement of the Fermi shells) Diffraction/Interference pattern Wigner function in phase space Number density in x

Figure: Scheme for the correspondence principle (cf. diffraction paterns in wave optics).

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Take-home message

• The particle statistics of free fermions defines a determinantal point processes;
• We studied the particle statistics of free fermions in certain classes of

stationary states (block projection processes). A particular case is the ground state that describes the eigenvalue statistics of Gaussian random matrices;

• Various asymptotics can be analysed using the Christoffel-Darboux formula;
• The scaling limit in the bulk is in general not determinantal (fermionic) but

rather α-determinantal with −1/α ∈ N; the limit process can be interpreted as a union of −1/α independent sine processes.

References

• O. Macchi, Adv. Appl. Probab. 7, 83 (1975).
• D. Vere-Jones, Linear Algebra Appl. 63, 267 (1988).

M.L. Mehta, Random matrices (1991).

• T. Shirai and Y. Takahashi, J. Funct. Anal. 205, 414 (2003).
• K. Johannson, Probab. Theory Relat. Fields 138, 75 (2007).
• S. Torquato, A. Scardicchio and C.E. Zachary, J. Stat. Mech. P11019 (2008).
• M. Ledoux, Commun. Stoch. Anal. 2, 27 (2008).
• E. Vicari, Phys. Rev. A 85, 062104 (2012).
• V. Eisler, Phys. Rev. Let. 111, 080402 (2013).
• D. S. Dean, P. Le Doussal, S.N. Majumdar and G. Schehr, Phys. Rev. Let. 114, 110402 (2015).
• F. D. Cunden, F. Mezzadri and N. O’Connell, J. Stat. Phys. 171, 768 (2018).

D.S. Dean, P. Le Doussal, S.N. Majumdar and G. Schehr, J. Phys. A: Math. Theor. 52, 144006 (2019). F.D. Cunden, S.N. Majumdar and N. O’Connell, J. Phys. A: Math. Theor. 52,165202 (2019).

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