Improved lattice actions & operators for non-relativistic - - PowerPoint PPT Presentation

Improved lattice actions & operators for non-relativistic fermions

Joaquín E. Drut

Los Alamos National Laboratory

EMMI workshop GSI, Darmstadt, April 2012

JETLab (Duke)

Ultracold Gases Astrophysics High-Energy Physics, QCD, Low-Energy NP Condensed Matter Physics Materials Science

JETLab (Duke)

Ultracold Gases Astrophysics (mostly neutron stars) Condensed Matter Physics Materials Science High-Energy Physics, QCD, Low-Energy NP

Strongly correlated quantum many-body systems

Ultracold Atoms

JET Lab (Duke) MIT

S-wave scattering length Inter-particle distance Range of the interaction

Spin 1/2 fermions, at unitarity

r0 → 0 ≪ n−1/3 ≪ |a| → ∞

The unitary limit

S-wave scattering length Inter-particle distance Range of the interaction

As many scales as a free gas! Qualitatively

r0 → 0 ≪ n−1/3 ≪ |a| → ∞

kF = (3π2n)1/3

εF = 2 2m(3π2n)2/3

Every dimensionful quantity should come as a power

• f times a universal constant/function.

εF

Spin 1/2 fermions, at unitarity

The unitary limit

S-wave scattering length Inter-particle distance Range of the interaction

As many scales as a free gas! Qualitatively

r0 → 0 ≪ n−1/3 ≪ |a| → ∞

kF = (3π2n)1/3

εF = 2 2m(3π2n)2/3

Every dimensionful quantity should come as a power

• f times a universal constant/function.

εF

Spin 1/2 fermions, at unitarity

The unitary limit

S-wave scattering length Inter-particle distance Range of the interaction

As many scales as a free gas! Qualitatively Quantitatively

r0 → 0 ≪ n−1/3 ≪ |a| → ∞

kF = (3π2n)1/3

εF = 2 2m(3π2n)2/3

Every dimensionful quantity should come as a power

• f times a universal constant/function.

εF

?

Spin 1/2 fermions, at unitarity

The unitary limit

Normal Normal

T

BEC Superfluid

?

BCS Superfluid

Unitary regime

?

Tc Tc

1/kF a

1 ≪ kF |a|

The BCS-BEC Crossover

Energy update (ground state)

Ground state energy per particle

Endres et al.

Drut, Lähde, Wlazlowski, Magierski, arXiv:1111.5079

Finite T equation of state (theory & experiment)

Energy update (finite temperature)

Experiment: Zwierlein et

• al. (MIT)

Accepted PRA(R)

Energy update (finite temperature)

Drut, Lähde, Wlazlowski, Magierski, arXiv:1111.5079 Experiment: Zwierlein et

• al. (MIT)

Finite T equation of state (theory & experiment)

Accepted PRA(R)

The Tan relations and the “contact”

k → ∞

nk → C/k4

Momentum distribution tail

• S. Tan, Annals of Physics 323, 2952 (2008).
• E. Braaten and L. Platter,
• Phys. Rev. Lett. 100, 205301 (2008).

Energy relation

k → ∞

nk → C/k4

Momentum distribution tail

• S. Tan, Annals of Physics 323, 2952 (2008).
• E. Braaten and L. Platter,
• Phys. Rev. Lett. 100, 205301 (2008).

The Tan relations and the “contact”

Energy relation Short distance density-density correlator

k → ∞

nk → C/k4

Momentum distribution tail

• S. Tan, Annals of Physics 323, 2952 (2008).
• E. Braaten and L. Platter,
• Phys. Rev. Lett. 100, 205301 (2008).

The Tan relations and the “contact”

Energy relation Adiabatic relation Pressure relation

k → ∞

nk → C/k4

Momentum distribution tail

• S. Tan, Annals of Physics 323, 2952 (2008).
• E. Braaten and L. Platter,
• Phys. Rev. Lett. 100, 205301 (2008).

The Tan relations and the “contact”

Short distance density-density correlator

Experiment

• J. T. Stewart et al

PRL 104, 235301 (2010)

Plateau seen both in theory and experiment! T/TF = 0 - 0.5

Momentum distribution

2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 4 4.5

3 !2 (k/kF)4 n(k) k/kF

Nx = 10, T/"F = 0.186 0.231 0.321 0.001 0.01 0.1 1 1 2 3 4 5 n(k) k/kF T/"F = 0.186 C/k4

Theory (lattice)

• J. E. Drut, T. A. Lähde, T. Ten
• Phys. Rev. Lett. 106, 205302 (2011)

Growth at low T

T-matrix: Palestini et al.

PRA 82, 021605 (2010)

Decrease at high T Maximum around T≅0.4TF

Virial expansion:

Yu, Bruun & Baym PRA 80, 023615 (2009) Hu, Liu, & Drummond, arXiv:1011.3845

Improved T-matrix: Enss et al.

doi 10.1016/j.aop.2010.10.002

Finite density effects? What happens in the crossover?

What do we know so far?

• J. E. Drut, T. A. Lähde, T. Ten
• Phys. Rev. Lett. 106, 205302 (2011)

Recent technical developments

Dealing with systematic effects

Finite volume Finite lattice spacing

Related to each other Induce finite-range effects In general we have... But we want...

  

... at unitarity

Dealing with systematic effects

Finite volume Finite lattice spacing

Related to each other Induce finite-range effects In general we have... But we want...

Canʼt do that with only one parameter!

  

Point-like interaction Transfer matrix ... at unitarity Effective range remains finite!

Dealing with systematic effects

We need a “richer” HS transformation Typically... Now... Endres et al. multiple papers.

e.g. using Lüscherʼs formula

s-wave phase shift Energy eigenvalues in a box (no lattice)

Highly improved actions

Dealing with systematic effects

p cot δ = S(E) πL

E = p2/m

e.g. using Lüscherʼs formula Highly improved actions

s-wave phase shift Energy eigenvalues in a box (no lattice) (scattering experiment information) (theory information)

Dealing with systematic effects

p cot δ = S(E) πL

E = p2/m

e.g. using Lüscherʼs formula Highly improved actions

s-wave phase shift Energy eigenvalues in a box (no lattice) (scattering experiment information)

Decide what scattering parameters you need

(theory information)

Dealing with systematic effects

p cot δ = S(E) πL

E = p2/m

e.g. using Lüscherʼs formula Highly improved actions

s-wave phase shift Energy eigenvalues in a box (no lattice) (scattering experiment information)

Decide what scattering parameters you need Tune your Hamiltonian accordingly

(theory information)

Dealing with systematic effects

p cot δ = S(E) πL

E = p2/m

e.g. using Lüscherʼs formula Highly improved actions

s-wave phase shift Energy eigenvalues in a box (no lattice) (scattering experiment information)

Decide what scattering parameters you need Tune your Hamiltonian accordingly Profit!

(theory information)

Dealing with systematic effects

p cot δ = S(E) πL

E = p2/m

Highly improved actions Adding more parameters to the transfer matrix and tuning via Lüscherʼs formula... JED arXiv:1203.2565 Endres et al. multiple papers. Improved transfer matrix

Dealing with systematic effects

Highly improved actions & operators Adding more parameters to the transfer matrix and tuning via Lüscherʼs formula... JED arXiv:1203.2565 Energy

Dealing with systematic effects

Highly improved actions & operators Adding more parameters to the transfer matrix and tuning via Lüscherʼs formula... JED arXiv:1203.2565 Contact

Dealing with systematic effects

A more direct way to the contact...

At T=0... At finite T... In both cases we need

Results: GS Energy

JED arXiv:1203.2565

Results: GS Contact

JED arXiv:1203.2565

Where do we go from here?

We have implemented these improved actions and

• perators in our finite-temperature codes.

We are reassessing our previous calculations in the light of new ones done with these new tools. We are simultaneously pursuing the calculation of response functions (specific heat, compressibility, susceptibility, viscosities).

Summary & conclusions

Strongly interacting Fermi gases have universal properties Studying the universal regime requires non-perturbative numerical approaches such as Quantum Monte Carlo and Lattice QCD-type tools. Cond-mat, Nucl-th, Hep-th, Hep-lat are all interested in these problems! (again universality)

Summary & conclusions

Strongly interacting Fermi gases have universal properties Studying the universal regime requires non-perturbative numerical approaches such as Quantum Monte Carlo and Lattice QCD-type tools. Cond-mat, Nucl-th, Hep-th, Hep-lat are all interested in these problems! (again universality) Much is known (much more than shown here) but much remains to be done! New tools are needed... Precise determination of equilibrium and linear-response quantities is largely in its infancy (just a couple of exceptions).

We are entering a “precision” era!

Precision is required to understand the physics, in some cases even qualitatively!

Thank you!