Fermions on a Lattice in the Unitary Regime at Finite Temperatures Aurel Bulgac, Joaquin E. Drut and Piotr Magierski University of Washington, Seattle, WA Also in Warsaw From a talk of J.E. Thomas (Duke) Outline What is the unitary regime? The two-body problem, how one can manipulate the two-body interaction? Feshbach resonance Brief overview of existing theoretical understanding Path integral Monte Carlo for many fermions on the lattice at finite temperatures Conclusions What is the Holy Grail of this field? Fermionic superfluidity! erconductivity and superfluidity in Fermi syste 20 orders of magnitude over a century of (low temperature) physics Dilute atomic Fermi gases 10-9 eV Liquid 3He

eV Metals, composite materials 10-2 eV Nuclei, neutron stars 106 eV Tc 10-12 Tc 10-7 Tc 10-3 Tc 105 units (1 eV 104 K) What is the unitary regime? A gas of interacting fermions is in the unitary regime if the average separation between particles is large compared to their size (range of interaction), but small compared to their scattering length. The system is very dilute, but strongly interacting! n r0 3 1 n |a|3 1 n - number density r0 n-1/3 F /2 |a|

r0 - range of interaction a - scattering length Feshbach resonance 2 2 p hf Z H (Vi Vi ) V0 (r ) P0 V1 (r ) P1 V d 2 r i 1 ahf e n hf V 2 S S , V Z (e S ze n S zn ) B Tiesinga, Verhaar, Stoof Phys. Rev. A47, 4114 (1993) Regal and Jin Phys. Rev. Lett. 90, 230404 (2003) Channel coupling

Bertsch Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin fermions interacting via a zero-range, infinite scattering-length contact interaction. Why? Besides pure theoretical curiosity, this problem is relevant to neutron stars! In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not! A number of people argued that under such conditions fermionic matter is unstable. - systems of bosons are unstable (Efimov effect) - systems of three or more fermion species are unstable (Efimov effect) Baker (winner of the MBX challenge) concluded that the system is stable. See also Heiselberg (entry to the same competition) Carlson et al (2003) Fixed-Node Green Function Monte Carlo and Astrakharchik et al. (2004) FN-DMC provided the best theoretical estimates for the ground state energy of such systems. Thomas Duke group (2002) demonstrated experimentally that such systems are (meta)stable. Expected phases of a two species dilute Fermi system BCS-BEC crossover T High T, normal atomic (plus a few molecules) phase Strong interaction weak interaction

BCS Superfluid a<0 no 2-body bound state ? weak interactions Molecular BEC and Atomic+Molecular Superfluids a>0 1/a shallow 2-body bound state halo dimers Early theoretical approach Eagles (1969), Leggett (1980) gs u k +v k ak a k vacuum BCS wave function k 1 m 1

4 2 a k 2 k 2E k k n 2 1 Ek k 8 exp F e2 2 k a F E k ( k ) 2 2 2k 2 k 2m

1 u 2k v 2k 1, v 2k = 1 k 2 Ek gap equation number density equation pairing gap quasi-particle energy Consequences: Usual BCS solution for small and negative scattering lengths, with exponentially small pairing gap For small and positive scattering lengths this equations describe a gas a weakly repelling (weakly bound/shallow) molecules, essentially all at rest (almost pure BEC state) r1 , r2 , r3 , r4 ,... A (r12 ) (r34 )... In BCS limit the particle projected many-body wave function has the same structure (BEC of spatially overlapping Cooper pairs) For both large positive and negative values of the scattering length these equations predict a smooth crossover from BCS to BEC, from a gas of spatially large Cooper pairs to a gas of small molecules

What is wrong with this approach: The BCS gap (a<0 and small) is overestimated, thus the critical temperature and the condensation energy are overestimated as well. In BEC limit (a>0 and small) the molecule repulsion is overestimated The approach neglects of the role of the meanfield (HF) interaction, which is the bulk of the interaction energy in both BCS and unitary regime All pairs have zero center of mass momentum, which is reasonable in BCS and BEC limits, but incorrect in the unitary regime, where the interaction between pairs is strong !!! (this situation is similar to superfluid 4He) Fraction of non-condensed pairs (perturbative result)!?! nex 8 3 nm amm , n0 3 n nm , 2 amm 0.6a

From a talk of Stefano Giorgini (Trento) Fixed-Node Green Function Monte Carlo approach at T=0 BCS Gorkov 8 exp F e2 2 k a F 2 e 7/3

F exp 2 k a F 3 E FG F 5 Carlson et al. PRL 91, 050401 (2003) Chang et al. PRA 70, 043602 (2004) Finite Temperatures Grand Canonical Path-Integral Monte Carlo H N 2 2

3 T V d x ( x ) ( x ) ( x ) ( x ) g

d x n ( x ) n ( x ) 2m 2m d 3 x n ( x ) n ( x ) ,

ns ( x ) s ( x ) s ( x ), s , 3 Trotter expansion (trotterization of the propagator) Z ( ) Tr exp H N Tr exp H N 1 Tr H exp H N Z (T ) 1 N (T ) Tr N exp H N Z (T )

E (T ) N , 1 N T No approximations so far, except for the fact that the interaction is not well defined! How to implement the path integral?

Put the system on a spatio-temporal lattice! A short detour Let us consider the following one-dimensional Hilbert subspace (the generalization to more dimensions is straightforward ) P 2 P projector in this Hilbert subspace sin (x y) l dk x P y exp[ik(x y)] , (x y) 2 l l (x) P x x , | (x ) (x ) K

(x nl ) sin N (x) c (x) O(exp( cN )) (nl ) l 1 n (x nl ) l 1 1 c dx (x) (x) (x ), x nl K K Littlejohn et al. J. Chem. Phys. 116, 8691 (2002) Schroedinger equation N (x) d F (x) O(exp( cN )) 1 1

F (x) (x), x nl, K F | F F |T | F V (x ) d Ed Recast the propagator at each time slice and put the system on a 3d-spatial lattice, in a cubic box of side L=Nsl, with periodic boundary conditions exp H N exp T N / 2 exp( V ) exp T N / 2 O( 3 )

Discrete Hubbard-Stratonovich transformation exp( V ) x 1 1 ( x ) An ( x )

1 ( x ) An ( x ) , ( x ) 1 2 -fields fluctuate both in space and imaginary time mkc m 1

, 2 2 2 4 a g 2 kc l Running coupling constant g defined by lattice A exp( g ) 1 n(k) 2/LL L box size l - lattice spacing kmax=/Ll k How to choose the lattice spacing and the box size

ky /l /l kx 2 2 F , , T 2ml 2 22 2 mL2 22 2 F , mL2 2/L L N sl Momentum space 2 p L Z(T ) D (x, ) T rU({ })

x, U({ }) T exp{ [h({ }) ]} D (x, )T rU({ }) E (T ) x, Z(T ) One-body evolution operator in imaginary time ({ }) T r HU T rU({ }) T rU({ }) {det[1 U({ })]}2 exp[ S({ })] 0 No sign problem! U({ })

* (y), n (x, y) n (x, y) k(x) l k,l kc 1 U ({ }) k l exp(ik x) k(x) V All traces can be expressed through these single-particle density matrices More details of the calculations: Lattice sizes used from 63 x 300 (high Ts) to 63 x 1361 (low Ts) 83 running (incomplete, but so far no surprises) and larger sizes to come Effective use of FFT(W) makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large matrices Update field configurations using the Metropolis importance sampling algorithm Change randomly at a fraction of all space and time sites the signs the auxiliary fields (x,) so as to maintain a running average of the acceptance rate between 0.4 and 0.6

Thermalize for 50,000 100,000 MC steps or/and use as a start-up field configuration a (x,)-field configuration from a different T At low temperatures use Singular Value Decomposition of the evolution operator U({}) to stabilize the numerics Use 100,000-2,000,000 (x,)- field configurations for calculations MC correlation time 250 300 time steps at T Tc a = Superfluid to Normal Fermi Liquid Transition Normal Fermi Gas (with vertical offset, solid line) Bogoliubov-Anderson phonons and quasiparticle contribution (dot-dashed lien ) Bogoliubov-Anderson phonons contribution only (little crosses) People never consider this ??? Quasi-particles contribution only (dashed line) 4 3 3 4 T Ephonons (T ) F N ,

5 16 s3/ 2 F s 0.44 3 5 2 3T Equasi-particles (T ) F N exp 4 5 2 F T 2 e 7/3 F exp 2 k a

F - chemical potential (circles) C s (Tc ) 2 C n (Tc ) (2.43 in BCS) S E T 3 F (n)N e 5 F (n) N kF3 2kF2 n 2 , F (n) V 3 2m T

2 S N , P (n)n ( n ) 3 F E (n)nV Conclusions Fully non-perturbative calculations for a spin many fermion system in the unitary regime at finite temperatures are feasible (One variant of the fortran 90 program, version in matlab, has about 500 lines, and it can be shortened also. This is about as long as a PRL! ) Apparently the system undergoes a phase transition at Tc = 0.22 (3) F

Below the transition temperature both phonons and (fermionic) quasiparticles contribute almost equally to the specific heat