Advanced condensed matter physics Keio Univ. BCS-BEC Crossover in a Superfluid Fermi Gas A unified description of Fermi and Bose superfluids and New Material Science Yoji Ohashi Department of Physics, Keio University, Japan Ultracold Atom Physics Atoms are trapped in a magnetic/optical potential, and are cooled down to <O(mK). Then, one can observe quantum phenomena, such as superfluidity. History of Fermi and Bose superfluids Fermi superfluids liquid 4He (K. Onnes) 1908 Superconductivity in Hg (K. Onnes) 1911 1924~1925 Bose superfluids Theoretical prediction by Bose and Einstein 1932 Superfluid 4He (W. Keesom) 1938 “Superfluidity” (Kapitza) BCS theory (Bardeen, Cooper, Schrieffer) 1957 Superfluid 3He (Osheroff) 1972 High-Tc superconductor (Bednorz and Muller) 1986 1995 BEC in ultracold Bose gas 2004 BCS-BEC crossover in 40K and 6Li Fermi gases (Jin, Ketterle, et al) History of Fermi and Bose superfluids Fermi superfluids liquid 4He (K. Onnes) 1908 Superconductivity in Hg (K. Onnes) 1911 1924~1925 Bose superfluids Theoretical prediction by Bose and Einstein 1932 Superfluid 4He (W. Keesom) 1938 “Superfluidity” (Kapitza) BCS theory (Bardeen, Cooper, Schrieffer) 1957 Superfluid 3He (Osheroff) 1972 High-Tc superconductor (Bednorz and Muller) 1986 1995 BEC in ultracold Bose gas 2004 BCS-BEC crossover in 40K and 6Li Fermi gases (Jin, Ketterle, et al) BCS-BEC crossover Unified phenomenon to describe Fermi and Bose superfluids which have been studied independently in 20th century. Character of fermion superfluidity continuously changes from the weak-coupling BCS type (such as superconductivity) to the Bose-Einstein condensation (such as superfluid 4-He) of tightly bound molecular bosons, as one increases the strength of a pairing interaction between Fermi atoms. In cold atom physics, we can easily tune the interaction between particles by using a Feshbach resonance. interaction 40K |9/2,-9/2> |9/2,-5/2> (JILA) BCS-BEC crossover in a superfluid Fermi gas Is this class useful for your work? Related research fields: cold atom physics laser physics (quantum electronics) condensed matter physics, especially, superconductivity and strongly correlated electron systems superfluidity, such as liquid 4-He nuclear physics (neutron rich nucleus) elementary particle physics (color superconducvity; di-quark state) astrophysics (BCS-BEC crossover in neutron star) quantum chemistry (ultracold molecules) The cold Fermi gas system is expected as a useful quantum simulator Impoprtant material parameters in material science interaction band structure (crystal lattice) particle (electron) density “obstacles” in material science defect, impurity High controllability of cold atom gas Impoprtant material parameters in material science interaction cold atom gases tunable by Feshbach resonance band structure (crystal lattice) particle (electron) density (JILA) “obstacles” in material science defect, impurity High controllability of cold atom gas Impoprtant material parameters in material science interaction band structure (crystal lattice) particle (electron) density “obstacles” in material science defect, impurity cold atom gases tunable by Feshbach resonance various crystal lattice by optical lattice High controllability of cold atom gas Impoprtant material parameters in material science interaction cold atom gases tunable by Feshbach resonance band structure (crystal lattice) various crystal lattice by optical lattice particle (electron) density tunable Particle statistics (fermion、boson、mixture) “obstacles” in material science defect, impurity High controllability of cold atom gas Impoprtant material parameters in material science interaction cold atom gases tunable by Feshbach resonance band structure (crystal lattice) various crystal lattice by optical lattice particle (electron) density tunable Particle statistics (fermion、boson、mixture) “obstacles” in material science defect, impurity ideal crystal without defects and impurities High controllability of cold atom gas search for new material accidental discovery! (electron system) High controllability of cold atom gas theoretical prediction atom electron search for new material We can confirm the prediction by using cold atom gas loaded on an artificial optical lattice. accidental discovery! + (electron system) optimal condition can be also examined. electron atom Search for a material to realize the prediction Advanced condensed matter physics Starting from the review of statistical mechanics and quantum physics, I will explain the physics of BCS-BEC crossover phenomenon. The real BCS-BEC crossover has been observed in cold Fermi gases, but this phenomenon itself is very general, being widely applicable to various fields. In this lecture, I will explain this interesting physics from general theoretical viewpoint so that you can get something useful from this lecture series. Advanced condensed matter physics (2014) Grading A,B,C,D I will check your attendance at the beginning of each lecture. You need to attend more than 70% of lectures.* At the end of this course, you need to submit a report, where you solve some problems related to the BCS-BEC crossover. I will show the problems during lectures. At the submission, we are required to give me a brief explanation about your solutions (oral interview!). *When you cannot attend a lecture due to an inevitable reason (eg., illness), please let me know after you come back. Then I will treat the absence as “attendance”. Chapter 1. Experimental Overview Recent Development in cold atom physics Trap potential 1995 BEC in Bose atom gases JILA Recent Development in cold atom physics Trap potential 1995 Optical lattice BEC in Bose atom gases 2002 Mott Transition in Superfluid Bose gases Greiner, Nature 2002 Recent Development in cold atom physics Trap potential Feshbach resonance 1995 Optical lattice BEC in Bose atom gases 2002 Mott Transition in Superfluid Bose gases 2004 Superfluid Fermi gas and BCS-BEC crossover JILA, MIT (2004) Recent Development in cold atom physics Trap potential Feshbach resonance 1995 Optical lattice BEC in Bose atom gases 2002 Mott Transition in Superfluid Bose gases 2004 2006 Superfluid Fermi gas and BCS-BEC crossover Superfluid Fermi gas in optical lattice Ketterle (2006) Recent Development in cold atom physics Trap potential Feshbach resonance 1995 Lin Nature 471 (2011) 83 Optical lattice BEC in Bose atom gases 2002 Mott Transition in Superfluid Bose gases 2004 2006 Artificial Gauge field 2008 Superfluid Fermi gas and BCS-BEC crossover Superfluid Fermi gas in optical lattice Synthetic spin-orbit interaction Recent Development in cold atom physics Trap potential Feshbach resonance 1995 Optical lattice BEC in Bose atom gases 2002 Mott Transition in Superfluid Bose gases 2004 2006 Artificial Gauge field 2008 2014 Superfluid Fermi gas and BCS-BEC crossover Superfluid Fermi gas in optical lattice Synthetic spin-orbit interaction 2D-Fermi superfluid (KT-transition) Jochim arXiv14095373 (2014) Fermion Superfluidity in 40K Fermi gas | 9 / 2, 7 / 2 | 9 / 2, 9 / 2 TF 0.35m K N 105 C. A. Regal, et al. PRL 92 (2004) 040403. Tc / TF ~ 0.08 0.2 104 102 (metal ) Condensate fraction (=the number of Bose condensed Cooper pair bosons) Cooper pair | 9 / 2, 7 / 2 | 9 / 2, 9 / 2 superconductivity Superfluid Fermi gas Fermion superfluidity in 6Li Fermi gas |1/ 2, 1/ 2 |1/ 2, 1/ 2 M. Zwierlein, et al. PRL 92 (2004) 120403. Single-particle excitations in a superfluid 6Li Fermi gas rf-tunneling current spectroscopy BEC unitarity limit BCS C. Chin , et al. Science 305 (2004) 1128. photon a I ( ) Binding energy of a Cooper pair boson Vortex phase in a 6Li superfluid Fermi gas M. Zwierlein, et al., PRL 2005 (This figure shows molecular density profiles.) The observation of vortices is a clear evidence of “superfluidity” in superfluid Fermi gases. Collective (surface) mode in a 6Li Fermi gas M. Bartenstein, et al., PRL 92 (2004) 203201 The gas cloud behaves like a macroscopic single wavefunction. optical lattice artifical lattice produced by standing wave of laser light standing wave of laser light E( x) E0 sin(kx) atom gas Stark shift by s-p dipole transition periodic potential V ( x) p s H H 0 eEx alkali metal atom (Li, Na, K,…) E 2 | E ( x) |2 sin 2 (kx) | s | eEx | p |2 Es E p E2 optical lattice artifical lattice produced by standing wave of laser light standing wave of laser light E( x) E0 sin(kx) atom gas Stark shift by s-p dipole transition periodic potential V ( x) 40K (Fermion) py 2 | E ( x) |2 sin 2 (kx) observation of the Brillouin zone px V / Er 5 7 Er (2 / laser )2 / 2m 8 12 Kohl et al., PRL 94 (2005) 080403. Superfluid-Mott insulator transition in 87Rb Bose gas 87Rb-Bose Bose condensate at q=0 gas in 3D optical lattice 1st Brillouin zone U/t =0 py px 87 Rb :| F 2, mz 2 N 2 105 optical 852[nm](65 65 65) V 20Er Er (2 / )2 / 2m large U/t Greiner et al., Nature 415 (2002) 39 h superfluid phase Mott phase U / zt ~ 6 (z=6: coordination number) Theory: U / zt 5.83 Sheshadri et al, Europhys. Lett. 22, 257 (1993) Superfluid 6Li gas in an optical lattice Bose condensed Cooper pairs 0 laser atom gas py px B[G] Laser intensity -12 33 83 J. K. Chin, et. al. (MIT), Nature 443 (2006) 961. 6Li |1/ 2, 1/ 2 |1/ 2, 1/ 2 |1/ 2, 1/ 2 TF 1.4m K N 4 105 |1/ 2, 1/ 2 Chapter 2. Quantum statistics of particle: Difference and similarity between Fermion and Boson Fermi atom and Bose atom atom = proton(F) + neutron(F) + electron (F) =NF Fermi atom: NF= odd Bose atom : NF= even In cold atom physics, in most experiments, alkali atoms are used. In particular, the Fermi superfluid has been realized in K and Li gases. isotope proton neutron electron sum 6Li 7Li 39K 40K 41K 3 3 19 19 19 3 4 20 21 22 3 3 19 19 19 9 10 58 59 60 statistics Fermi Bose Bose Fermi Bose Alkali atoms zA with z being even are Fermions, and those with z being odd are Bosons (which generally holds!). Atomic hyperfine state Atomic spin state (=hyperfine state) = nuclear spin +electron spin I S F alkali atom: S=1/2 F I 1/ 2 In metallic superconductivity, an up-spin electron and down-spin electron form a Cooper pair. In cold Fermi gases, two atomic hyperfine states work as pseudospin-up and –down. 6Li (I=1): F=3/2, 1/2 | F , Fz 1/ 2, 1/ 2 40K (I=4): F=9/2, 7/2 | F , Fz 9 / 2, 9 / 2 9 / 2, 7 / 2 Fermi and Bose statistics 1 While the boson wave function is symmetric with respect to the exchange of two particles, the fermion wave function is antisymmetric. x (r , ) n,m: quantum labels fermions ( x1 , x2 ) n ( x1 )m ( x2 ) n ( x2 )m ( x1 ) bosons ( x1 , x2 ) n ( x1 )m ( x2 ) n ( x2 )m ( x1 ) ( x2 , x1 ) ( x1 , x2 ) ( x2 , x1 ) ( x1 , x2 ) More than two fermions cannot occupy the same quantum state. Pauli’s exclusion principle ( x1 , x2 ) n ( x1 )n ( x2 ) n ( x2 )n ( x1 ) 0 Thus, in a two-component Fermi gas (consisting of atoms with two hyperfine states), the maximum occupation number of each momentum state is two. Fermi and Bose statistics 2 Even when two fermions are in different quantum states, they cannot locate at the same (generalized) position x=(r,). ( x1 x2 x) n ( x1 x)m ( x2 x) n ( x2 x)m ( x1 x) 0 In particular, when their spin states are the same, they cannot meet each other at the same spatial position r. statistical repulsion (This kind of repulsive effect is absent in the boson case.) Bosons tend to occupy the same quantum state (q=0) at low Temperatures (statistical attraction). Occupation of atoms in the ground state (T=0) Free Fermi gas with two hyperfine states Free Bose gas F ( TF ) (kB 1) Fermi energy Fermi temperature py py pF px Pauli’s exclusion principle (= statistical repulsion) px Bose-Einstein condensation (= statistical attraction) Occupation of atoms at finite temperautres Fermions: Fermi distribution function f ( m ) 1 e ( m ) 1 ( 1/ T ) The Fermi chemical potential m can take positive and negative values. Bosons: Bose distribution function nB ( m ) 1 e ( m ) 1 Since the distribution function must be positive, the Bose chemical potential have to be larger than the lowest energy of atomic states (m >0). (Otherwise, the Bose distribution Becomes unphysically negative!) For a free Bose gas with =p2/2m, we find m<0. Finite temperatures 1: Fermion The occupation of Fermi atoms obeys the Fermi distribution function. T T=0 1 1 T >0 f ( m ) ( m ) e 1 F ( TF ) 0 The chemical potential is determined from the equation for the number n of Fermi atoms: n 2 f ( p m ) (V = 1) p At T=0, we find m F , and n 2 F 0 pF3 d ( ) 2 3 pF2 F 2m ( 1) pF: Fermi momentum inter-particle distance d ~ n 1/ 3 ~ 1 / pF Finite temperatures 1: Fermion In a bulk system, we can replace the momentum sum with the momentum/energy integration (useful!). n 2 f ( p m ) (V = L3=1) p When we impose the periodic boundary condition, the momenta px,py, and pz are discretized as 2 px nx , nx 0, 1, 2,...... L p 1 1 2 L L 1D: g ( p ) p g ( p )p p dpg ( p) 2 dpg ( p) x x px px L V 1 3D: g ( p ) ( 2 ) dp g ( p) 8 dp g ( p) 8 dp g ( p) (V L3 1) 3 3 3 3 3 3 p p2 2m 1 8 3 3 dp g ( ) 4 8 3 2 p dp g ( ) m 2m d g ( ) 2 2 Finite temperatures 1: Fermion At finite temperatures, the Fermi edge at F is smeared in the region ~[F-T, F+T]. Thus, effects of the Fermi statistics (giving the step function at T=0), become weak when T ~ TF . At finite temperatures, m deviates from the Fermi energy. At low temperatures, we obtain (Sommerfeld expansion) 2 2 T m / F m (T ) F [1 ] 12 TF When m<0 (which occurs around T~TF), f(-m) reduces to the classical Boltzman factor. f ( m ) ~ e |m|e T / TF Namely, the Fermi energy is the characteristic temperature where effects of the quantum Fermi statistics become important. Finite temperatures 2: Boson The occupation of Bose atoms is described by the Bose distribution function. nB ( m ) 1 e ( m ) 1 nB f 0 The chemical potential is determined from the equation for the number of Bose atoms: n nB ( p m ) (V = 1) p where m must be lower than the lowest energy, m 0 . This equation is not satisfied below a certain temperature TBEC, where TBEC is the phase transition temperature of the Bose-Einstein condensation . At TBEC, we find that m=0, and TBEC 2 n2 / 3 ( (3/ 2)) 2 / 3 m (3/ 2) 2.612 Finite temperatures 2: Boson Below Tc, the condensate fraction n0 , which describes the number of Bose-condensed bosons, is obtained from n n0 nB ( p ) p p2 p , 2m m / Tc m 0 At T=0, the second term vanishes, so that all the atoms are Bosecondensed. T / Tc In the high temperature region, m is negatively large, and we obtain the classical Boltzmann distribution function: nB ( m ) ~ e |m|e Similarity between Fermion and Boson characteristic temperature Fermion Boson TF thermal de Broglie length TBEC quantum mechanical size of a particle p2 T 2m , p h 2 ( 1) 2 / 2mT (T 2 / mT ) Fermion: T=TF: ~ 1/ pF ~ 1/ n1/ 3 ~ d F Boson: T=TBEC: ~ 1/ n1/ 3 ~ d B TF and TBEC have the same physics that, around these temperatures, quantum mechanical size of a particle is comparable to the interparticle distance, and the ‘classical particle picture’ is no longer valid. From classical to quantum regime T Tc T TF or TBEC T d ~ 1/ n1/ 3 T ~ d ~ 1/ n1/ 3 Boson/Fermion thermal de Broglie length T h / 2 mT The classical particle picture breaks down and the quantum wave function picture becomes necessary. Thermal de Bloglie length T 2 / mT Thermal de Broglie length:quantum-statistical size of a particle p h 2 2 p ( 1) (de Bloglie length) Including thermal fluctuations, we take the statistical average of the de Bloglie length, which gives T 2 d p pe dpe 2 mT 2 p 2m p2 2m 2 0 0 dppe dpp 2e p2 2m 2 p 2m 2 2mT 0 0 dxxe x 2 2 x2 dxx e Thermal de Bloglie length Thermal de Broglie length:quantum-statistical size of a particle T 2 / mT ipr wavefunction: (r) e Including thermal fluctuations, we take the statistical average of this Wavefunction. Then one has, eipr T ipr d p e e dpe 波動関数の広がり 2 T mT p2 2m p2 2m e mT 2 x 2 ~ T Chapter 3. Pair formation and Fermi surface effect Bose condensation in Fermi gases Boson BEC Fermi degeneracy Fermion To form molecules, a pairing interaction is necessary. BEC of molecules (=bosons) Pair formation 1: two Fermi particle ( , ) case Assume two fermions interacting with each other. The system is described by the Hamiltonian ( x (r, )) 1 [12 22 ]( x1 , x2 ) U (r1 r2 )( x1 , x2 ) E( x1 , x2 ) 2m When the interaction is spin-independent, the wavefunction has the form (r1 , r2 , 1 , 2 ) (r1 , r2 ) 1 , 2 (r1 , r2 ) g (k )eik (r1 r2 ) (center of mass momentum =0) k Spin singlet: Spin triplet : , , g (k ) g (k ) g (k ) g (k ) (The assumed wavefuntction must be antisymmetric!) k2 g (k ) Vkk ' g (k ') Eg (k ) m k' Vkk ' d 3rV (r)ei (k k ')r Pair formation 1: two Fermi particle ( , ) case For a contact-type interaction (U (r) U (r) ), we obtain k2 ( E ) g (k ) U g (k ') m k' = k2 E 2 k m 0 spin-triplet C spin-singlet In the spin-triplet case, two fermions never meet at the same spatial position due to the statistical repulsion, so that the contact interaction does not work at all. singlet case: g (k ) U 2 k E m g (k ') k' k g (k ) k U k2 E m k g (k ') ' 1 1 1U U d ( ) 0 2 E k 2 k E ( ) m 2m 2 2 density of states (DOS) A bound state is obtained when this equation has a solution with E<0. We note that the RHS has a ultraviolet divergence ( ). Pair formation 1: two Fermi particle ( , ) case Introducing a cutoff, we solve the following equation: 1U k 1 1 U d ( ) 0 2 k E 2 E 1U k c 1 1 U d ( ) 0 2 k E 2 E For E<0, we obtain 1 |E| 1 2c 1 Tan U (c ) 2c |E| 1 solution! 1/ U (c ) |E| There is a threshold value of the interaction U (lower limit) to form a molecule in this two particle case. U (c ) 1 (c ) m 2m c 2 2 Discussion: what is c? U (c ) 1 (c ) m 2m c 2 2 This comes from introducing the contact interaction U (r) U (r) U assumed contact interacion |U()| real interaction c To evaluate the value of the cutoff, we need to know the details of the interaction in the high-energy region. However, we usually do not know what happens in the highenergy region, and we do not want to know such a high-energy physics in considering the interesting low energy physics. Renormalization (Regularization) To avoid introducing the unknown cutoff c, we introduce the two-particle s-wave scattering length as. Following the standard scattering theory, we calculate the scattering matrix (t-matrix) in terms of the contact interaction. The scattering length is then obtained as the low-energy limit of the t-matrix. 4 as (0, 0) m = + U c U U + U 1 U (U ) (U ) ..... p 0 2 U U +……. U c 1U p 1 2 Since the scattering length is observable experimentally, we use this equation to determining the value of the cutoff c. Renormalization (Regularization) c 1 1 1U U d ( ) 0 2 E k 2 k E c c c 1 1 1 1 1U U U 2 E 2 E 2 k k k 2 k k k k c c 1 1 1 1U U k 2 E 2 k 2 k k k 1 1 4 as c 1 c m k 1 k 2 k E 2 k 1U k 2 k c U 4 as m U c 1U p 1 2 1 1 2 k E 2 k This summation converges without the cutoff! ~ d 0 |E| ( | E |) Renormalized theory for the two-body bound state problem 4 as c 1 m k 1 1 4 as 1 1 d ( ) 0 2 E 2 m 2 E 2 k k This renormalized equation is written by using only physical parameters. Molecular formation: as 0. weak U as 1 0 Bound state energy E 1 mas2 Bound state wavefunction E 0 strong U molecular size= as a 1s 0 E molecular size 1 mas2 Renormalized theory for the two-body bound state problem In a two-fermion system, a bound state is not always formed even when they are interacting attractively. weak U as 1 0 strong U 0 a 1s 0 E molecular size 1 mas2 Pair formation 2: Cooper problem py U px many free Fermions ( Fermi surface) + two attractively interacting Fermions 1 [12 22 ]( x1 , x2 ) U (r1 r2 )( x1 , x2 ) ( E 2 F )( x1, x2 ) 2m The energy is measured from the Fermi level. Pair formation 2: Cooper problem py U px many free Fermions ( Fermi surface) + two attractively interacting Fermions (r1 , r2 ) k kF g (k )eik (r1 r2 ) Fermion states inside the Fermi surface are occupied. U (r1 r2 ) U (r1 r2 ) A bound state is obtained in the spin-singlet state. Pair formation 2: Cooper problem In the singlet case, the bound state equation is found to be 1U k kF 1 2 k 2 F E As in the two-fermion problem, this equation involves the unphysical ultraviolet divergence. To avoid this, we introduce the scattering length for the interaction renormalized down to the Fermi level: 4 as ( F ) m U 1U c k kF 1 2 , 4 as 4 as ( F ) V m c F m 4 as F 1 1 1 1V V 1 m k 0 2 k 0 2 k 0 2 The renormalized (regularized) equation for the bound state is then given by 4a s ( F ) 1 1 1 d ( ) m 2 2 E 2 F F The important thing is that this equation always has a bound state solution (E<0), Irrespective of the value of the scattering length as. Pair formation 2: Cooper problem weak-coupling case (|E|<< F) strong-coupling case (|E|>> F) E 8 F e 1 1 E ~ 2 2 mas ( F ) mas k F as ( F ) 1 2 kF as e kF ( R) 1 e R / 2 R as molecular size Fermi surface +two Fermions Two Fermions 0 (kF as )1 Fermi surface effect (Cooper instability) Bound state energy Pair formation 2: Cooper problem weak U strong U molecular size Fermi surface +two Fermions Two Fermions 0 (kF as )1 Fermi surface effect (Cooper instability) many-body bound state Bound state energy two-body bound state Essence of the BCS-BEC crosover phenomenon T Tc T Tc T d ~ 1/ n1/ 3 T ~ d ~ 1/ n1/ 3 boson thermal de Broglie length T h / 2 mT Phase diagram in the BCS-BEC crossover Eg T Fermi gas Molecular Bose gas Tc Superfluid phase strong U weak U BCS BEC Tc strong U weak U JILA 2004 Phase diagram in the BCS-BEC crossover Eg unitarity limit T Fermi gas crossover region (kF as )1 1 1 0 +1 (kF as )1 Molecular Bose gas Tc Superfluid phase (kF as )1 1 Phase diagram in the BCS-BEC crossover NOTE: interatomic distance ~ kF-1 (kF: Fermi momentum) unitarity limit T Fermi gas Molecular Bose gas Tc Superfluid phase crossover region (kF as )1 1 1 0 +1 (kF as )1 (kF as )1 1 two-body bound state A molecule is formed in a two fermion system. molecular size: ~ as Phase diagram in the BCS-BEC crossover NOTE: interatomic distance ~ kF-1 (kF: Fermi momentum) T crossover region Fermi gas Molecular Bose gas (molecular size) ~ kF1e Tc 2 kF as Superfluid phase ~ as ~ kF1 (kF as )1 1 1 many-body bound state A Fermi surface is necessary to form a Cooper pair. 0 +1 (kF as )1 (kF as )1 1 two-body bound state A molecule is formed in a two fermion system. Chapter 4. tunable interaction associated with a Feshbach resonance pairing interaction mediated by boson phonon ‘phonon’ mechanism superconductivity Phonon, AF spin fluctuations superfluid 3He Ferromagnetic spin fluctuations molecule Feshbach mechanism superfluid Fermi gas 40K, 6Li Conventional phonon mechanism in superconductivity p 'q p q D effective interaction mediated by phonon Veff p 'q p q J q J phonon phonon p' p p p' J is an electron-phonon coupling constant. Evaluating these scattering processes, we obtain ( p, p ' p q, p ' q) J 2 1 J2 1 ( p p ' ) ( p q p ' D ) ( p p ' ) ( p p ' q D ) 1 1 J2 J2 . ( p p q ) D ( p ' p 'q ) D For electrons in the low energy region (near the Fermi surface), satisfying | p pq | D | | , we obtain the phonon-mediated attractive pairing interaction p p 'q D Veff 2 J2 D 0 Feshbach resonance open channel Zeeman energy B closed channel Atomic hyperfine states in the closed channel are different from those in the open channel. open F , Fz 9 / 2, 9 / 2 9 / 2, 7 / 2 close F , Fz 9 / 2, 9 / 2 7 / 2, 7 / 2 40K The Zeeman energy of the open-channel is different from that of the closed channel. This is because of the different magnitude of the electron Bohr magneton and the nuclear one. Tunable interaction associated with a Feshbach resonance Fermi atom g 2n g 1 Veff g 2n 2 molecule 2n is referred to as the threshold energy of the Feshbach resonance. (JILA) Near the resonance field, 2n=(B-B0). Including a residual weak interaction part U, we obtain the Feshbach mediated pairing interaction, given by 40K |9/2,-9/2> |9/2,-5/2> Veff U g 2 (2n) 1 1 U g 2 2n ( B B0 ) Tunable interaction associated with a Feshbach resonance Fermi atom g 2n g 1 Veff g 2n 2 molecule This interaction is obtained within the second order perturbation theory, but the value of Veff becomes very large when 2n~0. question Don’t we need to consider higher order corrections beyond the second order perturbative calculation? Tunable interaction associated with a Feshbach resonance Fermi atom g 2n g 1 Veff g 2n 2 molecule This interaction is obtained within the second order perturbation theory, but the value of Veff becomes very large when 2n~0. question Don’t we need to consider higher order corrections beyond the second order perturbative calculation? answer Higher order corrections are formally included when parameters are replaced by renormalized ones. Renormalized Feshbach induced pairing interaction 4 as (0, 0) m + U + U U + + U U U +……. + + g2 ~ U 2 ~ Veff g 2n U ~ c 2 c 1 g 1 2n 1 Veff 1 (U ) 2n p 2 p 2 renormalized parameters c ~ U U c 1 1U p 2 ~ g g c 1 1U p 2 ~ 2n 2n g 2 1 p 2 c 1U p 1 2 Chapter 5. Many-body Hamiltonian in the second quantization representation second quantization The wavefunction of a many-fermion system can be written as, by using the Slater determinant, 1 ( x1 ) 1 ( x2 ) 1 ( x3 ) 1 (1) 1 ( x1 )2 ( x2 )3 ( x3 ) 2 ( x1 ) 2 ( x2 ) 2 ( x3 ) 3! P 3 ( x1 ) 3 ( x2 ) 3 ( x3 ) P This kind of wavefunctions can be written more simply when one uses the number representation. 1 2 3 4 5 6 7 n1 , n2 , n3 , n4 , n5 , n6 , n7 ....... 0110101... To include the Fermi statistics, we write this wavefunction in the second quantization form: 1110000 c1†c2†c3† 0 The creation operators c †j satisfy the (Fermi) anticommutation relation [c , c ] c c c c 0 † j † i † † j i † † i j c†j c†j 0 0 (exclusion principle) second quantization † c †j is called the creation operator. From the hermite conjugate of 1 c1 0 , We obtain 1 (c1† 0 )† 0 c1. When 1 is normalized, they satisty 1 1 1 0 c1 c1† | 0 From this, we can easily understand why c is called the annihilation operator. (The vacuum state |0> is (exactly speaking) defined as the state which vanishes when any cj acts on it. c j 0 0 ) [ci , c†j ] ij [c†j , ci† ] [c j , ci ] 0 In the bosonic case, the creation and annihilation operators satisfy the following commutation relation, refecting the that the wavefunction is symmetric in terms of particle exchange. [bi , b†j ] ij [b†j , bi† ] [b j , bi ] 0 second quantization c1†c2†c3† 0 When we use the second quantized wavefunction, we need to rewrite the Hamiltonian in the form appropriate to this scheme. one-particle part (kinetic term, potential) F fi F f nm cn†cm i , i 2m i , i 2m V Vi V (ri ) mn i H 0 H 0 i H 0 p c†p c p i p , two-particle part (interaction) 1 G gij 2 i , j (i j ) Vint 1 U (ri rj ) 2 i j 1 G mn g ij cm† cn†c j ci 2 i , j ,m,n Vint U c p , p ', q † † p q p ' q p ' p c c c Vint U 2 u(r r ) i j (volume=1) i j second quantization In the second quantization scheme, the “particle picture” revives. H 0 p c†p c p p , number operator 1 n p c†p c p 0 N c†p c p p Vint U c ( p ) ( p ) total number operator † † p q p ' q p ' p c c c p , p ', q U p' p Microscopic models to describe the Fermi superfluids BCS model (single-channel) H ( p m )cp† cp U p c † † p q p ' q p ' p c c c p ,p ',p In this model, the pairing interaction is simply treated as the constant parameter U, without specifying its origin. Coupled fermion-boson model (two-channel) . H ( p m )cp† cp U p p ,p ',p cp†qcp†'qcp 'cp ( qB 2n 2m )bq†bq q g bq†cpq / 2cpq / 2 c†pq / 2c†pq / 2bq p ,q Fermion Feshbach resonance Boson 2n N NF 2NB H m N 1 U eff n Un ( g n ) 2n 2m 2 crossover region narrow F.R. g n F U eff n ~ F broad F.R. g n F Boson Fermion n ~ F 2n Feshbach molecules appear as real particles in the crossover region. n F Feshbach molecules only appear in the virtual process to produce Ueff. (40K, 6Li) BCS CFB BCS CFB Chapter 6. Ground state of a superfluid Fermi gas and BCS-BEC crossover From the Cooper problem to the BCS state Cooper problem (r1 , r2 ) F g k k F ik ( r1 r2 ) † † e F g c c k k k k F † p c p pF , k ~ 0 ( g p1 c c p1 † p1 † p1 ~ )( g p2 c † p2 p2 † p2 c ~ )....( g p N / 2 cp†N / 2 c†p N / 2 ) 0 pN / 2 ~ g p ( pF p) ( g k c c † † k k k ~ )( g p1 c c p1 † † p1 p1 ~ )( g p2 c p2 † † p2 p2 c ~ )....( g p N / 2 cp†N / 2 c†p N / 2 ) 0 pN / 2 many Cooper pairs ( g p1 cp†1 c†p1 )( g p2 cp†2 c†p2 )....( g p N / 2 cp†N / 2 c†p N / 2 ) 0 p1 p2 pN / 2 BCS ground state BCS C 1 g k ck†c†k 0 k | > is obtained from ‘N-particle terms’ in |BCS >. Setting g=v /u and C uk vk , k we obtain the famous expression, BCS uk vk ck†c†k 0 k BCS BCS 1 uk2 vk2 1 The BCS state involves terms with the different numbers of fermions, which looks strange. However, it can be shown that the fluctuation around the mean-value <N> is very small when N is large (=106~108 >>1 in cold atom physics). N N N 2 N2 N 1 N BCS ground state BCS uk vk ck†c†k 0 k uk2 vk2 1 u and v are determined by minimizing the ground state energy Eg BCS H BCS BCS The resulting “optimized u and v” are given by ( p p m ) p 1 u p (1 ), 2 2 2 p 2 p 1 v p (1 ) 2 2 2 p 2 Here, is the (Fermi) superfluid order parameter, having the form U BCS c c † † k k p BCS U u p v p U p p 2E p Thus, the order parameter is determined by the BCS gap equation 1 1U p 2E p E p p2 2 . Regularization of the BCS gap equation The BCS gap equation involves the ultraviolet divergence. In super conductivity, the phonon-mediated pairing interacrtion has a physical cutoff associated with the upper limit of the phonon frequency (=Debye frequency D ~ 1000[ K ] ) . In cold Fermi gases, in contrast, there is no physical cutoff like the Debye cutoff, so that we have to regularize the theory to eliminate the divergence. c 1 1 1 1U U 2 p p 2E p p 2E p c 4 as 1 m c 1 U p 2 p 1 1 p 2 E p 2 p This summation now converges. BCS superfluid theory at T=0 Gap equation number equation (equation of state) 4 as 1 m 1 1 p 2 E p 2 p N BCS c c p p † p (cutoff-free) p BCS 2 v 1 E p p p 2 p and m are determined self-consistently from these coupled equations. NOTE In the usual (weak-coupling) BCS theory, the chemical potential can be taken to be equal to F. However, from general point of view, it should be determined by the equation for the number of fermions. Indeed, the chemical potential is found to remarkably deviates from the Fermi energy when the pairing interaction is strong. BCS-BEC crossover theory at T=0 (Leggett theory) 4 as 1 m 1 1 p 2 E p 2 p Weak coupling BCS limit: 8 Fe 2 e m F , Strong-coupling BEC limit: 2 p (kF as )1 2 k F as p N 2 v 1 E p p p 16 | m |1/ 4 F3/ 4 , 3 The chemical potential is at The Fermi level, as expected. (kF as )1 m 1 0 2 2mas Binding energy of a Cooper pair molecule obtain from the Cooper problem: (BCS) Ebind 1 mas2 (BEC) BCS-BEC crossover theory at T=0 (Leggett theory) 4 as 1 m 1 1 p 2 E p 2 p Weak coupling BCS limit: 8 2 Fe e m F , Strong-coupling BEC limit: 2 p (kF as )1 2 k F as p N 2 v 1 E p p p 16 | m |1/ 4 F3/ 4 , 3 (kF as )1 m 1 0 2 2mas Binding energy of a Cooper pair molecule obtain from the Cooper problem: Ebind 8 F e kF as ( F ) (BCS) Ebind 1 mas2 (BEC) BCS-BEC crossover at T=0 (self-consistent solution) BCS / F BEC The chemical potential gives the size of the Fermi surface. The chemical potential gives the value of the binding energyof a molecule. m / F BEC BCS (kF as )1 BEC Fermi superfluid = molecular BEC ? (1) superfluid order parameter U BCS ck†c† k BCS p (2) ground state energy BEC regime: EG BCS H BCS | m | N 1 2 N Ebind N B mas 2 All atoms form Cooper pairs with Ebind. BCS regime: EG EN 1 ( F ) 2 3 N 2 8 F Only a small fraction ( N ( / F ) ) of atoms form Cooper pairs with Ebind=. The molecular picture is not good in the BCS regime. Fermi superfluid = molecular BEC ? BCS uk v k ck†c† k 0 k v u ~ uk ~ 0, vk ~ 1 p F F n p v 2p Energy levels deep inside the Fermi level are fully occupied (as in the case of a free Fermi gas F c†p 0 ). These occupying p , atoms do not contribute to the condensation energy. Cooper pairs near the Fermi surface (| p | )only contribute to the condensation energy. N pair ~ N F 3 EG EN N 8 F Fermi superfluid = molecular BEC ? (3) molecular picture in the BCS regime p n p BCS c c p p 1 BCS v 1 2 2 2 ( ) p F 2 p 1 np F 0 size of a Cooper pair p ~ 1/ F ~ ( pF p)2 / 2m uncertainty principle kF 1 2 kF as ~ e m k F Cooper problem Fermi superfluid = molecular BEC ? (4) condensate fraction = the number of Bose-condensed Cooper pair molecules g k c p c p BCS e 0 e n0 b0 (Boson BEC ground state) 0 q2 U H ( m )bq bq 2m 2 q Bose gas BEC b b p q p ' q b p 'b p p , p ', q b0 b0 (BEC order parameter) n0 b0 (condensate fraction in BEC) 2 b0 g c c ? † † k k k k vk † † ck c k k uk Fermi superfluid = molecular BEC ? [b0 , b0† ] g k2 (1 ck†ck c† k c k ). k Within the expectation value in terms of |BCS >, this commutation relation is evaluated as [b0 , b ] g (1 2v ) † 0 2 k B0 2 k k 1 g 2 k (1 2vk2 ) b0 , [ B0 , B0† ] 1. k Boson! When RHS is positive, This condition is always satisfied when m<0 (v2<0.5). This is realized in the strong-coupling BEC regime, where the molecular character would be OK. In this case, we have BCS e gk c p c p 0 e n0 g k2 (1 2vk2 ) (1 k k n0 B0† 0 k k2 2 BEC limit )2 k E N 2 Fermi superfluid = molecular BEC ? [b0 , b0† ] gk2 (1 2vk2 ) k B0 1 i | g k2 (1 2vk2 ) | b0 , [ B0 , B0 ] 1. † k In the BCS regime (m>0), RHS can be negative and one cannot introduce the approximate boson operator. In this regime, Bose-condensed pairs consist of particle pairs and hole pairs due to the presence of the Fermi surface. Fermi surface Fermi superfluid = molecular BEC ? BCS uk vk ck†c†k 0 uk c k ck vk uk vk ck†c†k F ( m ) m k F ( m ) ck† c† k 0 b vkk c kck ukk ck†c† k e m (‘Fermi vacuum’) m † 0 u m v m F (m ) e uk2 vk2 2 [b0 , b ] 2 (1 2vk ) 2 (1 2vk2 ) 0 m vk m uk † 0 † B0 n0 B0† F (m ) The first term is set to be equal to 0 when m<0. 1 uk2 vk2 2 2 (1 2vk ) 2 (1 2vk2 ) m vk m uk uk2 vk2 N 2 2 n0 2 (1 2vk ) 2 (1 2vk ) 2 F m vk m uk b0† , [ B0 , B0† ] 1. BCS limit Fermi superfluid = molecular BEC ? condensate fraction ODLRO n0 / N BCS (kF as )1 BEC condensed molecular bosons Particle pairs Hole pairs (Fermi surface effect) Particle pairs Condensate fraction (ODLRO) (1) Boson BEC 1 † (r1 ) (r2 ) † (r1 ) (r2 ) f (r1 , r2 ) This vanishes when | r1 r2 | In the BEC phase, the condensate fraction is given by the maximum O(N)-eigen-value of this density matrix: 1 n0 * (r1 ) (r2 ) O(1) normalized eigen funcion of 1 From these two expressions, it is reasonable to equate the red terms. n0 dr | 2 (r ) | Condensate fraction (ODLRO) (2) Fermi superfluid 2 † (r1 )† (r2 ) (r3 ) (r4 ) † (r1 )† (r2 ) (r3 ) (r4 ) f 0 when | (r1 , r2 ) (r3 , r4 ) | . 2 n0 (r1 , r2 ) (r3 , r4 ) O(1) * n0 dr1dr2 | † (r1 ) † (r2 ) |2 Uniform Fermi superfluid at T=0 (|BCS >) (r ) e ik r ck k BCS (r) (r ') BCS eik(r r ') BCS ckck BCS uk vk eik(r r ') k 2 n0 u v 2 k k 4 Ek 2 2 k k k uk2 vk2 2 2 n0 2 (1 2vk ) 2 (1 2vk ) m vk m uk Chapter 7. Excitations in a Fermi superfluid at T=0 Excitations in Fermi and Bose superfluids collective modes single-particle excitations Bose atom BEC Fermi superfluid Cooper-pair is a molecular Boson with a finite binding Energy. Mean field theory of Fermi superfluids (Bogoliubov) In the previous discussion, we have shown that the BCS state (describing the ground state of a Fermi superfluid) is characterized by the finite value of the BCS order parameter U ck† c† k p In constructing the mean field theory of the Fermi superfluid, we expect that the interaction term may be approximated by replacing the operator cc by its expectation value <cc>. Dividing the pair-operator ck† c† k into the mean value and fluctuations around it as ck†c† k ck†c† k Ak† we can write the interaction part of the BCS Hamiltonian as U c†p 'c† p 'c pc p U ( c†p 'c† p ' A† )( c pc p A) p, p ' p, p ' U 2 U ( c†p 'c† p ' Ap Ap ' c pc p Ap ' Ap ) † p, p ' || (*c†pc† p c pc p ) A† A U p 2 † Mean field theory of Fermi superfluids (Bogoliubov) H ( p m )c c p ( c c † p p * † † p p p c pc p ) U | |2 A A U † In the mean field approximation, we ignore this term, which describes pairing fluctuations. Actually, we may take D to be real, because even if | | ei , the phase factor can be eliminated by the U(1) gauge transformation, c p c p ei / 2 H ( p m )c c p (c c † p p p c , c p p † † p p † p p m | |2 c pc p ) U c p 2 ( p m ) ( p m ) c† p U p Mean field theory of Fermi superfluids (Bogoliubov) This mean-field Hamiltonian can be diagonalized by the so-called Bogoliubov transformation, c p u p † c v p p v p p u p † p Here, obeys the Fermi statistics, and u and v are the same as those appeared in |BCS>. H E p †p p W0 p 2 W0 [( p m ) E p ] U Ep ( p m )2 2 0 Ep is the excitation spectrum of quasi-particle described by (Bogolon). E p : †p 0 Bogolon †p 0 What is the vacuum state for the Bogolons? Answer: 0 BCS The vacuum for the Bogolons is not the vacuum for Fermi atoms! Ep ( p m )2 2 What is this excitation? Answer: single particle excitation associated with the breakup of a Cooper pair. Excitation gap in the Bogolon spectrum Eg (m 0) Eg m 2 2 | m | ( m 0) BCS regime BEC regime Binding energy of a Cooper pair Single particle excitations at T=0 [ F ] Eg |m| BCS Energy gap Eg BEC 2 m 2 |m| BCS: m>0 BEC: m<0 BEC limit (|m|>>) Note: The binding energy of a Cooper-pair is given by 2Eg. Single particle excitations at T=0 Single-particle excitations can be seen in the superfluid density of states. N ( ) ( E p ) d ( ) ( E ) 0 coherence peak at N ( ) (m ) 2 2 ( ( ) ) m 0 : BCS m 0 : BEC The coherecne peak is absent in the excitation spectrum in the BEC regime. 1. Single particle excitations at T=0 (BEC regime) Fermion band 0 m Excitation spectrum in the BEC regime is simply given by the density of states of a free Fermi gas. N () ( | m |) Observation of single particle excitations in a superfluid Fermi gas (BCS-BEC crossover region) Single-particle excitations can be observed by using the rf-tunneling current spectroscopy. superfluid 6Li Fermion band , 0 light Eg C. Chin , et al. Science 305 (2004) 1128. 2. collective excitations at T=0 (1. Boson BEC) Bogoliubov mode in a boson BEC q2 H ( m )bq bq U b p q b p' q b p ' b p 2m q p , p ', q b0 b0 n0 , + Bogoliubov approximation Un0 q2 H Eg ( m 2Un0 )bq bq 2m 2 q [ b p b p b pbp ] p m is chosen so that the ground state energy E g can be minimum, which gives m Un0 . Un q2 H Eg ( Un0 )bqbq 0 2m 2 q [ b p b p b pbp ] p Bogoliubov transformation U 2 n0 mn0 2 2. collective excitations at T=0 (1. Boson BEC) Bogoliubov mode in a boson BEC H E q q q q boson Eq q ( q 2Un0 ) Un0 q v q m (q 0) gapless Bogoliubov phonon This Goldstone mode is associated with the spontaneous breakdown of the continuous gauge symmetry in the BEC phase. b0 b0 n0 ei The boson BEC is dominated by collective excitations only. 2. collective excitations at T=0 (1. Boson BEC) E b0 Realized ground state | b0 | ei1 Another possible ground state | b0 | ei2 Gapless Goldstone mode 2. collective excitations at T=0 (2. Superfluid Fermi gas) BCS (uk vk ckck ) 0 i e i (b0 b0 n0 e ) BCS ' (uk vk ckck ei ) 0 . different state with the same energy (degenerate!) When the order parameter oscillates with q, the additional † † † pair amplitude, q c pq / 2c pq / 2 and q c pq / 2c pq / 2 are induced. The resulting mean-field Hamiltonian is H H BCS U ( q† c p q / 2c p q / 2 q c†p q / 2c† p q / 2 ) p ,q This perturbation again generates the oscillation of the order parameter. In the linear response theory, we obtain 2. collective excitations at T=0 (2. Superfluid Fermi gas) q (t ) U dt ' q (t ); (t ') † q 0 q† (t ) U dt ' q† (t );q† (t ') 0 Here, ..... q (t ') U dt ' q (t ); q (t ') q† (t ') 0 q (t ') U dt ' q† (t ); q (t ') q† (t ') 0 are the double-time Green’s functions, e.g., q (t );q† (t ') i (t t ') BCS | q (t '), q† (t ') | BCS In the frequency space W, q (W) U q ;q† q† (W) U q† ;q† W W q (W) U q ;q q (W) U q† ;q W W q† (W) q† (W) 2. collective excitations at T=0 (2. Superfluid Fermi gas) Mode equation: 0 1 U q ; q† U q† ; q† W W U q ;q W 1 U q† ; q W 2 U 1 U 1 22 (q, W) 21 (q, W) 12 (q, W) U 2 2 1 11 (q, W) 2 2 E E 22 (q, W) 1 E E ( E E )2 W2 p phason 2 E E 11 (q, W) 1 E E ( E E )2 W2 p ampliton iW 12 (q, W) 21 (q, W) E ( E E )2 W2 p E coupling mode 2. collective excitations at T=0 (2. Superfluid Fermi gas) In the long wave length limit, taking W = v q, we obtain the sound velocity in the entire BCS-BEC crossover at T=0, v 1 m BCS regime: v p 2 E 5 p 2E 3 p p 1 p E 3 E 3 p p 1 vF 3 U B n0 BEC regime: v M UB 2 2 / 3 Ep Anderson-Bogoliubov mode Bogoliubov phonon 4 aB 4 (2as ) 0 M M (M 2m) Effective repulsive interaction, given by aB=2as, looks working between Cooper-pair molecules in the BEC regime. 2. collective excitations at T=0 (2. Superfluid Fermi gas) v sound velocity 1 vF 3 v U B n0 M v / vF BCS (kF as )1 BEC NOTE: 1. In a charged Fermi superfluid, this sound mode remains only just below Tc (Carlson-Goldman mode). At T=0, the plasma only exists. 2. In a more sophisticated theory, it has been pointed out that the effective interaction between molecules is given by aB=0.6as. Chapter 8. BCS-BEC Crossover theory based on the Coupled Fermion-Boson Model (T=0) BCS-BEC crossover tuned by a Feshbach resonance H ( p m )c†p c p U p c†p qc†p 'qc p 'c p p , p ', q † † † ( qB 2n 2m )bq†bq g bq c p q / 2c p q / 2 c p q / 2c p q / 2bq p ,q q The fermion field and boson field lead to the two superfluid order parameters: U BCS ck†c† k BCS BCS order parameter m b0 b0† BEC molecular condensate p However, these are NOT independent, but related to each other due to the resonance between atoms and molecules. 1 g m 2n 2m U i 0 b 0 t b0 , H (2n 2m )m Thus, both order parameters are finite in the superfluid phase. g U BCS-BEC crossover tuned by a Feshbach resonance The mean field CFB Hamiltonian has the form ~ H ( p m )c c p (c†pc† p c pc p ) ( qB 2n 2m )bq†bq † p p q 0 p BCS-type with composite order parameter Free Bose gas ~ gm Ep ~ 2 ( p m ) 2 : Single-particle excitations ~ U BCS ck†c† k BCS U p p g2 1 (U ) 2n 2m p ~2 2 ( p m ) 2 ~ 1 ~2 2 ( p m )2 : gap equation for BCS-BEC crossover tuned by a Feshbach resonance g2 1 (U ) 2n 2m p 1 ~2 2 ( p m ) 2 The equation indicates that the effective pairing interaction associated with the Feshbach resonance is given by g2 U eff U 2n 2m Note that this expression is different from the two-particle result: U 2b eff g2 U 2n At T=0, thermally excited molecules are absent, so that the equation for the total number of Fermi atoms is given by N 2 | m | [1 2 p p m ~2 ( p m )2 ] Broad Feshbach resonance: g n F 2n 2m ~ 2 F crossover region Boson U eff g2 g2 U ~U 2n 2m 2n Fermion 2n m ~ 0 1 4 as m 1 [ p ~2 2 ( p m ) 2 N 2 | m | [1 2 p p m ~2 1 ] 2 g2 U 4 a 2n s m g 2 c 1 1 (U ) 2n p 2 ] ( p m )2 Single-channel BCS ~ model with Narrow Feshbach resonance: g n F crossover region 2n ~ 2m Boson Fermion In this case, one cannot eliminate the cutoff by using the two-body scattering length, because Ueff is different from Ueff2b. 2n The cutoff can be formally eliminated by introducing the ‘generalized scattering length,’ defined by g2 U 4 as 2n 2m c m g2 1 1 (U ) 2n 2m p 2 1 4 as m 1 [ p ~2 2 ( p m ) 2 N 2 | m | [1 2 p p m ~2 ( p m )2 1 ] 2 ] Chemical potential in the crossover region Boson m n Fermion m / F 2n U eff n / F g2 U 2n 2m The pairing interaction Ueff is always attractive and become strong as one decreases the threshold energy 2n. In the strong-coupling regime, 2m=2n is obtained. This means that the molecular chemical potential 2m is at the lowest boson energy (= BEC condition). Composite order parameter in the crossover region ~ / F / F ~ / F / F m / F m / F n / F n / F narrow resonance broad resonance Un 0.02 F , g n 0.17 F Un 0.02 F , g n 5 F ~ / F / F narrow m / F ~ / F / F broad m / F / F single-channel BCS (kF as )1 BEC [N ] NF n0 m2 narrow m2 n0 NF n0 m2 n0 broad m2 n0 single-channel BCS (kF as )1 BEC broad m / F narrow (kF as )1 As far as we consider quantities independent of the character of molecules (Cooper pairs or Feshbach molecules), the narrow FR and broad FR almost give the same results when scaled by the (generalized) scattering length. Chapter 9. BCS-BEC crossover at finite temperatures 1. Superfluid phase transition Breakdown of the mean-field theory at T>0 1. Single-channal BCS model The previous mean-field theory can be immediately extended to the case at T>0. A BCS A BCS †p p f ( E p ), 1 tr e H A Z p †p 1 f ( E p ), †p †p p p 0 E 2 2 Ep 1 1U tanh 2 p 2E p N c c p p † p p Ep 1 tanh E 2 p p ( 1/ T , kB 1) Breakdown of the mean-field theory at T>0 1. Single-channal BCS model The previous mean-field theory can be immediately extended to the case at T>0. A BCS A BCS †p p f ( E p ), 1 tr e H A Z p †p 1 f ( E p ), †p †p p p 0 E 2 2 U c†p c† p U (u p †p v p p )(v p p u p † p ) p p U u p v p †p p u 2p †p † p v 2p p p u p v p p † p p U u p v p 1 2 f ( E p ) p Breakdown of the mean-field theory at T>0 1. Single-channal BCS model At Tc, these equations reduce to 4 as 1 m (Tc 0) 1 ( m ) 1 2( m ) tanh 2 2 N 2 f ( p m ) Free Fermi gas! p m ~ F (T TF ) Breakdown of the mean-field theory at T>0 1. Single-channel BCS model In the weak-coupling BCS limit, Tc<< TF, and the equation of states gives m=F, as expected. In this case, the gap equation gives Tc 8 2 kF as 2 kF as e 0.61 e F F e2 Note: 8 (T 0) 2 F e 2 kF as e ~ 1.78 2 / Tc 2 / 3.54 (BCS universal constant) The fact that /Tc=O(1) means that, as one increases temperatures, the superfluid phase transition occurs when Cooper pairs are completely destroyed thermally. Breakdown of the mean-field theory at T>0 1. Single-channel BCS model In the strong-coupling limit, the gap equation gives 1 m 2mas2 This equals the binding energy of a Cooper-pair obtained at T=0. This means that molecules does not dissociate even at Tc. Thus, one expects that Tc is essentially the same as that for an ideal molecular Bose gas (NB=N/2, M=2m). 2 (n / 2) 2 / 3 Tc 0.218 F 2/3 ( (3 / 2)) (2m) N 2 f ( p m ) p The free Fermi gas expression cannot describe BEC transition. Breakdown of the mean-field theory at T>0 2. Coupled Fermion-Boson model p g2 1 1 (U ) tanh 2n 2m p 2 p 2 N 2 nB ( qB 2n 2m ) 2 f ( p m ) 2 N B 0 N F 0 q qB q 2 / 2M p Weak-coupling regime (2n>>2m): N B 0 0 m ~ F BCS Strong-coupling regime (2n~ 2m0): Gap eq.: m 0 N F 0 0 N nB ( bB ) 2 q ideal molecular Bose gas!! 2 (n / 2) 2 / 3 Tc 0.218 F ( (3 / 2)) 2 / 3 (2m) OK! Crossover region: Gap eq.: 2n 2m A finite energy gap exists in the molecular excitations even at Tc. Why does the mean field theory break down at T>0? The mean field theory ignores fluctuation effects. This approximation is valid at T=0, or at finite temperature in the weak interaction regime (weak-coupling BCS regime). However, it does not work when the interaction is strong (=BEC regime) at T>0, where thermally excited pairing fluctuations play crucial roles. We need to improve the BCS theory at finite temperaures going past the mean-field approximation. strong-coupling theory = mean field theory + fluctuation effects Gaussian fluctuation theory (Nozières and Schmitt-Rink: NSR) How to improve the theory (BCS model) In the strong-coupling limit, the gap equation gives 1 m 2mas2 This equals the binding energy of a Cooper-pair obtained at T=0. This means that molecules does not dissociate even at Tc. Thus, one expects that Tc is essentially the same as that for an ideal molecular Bose gas (NB=N/2, M=2m). 2 (n / 2) 2 / 3 Tc 0.218 F 2/3 ( (3 / 2)) (2m) N 2 f ( p m ) p The free Fermi gas expression cannot describe BEC transition. How to improve the theory (BCS model) In the strong-coupling limit, the gap equation gives 1 m 2mas2 This equals the binding energy of a Cooper-pair obtained at T=0. This means that molecules does not dissociate even at Tc. Thus, one expects that Tc is essentially the same as that for an ideal molecular Bose gas (NB=N/2, M=2m). 2 (n / 2) 2 / 3 Tc 0.218 F 2/3 ( (3 / 2)) (2m) N 2 f ( p m ) p The free Fermi gas expression cannot describe BEC transition. Strong-coupling theory at Tc (Nozières and Schmitt-Rink: NSR) Importance of thermal pairing fluctuations to consistently describe the BCS-BEC crossover at Tc. We calculate the thermodynamic potential W including pairing fluctuations. Then we derive the number equation using the identity W N m BCS model H ( p m )c†p c p U p c†p qc†p 'q c p 'c p H 0 H1 p , p ', q We treat the interaction perturbatively Perturbation theory in the quantum statistical mechanics d H1 ( ) R( ) T e 0 H H 0 H1 H1 ( ) e H0 H1e H0 e H e H 0 R( ) † c ( ) c ( 2 ) 1 2 † 1 T c( 1 )c ( 2 ) † c ( 2 )c( 1 ) 1 2 time-ordered product (+: boson, -: fermion) Z tre H W T log Z T log tr e Z0 tre H0 H0 tr e H0 R( ) R( ) W0 T log Z0 W 0 T log R( ) 0 Perturbation theory in the quantum statistical mechanics non-perturbative part p c†p c p W0 T log Z 0 T log tr e p NF 0 2T log(1 e p ) p W0 2 f ( p ) m p ( p p m ) (free Fermi gas) perturbative part W T log R( ) 0 T R( ) c 1 Linked cluster theorem: We may only take into account connected diagrams in this calculation. N W 2 m p n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m c Perturbation theory in the quantum statistical mechanics N W 2 m p n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m c causality ( i ) (1) n=1 W1 U d p , p ', q T c†p q / 2 ( )c† p q / 2 ( )c p ' q / 2 ( )c p ' q / 2 ( ) 0 c Wick’s theorem: One can decompose <cccc> into the sum of all possible combinations in terms of <cc><cc> T c†p q / 2 ( )c† p q / 2 ( )c p ' q / 2 ( )c p ' q / 2 ( ) p , p ', q T c†p q / 2 ( )c† p q / 2 ( ) T c†p q / 2 ( )c p ' q / 2 ( ) T c p ' q / 2 ( )c†p q / 2 ( ) 0 p , p ', q 0 p , p ', q p , p ', q 0 T c p ' q / 2 ( )c p ' q / 2 ( ) 0 T c† p q / 2 ( )c p ' q / 2 ( ) T c p q / 2 ( )c †p ' q / 2 ( ) 0 . 0 Perturbation theory in the quantum statistical mechanics N W 2 m p (1) n=1 W1 U n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m d p , p ', q U p ,q T c†p q / 2 ( )c† p q / 2 ( )c p ' q / 2 ( )c p ' q / 2 ( ) 0 d G p q / 2 ( )G p q / 2 ( ) c antisymmetric condition 0 Gp ( ') T c p ( )c†p ( ') c 0 Single-particle thermal Green’s function This is one of the most important functions in many-body quantum field theory. Perturbation theory in the quantum statistical mechanics Gp ( ') T c p ( )c ( ') † p 0 Fermi dolphin c p ( ) c ( ') † p Fermi sea (ground state) Perturbation theory in the quantum statistical mechanics Gp ( ) T c p ( )c†p (0) antisymmetric condition G p ( ) 1 e in 0 e p G ( ) c( )c † (0) G p (in ) G p (in ) d e 0 in 0 1 tr e H 0 e( ) H 0 ce ( ) H 0 c † Z0 1 tr e H 0 ce H 0 e H 0 c † Z0 1 tr e H 0 c†e H 0 ce H 0 Z0 1 tr e H 0 c( )c † G ( ) Z0 n T (2n 1) n [0, ] (1 f ( p )) n 0, 1, 2,...... Fermion Matsubara frequency 1 G p ( ) in p Perturbation theory in the quantum statistical mechanics N W 2 m p n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m c (1) n=1 W1 U p ,q d G p q / 2 ( )G p q / 2 ( ) 0 U ein n q ,n n 1 G p, p q / 2 (in in n )G p q / 2 (in ) U ein n (q, in n ) n q ,n n n n 2n T n 0, 1, 2,...... (q, in n ) 1 G p, p q / 2 (in in n )G p q / 2 (in ) n Physically, this polarization function describes pairing fluctuations. The sum of the fermion Matsubara frequencies can be carry out using the knowledge of the complex integration. How to sum up Matsubara frequencies Im[Z] in × × Re[Z] × × Pole of A(z) × C1 C × × 1 A(in ) n (q, in n ) 1 dzf ( z ) A( z ) 2 i C 1 2 i p dzf ( z) C f ( z) 1 e z 1 1 1 z in n p q / 2 z p q / 2 Taking the poles at z pq / 2 in n , and z pq / 2 , we obtain Perturbation theory in the quantum statistical mechanics N W 2 m p n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m (1) n=1 W1 UT ein (q, in n ) n n 2n T n q ,n n (q, in n ) p n 0, 1, 2,...... 1 f ( p q / 2 ) f ( p q / 2 ) in n p q / 2 p q / 2 (2) n=2 (a) connected type U 2G(1 2 )G( 2 1 )G(1 2 )G( 2 1 ) (b) disconnected type U 2G(1 1 )G(1 1 )G( 2 2 )G( 2 2 ) Gp (1 2 ) T c p (1 )c†p ( 2 ) 0 1 2 c Linked cluster theorem (a) connected type (b) disconnected type G(1 2 )G( 2 1 )G(1 2 )G( 2 1 ) G(1 2 ) 1 U G(1 1 )G(1 1 )G( 2 2 )G( 2 2 ) G( 2 1 ) 1 2 U 2 U U 1 2 We may only consider the connected type (linked cluster theorem). U2 2 W2 T (q, in n ) 2 q ,n n Perturbation theory in the quantum statistical mechanics N W 2 m p n 1 f ( p m ) T R( ) c N F 0 T T d H1 ( ) m 0 n 1 n ! m (3) n=3,4,5….. Among various diagram, we only take into account the following types, describing pairing fluctuations. W + + +…….. 1 T ein n (q, in n )2 T ein n log 1 U (q, in n ) q ,n n n 1 n q ,n n N W NF 0 T ein n log 1 U (q, in n ) m m q ,n n c Gaussian fluctuation theory for Fermi superfluid at Tc (NSR) 4 as 1 m p ( p m ) 1 1 tanh 2 2 p 2( p m ) 4 a in n s N 2 f ( p m ) T e log 1 m m p q ,n n 1 (q, in n ) p 2 p Tc and m are determined self-consistently for a given value of the pairing interaction. Gaussian fluctuation theory in the BCS regime (BCS limit: (kF as )1 1 ) 4 as 1 m p ( p m ) 1 1 tanh 2 2 p 2( p m ) 4 a in n s N 2 f ( p m ) T e log 1 m m p q ,n n 1 (q, in n ) p 2 p m F (Tc TF ) The resulting equation is the ordinary BCS gap equation at Tc in the weak-coupling superconductivity. Gaussian fluctuation theory in the BEC regime (BEC limit: (kF as )1 1 ) 4 as 1 m p ( p m ) 1 1 tanh 2 2 p 2( p m ) 4 a in n s N 2 f ( p m ) T e log 1 m m p q ,n n 1 m 2mas2 1 (q, in n ) p 2 p Expanding around =q=0, we obtain condition for BEC of N/2 Bose gas. N 1 2 mB 1 q2 in n e nB ( mB ) 2 q 2(2m) q ,n n b in n mB 2(2m) |m| 2 [1 2 ma | m |] 0 s 2 2mas BCS-BEC crossover behavior of Tc (Gaussian) 40 K :| 9 / 2, 7 / 2 | 9 / 2, 9 / 2 0.218TF BEC BCS C. A. Regal, et al. PRL 92 (2004) 040403. m / F 1/ 2mas2 Extension to the coupled Fermion-Boson model two hyper-fine states: H p cp c p Eqbqbq g bq†c pq / 2c pq / 2 h.c. U c pc pc p 'c p ' q2 Eq 2n 2M Feshbach resonance Fermion Boson 2n NFermi+2NBoson= N H-mN Extension to the coupled Fermion-Boson model m:fluctuation contribution to thermodynamic potential W W MF D0 U (q) + Nozieres and Schmitt-Rink Feshbach resonance W N N F0 N B0 T log 1 U eff (q)(q) m q 2 g U eff (0) U 2n 2m U eff (q) U g 2 D0 (q) Extension to the coupled Fermion-Boson model p g2 1 1 (U ) tanh 2n 2m p 2 p 2 g2 in n N 2NB0 NF 0 T e log 1 U (q, in n ) B m q ,n n in n ( q 2n 2m ) In the weak-coupling BCS regime, the number equation simply gives m=F, so that the ordinary BCS theory is obtained. In the strong-coupling BEC regime, the number equation reduces to the equation for Tc of a N/2 ideal Bose gas: N q2 nB ( ) 2 2(2m) b BCS-BEC crossover at Tc Theory Experiment (coupled fermion-boson model for the Feshbach resonance) 40 K Tc 0.218TF BCS BEC BCS BEC Regal, et al. PRL 92 (2004) 040403. Chapter 10. BCS-BEC crossover at finite temperatures 2. Superfluid phase below Tc Extended NSR in the superfluid phase below Tc H ( p m )c†p c p c†p qc†p 'qc p 'c p p p , p ', q ( p m )c†p c p (c†pc† p c pc p ) p MF (BCS) part p j (q) †p q j p p U 4 (q) (q) (q) (q) Fluctuation part 1 1 2 2 q generalized density operator 1 2 c p p † : Nambu field c p : amplitude fluctuations : phase fluctuations j : Pauli matrix Extended NSR in the superfluid phase below Tc gap equation (below Tc) Ep 1 1U tanh 2T p 2E p chemical potential U W ij i,j=1,2 ij (q, in n ) T tr i G(p q / 2, in in n ) j G(p q / 2, in ) p ,n G(p, in ) 1 in p 3 1 T 1 ˆ N NF tr log 1 U (q, in n ) 2 q ,n n m 2 11 12 ˆ (q) 21 22 BCS-BEC crossover in the superfluid phase below Tc m F F T TF (T ) ( 0) T / TF (k F aS ) 1 (k F aS ) 1 2 (k F aS ) 1 2 BCS T / Tc (k F as ) 1 Phase diagram in the BCS-BEC crossover Eg T Fermi gas Molecular Bose gas Tc Superfluid phase strong U weak U BCS BEC Tc strong U weak U JILA 2004 Advanced Condensed Matter Physics Supplementary materials What do we learn from Tc/TF ~ 0.2 ? Tc / TF 0.2 strong attractive interaction BCS-BEC crossover 0 weak attractive interaction weak attractive interaction strong attractive interaction What do we learn from Tc/TF ~ 0.2 ? The Fermi superfluid phase transiton temperature has this upper limit (which is given by the GOD)! Fermi gas: TF ~ 1μK metal: TF ~ 104 K Tc / TF 0.2 BCS-BEC crossover 0 Tc ~ 2000K<<300K weak attractive interaction Room temperature superconductivity (Tc~300K) is not prohibitted by nature! (Thanks, GOD!) strong attractive interaction BEC (Bose-Einstein condensation) = superfluid state Condition for superflow: There is no dissipation by elementary excitations Landau criterion v superflow Total mass of fluid: M v' q superflow elementary excitation Mv Mv ' q Total energy E’ of the fluid after the elementary excitation q is excited: 2 1 1 q 1 q E ' Mv '2 q M ( v ) 2 q Mv 2 v q q 2 2 M 2 2M Initial energy E of the flow BEC (Bose-Einstein condensation) = superfluid state v superflow v' q superflow elementary excitation Mv Mv ' q Total mass of fluid: M Total energy E’ of the fluid after the elementary excitation q is excited: 2 1 1 q 1 q E ' Mv '2 q M ( v ) 2 q Mv 2 v q q 2 2 M 2 2M Initial energy E of the flow Landau criterion Stability condition q2 v q q 0 2M q vq (M ) BEC (Bose-Einstein condensation) = superfluid state q vq Stability condition = Elementary excitation wavenumber Of excitation speed of fluid The ideal Bose gas does NOT satisfy this condition! 2 Excitation energy of an ideal Bose gas The system energy is lowered by excitations! q2 q q2 2m vq q vq ! 0 q How is the stable superflow realized ?? Stability condition = q vq If the excitation energy has a linear dispersion, then q V q 2 q2 q 2m vq 0 “the stable superflow” is obtained as far as v<V. q How is the linear dispersion realized ?? Bogoliubov A linear dispersion is obtained in the BEC phase when There is a repulsive interaction between bosons. 23Na Superfluid 4He Bose gas BEC q v q observed velocity Andrews et al. PRL 79 (1997) 553 How does q2 dispersion become q-linear? m …. 0 a 2a 3a 2 4a q2 2 q q 2m 5a X …. How does q2 dispersion become q-linear? Natural length = a m …. 0 a m 2 K L xj 2 j 2 2a 3a 4a 5a X …. 2 ( x x ) j j 1 j m x j K ( x j x j 1 ) K ( x j x j 1 ) iq t iq ( ja ) x j Ae e 4K qa q sin ~ m 2 K 4K 2 qa 2 2cos qa sin m m 2 2 q K aq m (q ~ 0) Linear dispersion! How does q2 dispersion become q-linear? m …. 0 a 2a 3a 4K qa q sin ~ m 2 4a K aq m 5a (q ~ 0) This is not a one-particle excitation but a collective motion. X …. The essesnse of the superfluid phase The phase of the wavefunction of each boson is aligned in the BEC. i1 | ( x1 ) | e i | ( x1 ) | e | ( x2 ) | e i2 | ( x2 ) | e i i3 | ( x4 ) | ei4 i | ( x4 ) | ei | ( x3 ) | e | ( x3 ) | e