The semiclassical method in interacting many body systems Path - - PowerPoint PPT Presentation

The semiclassical method in interacting many body systems

Path integrals, fields and particles Quirin Hummel, Benjamin Geiger, Juan Diego Urbina, Klaus Richter Quantum chaos: methods and applications, March 2015

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Motivation: The subject (fields)

Subject: Quantum Fields with large number of excitations (N → ∞) So far:

→ Finite (Trying to do scattering) → Lattice (Trying to get continuum) → Non-Relativistic (Dreaming about QCD) → Isolated (Trying to do Feynman-Vernon)

interactions

→ General

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Motivation: The goal (fields)

Goal: Study analytically non-perturbative effects in eff = 1/N Use only classical info (actions, stabilities, etc) from a classical (nonlinear) field equation Which properties?: dynamics (a van Vleck -Gutzwiller propagator) spectrum (a Gutzwiller trace formula) Thermodynamics How?: For the moment, using universality:

→ Due to single-particle chaos (Mesoscopic Boson Sampling) → Due to field chaos (Thomas Engl, Peter Schlagheck)

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Break: summarizing fields

Semiclassical propagator a la Gutzwiller: Write K(ψin, ψfin, t) = ´ D[ψ(s)]eiR[ψ(s)]/eff Define classical limit δψR[ψ(s)] = 0 and B.C. Evaluate in Stationary Phase Approximation Careful:

→ Coherent states (Bosons, Fermions,Klauder) → bad classical limit → Extra conditions (large densites, gauge symmetries)

van Vleck-Gutzwiller propagator (Bosons, Fermions) Gutzwiller trace formula (Bosons)

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Motivation: The subject (particles)

Subject: fixed number of identical particles (N ∼ O(10)) simple external potentials:

→ free quantum gases with periodic boundary conditions → quantum billiards → harmonic traps → other homogeneous potentials

isolated or in a thermal bath interactions between particles

→ model: contact-interaction

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Motivation: The goal (particles)

Goal: Study properties related to MB-spectrum analytically and non-perturbatively avoid numerical calculation of MB-energy levels and power series in interaction strength Which properties?: the spectrum itself canonical partition function equation of state spatial properties:

→ (non-local) pair correlations in (micro-)canonical ensemble → spatial particle densities near boundaries/impurities

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Motivation: The goal (particles)

Goal: Study properties related to MB-spectrum analytically and non-perturbatively avoid numerical calculation of MB-energy levels and power series in interaction strength Which properties?: the spectrum itself canonical partition function equation of state spatial properties:

→ (non-local) pair correlations in (micro-)canonical ensemble → spatial particle densities near boundaries/impurities

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

7 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

7 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

7 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

7 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

Esp independent particles = ⇒ EMB many-body

7 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Many-body(MB)-spectrum: General consideration

around ground state: mean-field approaches work

→ effectively independent particles (HF, . . . )

but: no single mean field for all excitations! (MCSCF, . . . ) Esp independent particles

✘✘ ❳❳

= ⇒ EMB many-body

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Simplification

Simplification: Forget about discreteness of spectrum!

50 100 150 200 xaxis yaxis

N 12

0.01 1 100 104

E[̺−1

0 ]

N(E) N = 1/2

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Overview

1 Motivation 2 Particle Exchange Symmetry

Method Thermodynamics

3 Contact-Interaction

Lieb-Liniger (LL) Model Method: Two Particles Results: Two Particles

4 Quantum Cluster Expansion (QCE)

Method QCE for LL Thermodynamics in QCE

5 Conclusion and Outlook

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Single particle(sp) Weyl expansion

1 particle, D-dim. billiard ̺sp(E) = FT t ˆ dqK(q, q; t)

• (E)

V = 0 V = ∞

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Single particle(sp) Weyl expansion

1 particle, D-dim. billiard ̺sp(E) = FT t ˆ dqK(q, q; t)

• (E)

V = 0 V = ∞

smooth part ¯ ̺ ↔ short time behaviour of K ¯ ̺sp(E) = const. · VD E

D 2 −1

• locally free

− const. · SD−1 E

D−1 2

−1

• reflection on flat boundary

+ . . . → basic geometric properties!

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

MB Weyl expansion for bosons (+) or fermions (−)

N identical particles

q = (q1, . . . , qN) ̺

(N) ± (E) = FT t

ˆ dq K

(N) ± (q, q; t)

• (E)

K

(N) ± (q, q; t) = 1

N!

• P

(±1)PK(N)(Pq, q; t) 2 particles, 1D

q Pq ±

first particle second particle

smooth part ¯ ̺±, non-interacting:

• Q. Hummel, J. D. Urbina and K. Richter, J. Phys. A: Math. Theor.

47, 015101 (2014):

P-contributions ↔ Weyl-like corrections

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

MB Weyl expansion for bosons (+) or fermions (−)

N identical particles

q = (q1, . . . , qN) ̺

(N) ± (E) = FT t

ˆ dq K

(N) ± (q, q; t)

• (E)

K

(N) ± (q, q; t) = 1

N!

• P

(±1)PK(N)(Pq, q; t) 2 particles

smooth part ¯ ̺±, non-interacting:

• Q. Hummel, J. D. Urbina and K. Richter, J. Phys. A: Math. Theor.

47, 015101 (2014):

P-contributions ↔ Weyl-like corrections

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

MB Weyl expansion for bosons (+) or fermions (−)

N identical particles

q = (q1, . . . , qN) ̺

(N) ± (E) = FT t

ˆ dq K

(N) ± (q, q; t)

• (E)

K

(N) ± (q, q; t) = 1

N!

• P

(±1)PK(N)(Pq, q; t) N particles

smooth part ¯ ̺±, non-interacting:

• Q. Hummel, J. D. Urbina and K. Richter, J. Phys. A: Math. Theor.

47, 015101 (2014):

P-contributions ↔ Weyl-like corrections

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

MB Weyl expansion for non-int. bosons (+) or fermions (−)

¯ ̺±(E) =

N

• l1,l2=1

Cl1,l2 V l1

D Sl2 D−1 El1D/2+l2(D−1)/2−1

12 fermions, 2D, counting function N(E) = ´ dE ¯ ̺(E)

105 110 115 120 1 100

50 100 150 200 0.01 1 100 104 106 xaxis yaxis

N 12

E[̺−1

0 ]

N(E) EGS EGS N = 1/2

N = 1/2 12 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

MB Weyl expansion for non-int. bosons (+) or fermions (−)

¯ ̺+(E) =

N

• l=1

Cl Ll El/2−1 7 bosons, 1D, counting function N(E) = ´ dE ¯ ̺(E)

5

1

15

2

xaxis yaxis

N 12

1 1

1

E[̺−1

0 ]

N(E) N = 1/2

12 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Switching between domains

Relation between E, t, β domains: energy domain E: time domain t:

• inv. temperature β:

(imaginary time β = i

t)

¯ ̺(E)

smooth part of DOS

tr K(t) = ´ dE¯ ̺(E)e− i

Et

short time propagation

Z(β) = LE[¯ ̺(E)](β)

”high“ temperature behaviour of canonical par- tition function

V/λD

T N ,

λT = √4πβ

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Ideal quantum gas in D-dim. canonical:

(our approach)

grand canonical:

(std. textbook)

Z(β) = N

l=1 zl

V

λD

T

l , λT = √4πβ ”high“ temperature ↔ no discreteness ln ZG = ∓

i ln(1 ∓ e−β(ǫi−µ))

≈ ∓ ´ dǫ¯ ̺sp(ǫ) ln(1 ∓ e−β(ǫ−µ)) = ± V

λD

T Li D 2 +1(±eβµ) , Lis(x) = ∞ k=1 xk ks

no discreteness as well! → ”equal footing“, different ensembles

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Equation of state (EOS) for a given N canonical:

(our approach)

grand canonical:

(std. textbook)

P(V, T, N) = kBT

V N

l=1 zll

• V

λD T

l

N

l=1 zl

• V

λD T

l P(V, T, µ) = ± kBT

λD

T Li D 2 +1(±eβµ)

< N >= ± V

λD

T Li D 2 (±eβµ) 15 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Equation of state (EOS) for a given N canonical:

(our approach)

grand canonical:

(std. textbook)

P(V, T, N) = kBT

V N

l=1 zll

• V

λD T

l

N

l=1 zl

• V

λD T

l P(V, T, µ) ∼ c1z+c2z2+c3z3+· · · , z = eβµ < N >∼ c1z + 2c2z2 + 3c3z3 + · · ·

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Equation of state (EOS) for a given N canonical:

(our approach)

grand canonical:

(std. textbook)

P(V, T, N) = kBT

V N

l=1 zll

• V

λD T

l

N

l=1 zl

• V

λD T

l P = nkBT(1 + a2nλD

T + a3n2λ2D T

+ · · · ) → high temperature! → convergence?

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Ideal Bose gas, 1D-box, EOS for N = 3

1 2 3 4 1 2 3 4 5 6 L ΛT P k BT ΛT

Symmetry1, D1, N3, nCO15

canonical exact virial 3rd order virial 5th order virial 9th order

1 2 3 4 5 L ΛT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n 0

16 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Ideal Bose gas, 1D-box, EOS for N = 5

2 4 6 8 1 2 3 4 5 6 L ΛT P k BT ΛT

Symmetry1, D1, N5, nCO24

canonical exact virial 8th order virial 11th order virial 14th order

2 4 6 8 L ΛT 1 2 3 4 5 n 0

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Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

Ideal Bose gas, 1D-box, EOS for N = 10

5 10 15 1 2 3 4 5 6 L ΛT P k BT ΛT

Symmetry1, D1, N10, nCO34

canonical exact virial 8th order virial 11th order virial 14th order

5 10 15 L ΛT 2 4 6 8 10 n 0

16 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

”. . . In view of the present experimental status the canonical and microcanonical descriptions of the BEC are of primary importance. Recent BEC experiments on harmonically trapped atoms of dilute gases deal with a finite and well defined number of particles. . . . “

• V. V. Kocharovsky et al, Advances in Atomic, Molecular and Optical Physics 53, 291 (2006)

17 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics: Canonical vs. grand canonical

• W. J. Mullin and J. P. Fernandez, Am. J. Phys. 71, 661 (2003):

18 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Overview

1 Motivation 2 Particle Exchange Symmetry 3 Contact-Interaction

Lieb-Liniger (LL) Model Method: Two Particles Results: Two Particles

4 Quantum Cluster Expansion (QCE) 5 Conclusion and Outlook

19 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Lieb-Liniger model

N bosons in 1D, xi ∈ [0, L], periodic boundary conditions ˆ H = −

N

• i=1

∂2 ∂x2

i

+ √ 8α

• i<j

δ(xi − xj) exactly solvable (Bethe ansatz): EI =

i k2 i

Lki = 2πIi − 2

N

• j=1

arctan ki − kj √ 2α

• ,

i = 1, . . . , N where I1 < · · · < IN are

• integer

N odd half-integer N even describes ultracold atoms

20 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 particles, 1D, contact-interaction

ˆ H = − ∂2 ∂q2

1

+ ∂2 ∂q2

2

• +

√ 8αδ(q1 − q2) , α > 0

smooth part (t → 0, small β): no reflections from boundary ⇓ CM (R): free propagator K0 rel (r): propagator Kδ of δ-potential placed in free space

21 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 particles, 1D, contact-interaction

distinguishable case: K(2)(q′, q; t) = K(CM) (R′, R; t) · K(rel)

δ

(r′, r; t)

22 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 particles, 1D, contact-interaction

distinguishable case: K(2)(q′, q; t) = K(CM) (R′, R; t) · K(rel)

δ

(r′, r; t) with K(rel)

δ

(r′, r; t) = K(rel) (r′, r; t) + K(rel)

α

(r′, r; t) , where K(rel)

α

(r′, r; t) = −α

1 2

4πit 1

2 ˆ ∞

du e−( α

2 ) 1 2 u+ i 8t (|r′|+|r|+u)2 22 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 particles, 1D, contact-interaction

distinguishable: K(2) = K(2) + K(CM) K(rel)

α

indistinguishable: K(2)

± = K(2) 0,± + 1 2(1 ± 1)K(CM)

K(rel)

α

tr K(2)(q, q; t) = tr K(2)(P12q, q; t) = + + No Feynman diagrams!

23 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

• 5

1

• 15

2

• 2

4 6

• 1

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 10−5

α → 0

(free bosons)

α = 10−5 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ✁ 1

15 ✁ 2

2 ✁ 4 ✁ 6 ✁

1 ✁ ✁

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 2

α → 0

(free bosons)

α = 2 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ✄ 1

15 ✄ 2

2 ✄ 4 ✄ 6 ✄

1 ✄ ✄

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 10

α → 0

(free bosons)

α = 10 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ✆ 1

15 ✆ 2

2 ✆ 4 ✆ 6 ✆

1 ✆ ✆

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 50

α → 0

(free bosons)

α = 50 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ✞ 1

15 ✞ 2

2 ✞ 4 ✞ 6 ✞

1 ✞ ✞

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 250

α → 0

(free bosons)

α = 250 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ✠ 1

15 ✠ 2

2 ✠ 4 ✠ 6 ✠

1 ✠ ✠

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 1000

α → 0

(free bosons)

α = 1000 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

5 ☛ 1

15 ☛ 2

2 ☛ 4 ☛ 6 ☛

1 ☛ ☛

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 106

α → 0

(free bosons)

α = 106 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

1 ✌

1 ✌

1

1

1

1

1 ✌

5 ✌ ✌ ✌ 5 ✌5 ✌ 51 ✌ ✌ 515 ✌

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 10

α → 0

(free bosons)

α = 10 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

1 ✍

1 ✍

1

1

1

1

1 ✍

5 ✍ ✍ ✍ 5 ✍5 ✍ 51 ✍ ✍ 515 ✍

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 250

α → 0

(free bosons)

α = 250 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

2 bosons, counting function N(E) =

´ dE ¯ ̺(E) ¯ N (2)

+ (E) = L2

8π θ(E) + √ 2L 4π √ Eθ(E)

• non−int.

−2 √ 2L 4π √ Eθ(E) + 2 √ 2L 4π ( √ E + α − √α)θ(E)

1 ✏

1 ✏

1

1

1

1

1 ✏

5 ✏ ✏ ✏ 5 ✏5 ✏ 51 ✏ ✏ 515 ✏

E[ 2

2m 4π L2 ]

N(E)

N = 2, α = 1000

α → 0

(free bosons)

α = 1000 α → ∞

(free fermions)

24 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Overview

1 Motivation 2 Particle Exchange Symmetry 3 Contact-Interaction 4 Quantum Cluster Expansion (QCE)

Method QCE for LL Thermodynamics in QCE

5 Conclusion and Outlook

25 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

path integral representation:

K(N)(qf, qi; t) = ˆ qf

qi

D[q(s)]

N

• k=1

e

i

• ´ t

m 2 [ ˙

qk(s)]2ds k<l

e− i

• ´ t

0 V (qk(s)−ql(s))ds

”Mayer functionals“ fkl:

1 + fkl := e− i

• ´ t

0 V (qk(s)−ql(s))ds

expand:

• k<l

(1 + fkl) = 1 +

• k<l

fkl + . . .

⇓

K(N)(qf, qi; t) = K(N) (qf, qi; t) +

• k<l

K(N−2) (qf

kl, qi kl; t)∆K(2)(qf kl, qi kl; t) + . . . 26 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

with permutations:

ˆ dNqK(N)(Pq, q; t)

P = id: · usual cluster expansion · direct p-integration in classical Z(β)

27 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

with permutations:

ˆ dNqK(N)(Pq, q; t)

P = id: · need ”quantum cluster expansion“ · yields traces like

• r

27 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

with permutations:

ˆ dNqK(N)(Pq, q; t)

P = id: · need ”quantum cluster expansion“ · yields traces like

• r

· δ-int.: solution in t/β-domain involves special function (Owen’s T-f.) · but: L-1

β [. . .] has elementary solution!

27 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

e.g. 3 bosons (

2 2mL2 = 1 4π)

¯ N (3)

0,+(E) =

2 9√πE3/2θ(E) + 1 √ 8Eθ(E) + 2 √ 27πE1/2θ(E) ∆1 ¯ N (3)

+ (E) = −1

2Eθ(E) − √ 2α π E1/2θE − 8 √ 27πE1/2θ(E) − 2√α √π θ(E) + √ 2 π (E + α) arctan

• E

α

• θ(E)

+ 8 π3/2 (E + α)(3E + 4α)−1/2 arctan

• 3 + 4α

E

• θ(E)

28 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✒ ✒

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 1

α → 0

(free bosons)

α = 1

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✔ ✔

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 5

α → 0

(free bosons)

α = 5

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✖ ✖

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 25

α → 0

(free bosons)

α = 25

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✘ ✘

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 125

α → 0

(free bosons)

α = 125

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✚ ✚

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 500

α → 0

(free bosons)

α = 500

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

5

1

15

2

1

2 ✜ ✜

4

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = ∞

α → 0

(free bosons)

α = ∞

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

1600 1650 1

1

1800

8000 8500 9000 9500 10 000

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 5

α → 0

(free bosons)

α = 5

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

1600 1650 1700 1750 1800 7500 8000 8500 9000 9500 10 000

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 50

α → 0

(free bosons)

α = 50

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

3 bosons, counting function N(E) =

´ dE ¯ ̺(E)

1600 1650 1700 1750 1800 7500 8000 8500 9000 9500 10 000

E[ 2

2m 4π L2 ]

N (3)(E)

N = 3, α = 500

α → 0

(free bosons)

α = 500

(”QCE“)

α → ∞

(free fermions ∼ Tonks-Girardeau)

29 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

more bosons (keep αN 2 constant)

50 100 150 200 200 400 600 800 1000 1200

E[ 2

2m 4π L2 ]

N (4)(E)

N = 4, α = 6.25

α → 0

(free bosons)

α = 6.25

(”QCE“)

α → ∞

(free fermions)

30 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

more bosons (keep αN 2 constant)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 4

α → 0

(free bosons)

α = 4

(”QCE“)

α → ∞

(free fermions)

30 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

more bosons (keep αN 2 constant)

50 100 150 200 1000 2000

5000

E[ 2

2m 4π L2 ]

N (6)(E)

N = 6, α = 2.78

α → 0

(free bosons)

α = 2.78

(”QCE“)

α → ∞

(free fermions)

30 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

more bosons (keep αN 2 constant)

600 620 640 660 680 700 5

E[ 2

2m 4π L2 ]

N (6)(E)

N = 6, α = 2.78

α → 0

(free bosons)

α = 2.78

(”QCE“)

α → ∞

(free fermions)

30 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 1

α → 0

(free bosons)

α = 1

(”QCE“)

α = 1

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 4

α → 0

(free bosons)

α = 4

(”QCE“)

α = 4

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 10

α → 0

(free bosons)

α = 10

(”QCE“)

α = 10

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 30

α → 0

(free bosons)

α = 30

(”QCE“)

α = 30

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 100

α → 0

(free bosons)

α = 100

(”QCE“)

α = 100

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

N bosons, ”quantum cluster expansion“

5 bosons, counting function N(E) =

´ dE ¯ ̺(E)

50 100 150 200 500 1000 1500 2000 2500

E[ 2

2m 4π L2 ]

N (5)(E)

N = 5, α = 300

α → 0

(free bosons)

α = 300

(”QCE“)

α = 300

(”QCE“ of TG)

α → ∞

(free fermions)

31 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

Scaling of canonical partition function in QCE (general) Z(V, β, N, α) =

N

• l=1

[ zl

• non−int.

+ ∆zl(βα)

• QCE

] V λD

T

l

32 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

Scaling of canonical partition function in QCE (general) Z(V, β, N, α) =

N

• l=1

[ zl

• non−int.

+ ∆zl(βα)

• QCE

] V λD

T

l ⇒ P(V, β, N, α) = kBT V N

l=1 l[zl + ∆zl(βα)]

V

λD

T

l N

l=1[zl + ∆zl(βα)]

V

λD

T

l → simple polynomial / rational structure remained

32 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 2

4 6 8 10 50 100 150 200 L ΛT Z

N3, s0.01, eTmaxConst130, eTmaxOffset4000

• 2

4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst150, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 10 20 30 40 50 L ΛT Z

N3, s0.01, eTmaxConst150, eTmaxOffset4000

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 0.6 0.8 1.0 1.2 1.4 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst150, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 10 20 30 40 50 L ΛT Z

N3, s0.05, eTmaxConst230, eTmaxOffset4000

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 0.6 0.8 1.0 1.2 1.4 L ΛT P k BT ΛT

N3, s0.05, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 10 20 30 40 50 L ΛT Z

N3, s0.1, eTmaxConst150, eTmaxOffset4000

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 0.6 0.8 1.0 1.2 1.4 L ΛT P k BT ΛT

N3, s0.1, eTmaxConst150, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 10 20 30 40 50 L ΛT Z

N3, s0.25, eTmaxConst150, eTmaxOffset4000

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 0.6 0.8 1.0 1.2 1.4 L ΛT P k BT ΛT

N3, s0.25, eTmaxConst150, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 10 20 30 40 50 L ΛT Z

N3, s0.5, eTmaxConst230, eTmaxOffset4000

• 3.0

3.5 4.0 4.5 5.0 5.5 6.0 0.6 0.8 1.0 1.2 1.4 L ΛT P k BT ΛT

N3, s0.5, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.01, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 0.0

0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT Z

N3, s0.01, eTmaxConst230, eTmaxOffset4000

• 0.0

0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.05, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.05, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.1, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.1, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.25, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.25, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.5, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.5, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact Dashed: Z = e−βE0 + e−βE1

l ¯

zl(V/λT)l → fully analytic!

33 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Conclusion and Outlook

Constructed analytic Weyl law / finite EOS for . . . 2 bosons with contact interaction (exact)

→ agreement with numerics

N bosons with contact interaction in terms of a first order ”quantum“ cluster expansion

→ agreement in regimes of ”weak“ and infinite interaction or high excitation, ”high“ temperature → improvement by splitting off lowest level

Projects: use 2-body info to all orders apply techniques to dynamical impurity model

(N non-int. fermions + 1 int. particle → is QCE ”complete“?)

N spin-1/2 fermions (Luttiger liquid physics) Harmonic confinement (not solvable!) 3,5,7,... body terms from Yang-Yang solution of Lieb-Linniger.

34 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.01, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

• id. cl. Gas,

Z = e−βE0 + 2e−βE1

35 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 0.0

0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT Z

N3, s0.01, eTmaxConst230, eTmaxOffset4000

• 0.0

0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.01, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

• id. cl. Gas,

Z = e−βE0 + 2e−βE1

35 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.25, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.25, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

• id. cl. Gas,

Z = e−βE0 + 2e−βE1

35 / 35

Motivation Particle Exchange Symmetry Contact-Interaction Quantum Cluster Expansion (QCE) Conclusion and Outlook

Thermodynamics of QCE

EOS for Lieb-Liniger model N = 3 βα = 0.01 0.05 0.1 0.25 0.5

• 1

2 3 4 5 5 10 15 20 25 30 L ΛT Z

N3, s0.5, eTmaxConst230, eTmaxOffset4000

• 1

2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 L ΛT P k BT ΛT

N3, s0.5, eTmaxConst230, eTmaxOffset4000

QCE, Tonks-Girardeau,

• id. Bose Gas,

exact

• id. cl. Gas,

Z = e−βE0 + 2e−βE1

35 / 35