Superfluidity of p-wave and s-wave atomic Fermi gases in optical lattices

a r X i v :c o n d -m a t /0508134v 2 [c o n d -m a t .s u p r -c o n ] 7 N o v 2005

Super?uidity of p -wave and s -wave atomic Fermi gases in optical lattices

M.Iskin and C.A.R.S′a de Melo

School of Physics,Georgia Institute of Technology,Atlanta,Georgia 30332,USA

(Dated:February 2,2008)

We consider p -wave pairing of single hyper?ne state and s -wave pairing of two hyper?ne states ultracold atomic gases trapped in quasi-two-dimensional optical lattices.First,we analyse super?uid properties of p -wave and s -wave symmetries in the strictly weak coupling BCS regime where we discuss the order parameter,chemical potential,critical temperature,atomic compressibility and super?uid density as a function of ?lling factor for tetragonal and orthorhombic optical lattices.Second,we analyse super?uid properties of p -wave and s -wave super?uids in the evolution from BCS to BEC regime at low temperatures (T ≈0),where we discuss the changes in the quasi-particle excitation spectrum,chemical potential,atomic compressibility,Cooper pair size and momentum distribution as a function of ?lling factor and interaction strength for tetragonal and orthorhombic optical lattices.

PACS numbers:74.25.Bt ,74.25.Dw,03.75.Ss,03.75.Hh

I.INTRODUCTION

Tunable optical lattices have been extensively used to study phase transitions in bosonic atomic gases 1,2,since they allow the controlled manipulation of the particle density n ,and of the ratio between the particle trans-fer energy t ,and the interparticle interaction strength V 3,4.This kind of control is not fully present in standard fermionic condensed matter systems,and has hindered the development of experiments that could probe sys-tematically the e?ects of strong correlations as a function of n and t/V .However,fermionic atomic gases like 6Li and 40K have been succesfully trapped,and their normal state and super?uid properties are beginning to be stud-ied 5,6,7,8,9.Because of the greater tunability of experi-mental parameters,novel super?uid phases may be more easily accessible in the experiments involving ultracold atomic gases.For instance,single hyper?ne state (SHS)ultracold atomic systems are ideal candidates for the ob-servation of novel triplet super?uid phases and for testing theoretical models that were proposed earlier.Thus,it is only natural to propose that optical lattices could be used to study the normal state and super?uid properties of ultracold fermionic systems as a function of n ,t/V and lattice symmetry.These systems are of broad inter-est not only for the atomic physics community but also for the nuclear,condensed matter and more generally for the many-body physics communities,where models for super?uidity have been investigated in various contexts.The interaction between induced dipole moments of atoms and the electric ?eld of laser beams is used to trap atoms,particularly alkali atoms,in optical lattices.Al-kali atoms have only one electron (S =1/2)out of closed shells.This electron is in a zero orbital angular momen-tum (L =0)channel,and its total angular momentum J =L +S gives J =1/2.The nuclear angular mo-mentum I and electron angular momentum J are com-bined in a hyper?ne state with total angular momentum F =I +J which gives F =I ±1/2for alkalis.Fur-thermore,the electron and nuclear spins are coupled by

the hyper?ne interaction that splits the atomic levels in the absense of magnetic ?eld H hf ∝I ·J .A weak mag-netic ?eld causes Zeeman splitting of the hyper?ne levels |F,m F >with di?erent m F ,which can be made to corre-spond to pseudo-spin labels.Trapping one 11,12,13,14,15,16,two 17,18,19,20,three 21and four 22hyper?ne states were considered by several authors.However,it is also known that trapping more hyper?ne states increases the num-ber of channels through which the gas can decay.There-fore,trapping one or two hyper?ne states of ultracold fermionic gases is experimentally more plausible.

For paired identical fermions,the Pauli exclusion prin-ciple requires the total pair wave function to be anti-symmetric.The total orbital angular momentum should be odd for pseudo-spin symmetric pairs and even for pseudo-spin anti-symmetric ones.Therefore in the case of trapping two hyper?ne states (THS),s -wave scatter-ing of atoms between fermions from di?erent hyper?ne states is dominant.Thus,one expects that the super-?uid ground state of such two-component Fermi gases to be s -wave and pseudo-spin singlet.Presently there is experimental evidence that 40K (Ref.23,24)and 6Li (Ref.25,26,27,28,29)can form weakly and tightly bound atom pairs,when the magnetic ?eld is swept through an s -wave Feshbach resonance.

However,the properties of SHS ultracold fermions and their possible super?uid behaviour are beginning to be investigated 30,31,32,33,34.These systems are probably the next frontier for experiments with ultracold atoms.When identical fermionic atoms are trapped in a single hyper?ne state,the interaction between them is strongly in?uenced by the Pauli exclusion principle,which pro-hibits s -wave scattering of atoms in identical pseudo-spin states.As a result,in SHS degenerate Fermi gases,two fermions can interact with each other at best via p -wave scattering.Thus,one expects that the super?uid ground state of such SHS Fermi gases to be p -wave and pseudo-spin triplet.

In the p -wave channel,if the atom-atom interactions are e?ectively attractive then the onset for the forma-

tion of Cooper-pairs in three dimensions occurs at a temperature 10,16T c ≈?F exp[?π/2(k F a sc )3]in the BCS regime,where ?F is the Fermi energy and a sc is the p -wave scattering length.Unfortunately,this temperature is too low to be observed experimentally.However,in the presence of Feshbach resonances 10,30,p -wave interac-tions can be enhanced,and the critical temperature for super?uidity is expected to increase to experimentally accessible values.On the other hand,we show that the pseudo-spin triplet (p -wave)weak coupling limit in op-tical lattices (like in the singlet case 8)may be su?cient to produce a super?uid critical temperature that is ac-cessible experimentally.Furthermore,several interesting super?uid properties can be investigated in p -wave sys-tems as the system is tuned from the BCS to the BEC regimes 10,11,12,13,14,15,16,17,18.In this paper,we address super?uid properties of ultracold fermionic atoms in op-tical lattices for both SHS p -wave and THS s -wave states as a function of atom ?lling factor,interaction strength,temperature and lattice symmetry.

The rest of the paper is organized as follows.In Sec.II,we discuss

the e?ective action method and the saddle point approximation for a lattice Hamiltonian with at-tractive interactions in the p -wave and s -wave channels.We also derive the order parameter,critical temperature and number equations in the saddle point approximation and discuss pseudo-spin triplet states in the d -vector for-malism in this section.In Sec.III,we analyse super?uid properties of SHS p -wave and THS s -wave symmetries in the strictly weak coupling BCS regime.There,we discuss the order parameter,chemical potential,critical temper-ature,atomic compressibility,and super?uid density as a function of ?lling factor for tetragonal and orthorhom-bic optical lattices.In Sec.IV,we analyse super?uid properties of SHS p -wave and s -wave super?uids in the evolution from BCS to BEC regime at low temperatures (T ≈0).There,we discuss changes in quasi-particle ex-citation spectrum,chemical potential,atomic compress-ibility,Cooper pair size and momentum distribution as a function of ?lling factor and interaction strength for tetragonal and orthorhombic optical lattices.Finally,a summary of our conclusions is given in Sec.V.

https://www.doczj.com/doc/db15132416.html,TTICE HAMILTONIAN

In this manuscript,we consider quasi-two-dimensional

optical lattices with a periodic trapping potential of

the form U (r )= i U 0,i cos 2

(k i x i ),with U 0,z ?min {U 0,x ,U 0,y },which strongly suppresses tunneling along the ?z direction.This is a non-essential assumption,which just simpli?es the calculations,but still describes an experimentally relevant situation.Here x i =x,y,or z labels the spatial coordinates,k i =2π/λi is the wave-length,and U 0,i is the potential well depth along direction ?x

i ,respectively.The parameters U 0,i are proportional to the laser intensity along each direction,and it is typically several times the one photon recoil energy E R such that

tunneling is small and the tight-binding approximation can be used.

Thus,in the presence of magnetic ?eld h ,we consider the following quasi-two-dimensional lattice Hamiltonian (already in momentum space)for an SHS p -wave Fermi gas H =

k

ξ(k )a ?k ↑a k ↑+

1

2

k ,k ′,σ,q

V s b ?k ,q b k ′,q ,

(4)

where the pseudo-spinsσ=(↑,↓)label the trapped hy-per?ne states represented by the creation operator a?

kσ

,

and b?

k,q =a?

k+q/2,σ

a?

?k+q/2,?σ

.Here,V s=?|V0,s|is

the attractive s-wave interaction.

In the following section,we describe the e?ective ac-tion method for the SHS p-wave Hamiltonian in Eq.(1) and we discuss its saddle point order parameter and num-ber equations,as well as the critical temperature.This method can be applied to the s-wave Hamiltonian in Eq.(4),and similar equations can be derived for s-wave case,but we do not repeat this derivation here.

A.E?ective Action Formalism

In the imaginary-time functional integration formal-ism35(β=1/T and unitsˉh=k B=1),the partition function can be written as

Z= D[a?,a]e?S(5) with an action given by

S= β0dτ k a?k↑(τ)(?τ)a k↑(τ)+H(τ) .(6)

The Hamiltonian given in Eq.(1)can be rewritten in the form

H(τ)= kξ(k)a?k↑(τ)a k↑(τ)+ q B?q(τ)V

2ξ(k)δp,p′?Tr ln

β

2

)σ?+Γ?(

p+p′

2 qˉΛ?(q)F?1(q)ˉΛ(q),(11)

where S0is the saddle point action,the4-component vec-

torˉΛ?(q)is such thatˉΛ?(q)=[Λ?(q),Λ(?q)]and F?1(q)

is the inverse?uctuation propagator.The?uctuation

term in the action leads to a correction to the thermo-

dynamic potential,which can be written as?gauss=

?0+??uct with??uct=β?1 q ln det[F?1(q)/2].In

weak coupling for all temperatures and for all couplings

at low temperatures,it can be shown that39Gaussian

corrections S?uct to the gaussian action S gauss are small

in the quasi-two-dimensional case,and therefore,we con-

sider only the saddle point action

S0=2β??0V?1?0+ p β2G0?1 ,(12)

where the inverse Nambu propagator becomes

G?10=iw?σ0?ξ(k)σ3+??0Γ(k)σ?+Γ?(k)?0σ+.(13)

Notice that,this will not be the case close to the critical

temperatures T c for intermediate and strong interactions,

since the inclusion of?uctuation corrections change the

number equation considerably and are necessary to pro-

duce correct behaviour38.

B.Saddle-Point Equations

The saddle point conditionδS0/δ??0=0leads to a

matrix equation for the order parameter

?0=M?0,(14)

where matrix M has the following matrix elements

M ij= k V0,i sin(k i a i)sin(k j a j)2.(15)

Here,we introduce the quasi-particle energy

E(k)=(ξ2(k)+|?(k)|2)1

ξ(k)

tanh

ξ(k)

Both the order parameter and critical temperature equations have to be solved simultaneously with the num-ber equation N =???/?μwhere ?is the full thermo-dynamic potential.This leads to two contributions to the number equation N ≈N gauss =N 0+N ?uct .Here,N 0=???0/?μis the saddle point number equation,where ?0=S 0/βis the saddle point thermodynamic potential,and N ?uct =????uct /?μis the ?uctuation contribution.In the low temperature limit (T ≈0)for any coupling,or in the weak coupling limit for any tem-perature (T ≤T c )the ?uctuation contribution

N ?uct =?T

q

?[det F ?1(q )]

det F ?1(q )

(19)

is small and negligible compared to the saddle point value N 038.Thus,the number equation becomes N ≈N 0,given by

N = k

n (k ),(20)

where n (k )is the momentum distribution de?ned by

n (k )=

1

E (k )

tanh

βE (k )

2E (k )

tanh

βE (k )

ξ2(k )+|?0,s |2is the quasi-particle en-ergy spectrum.The critical temperature is determined when ?0,s =0,which leads to

1=

k

V 0,s

2T c

.(23)

Both order parameter and critical temperature equations have to be solved simultaneously with the number equa-tion

N = k σ

n (k ),(24)

where the momentum distribution n (k )is de?ned in

Eq.(21).

C.

Spin Triplet (SHS p -wave)States

In general,the pseudo-spin triplet order parameter can be written in the standard form 36

O (k )=

?d 1(k )+id 2(k )d 3(k )

d 3(k )d 1(k )+id 2(k )

,(25)where d (k )is an odd vector function of k .In our SHS p -wave interaction O ↑↑(k )=?(k ),therefore,d 3(k )=0and d 1(k )=?id 2(k )which leads to d (k )=d 1(k )(1,i,0).Within the irreducible representations of the D 4h (D 2h )group in the tetragonal (orthorhombic)lattices,37our exotic SHS p -wave state corresponds to the 3E u (n )rep-resentation with a d -vector given by

d (k )=f (k )(1,i,0),

(26)

where f (k )=AX +BY ,and X and Y are sin(k x a x )

and sin(k y a y ),respectively.Notice that,this state also breaks time-reversal symmetry,as expected from a fully spin-polarized state.

In the tetragonal lattice,the stable solution for our model corresponds to the case A =B =0,and thus to the 3E u (d )representation,where spin-orbit symme-try is preserved,but both spin and orbit symmetries are independently broken.In the orthorhombic lattice,the stable solutions correspond to either A =0,B =0or A =0,B =0,thus leading to the 3E u (b )representation.Notice that,depending on the lattice anisotropy,there are three distinct solutions to the order parameter equa-tion Eq.(14)in relation to Eq.(26).The ?rst solution (case 1)occurs when ?0,x =0and ?0,y =0,which cor-responds to a d -vector with A =0and B =0.Similarly,a second solution (case 2)occurs when ?0,x =0and ?0,y =0,and it corresponds to a d -vector with A =0and B =0).Third and ?nal solution (case 3)occurs when ?0,x =?0,y =0,which corresponds to a d -vector with A =B =0.In a tetragonal lattice both directions are degenerate (case 3),but even small anisotropies in the optical lattice spacings,transfer energies (t x ,t y ),or optical lattice potentials (U 0,x ,U 0,y )lifts the degeneracy and throws the system into either case 1or 2.

III.

SUPERFLUID PROPERTIES IN THE WEAK COUPLING BCS REGIME

Our main interest is in tetragonal (square)lattices,however,we also want to investigate the e?ects of small anisotropies in optical lattice lengths.For this purpose we investigate ?ve di?erent cases that may be encoun-tered experimentally.

Case (I)corresponds to lattice spacings a x =a y ,to transfer integrals t x =t y ,and to interaction strengths V 0,x =V 0,y .Case (II)corresponds to a x =a y ,t x =t y and V 0,x =V 0,y .Case (III)corresponds to a x =a y ,t x =t y and V 0,x =V 0,y .Case (IV)corresponds to a x =a y ,t x =t y and V 0,x =V 0,y .Case (V)corresponds to a x =a y ,t x =t y and V 0,x =V 0,y .Notice that cases (II)and (IV)are equivalent except for a unit cell volume normalization.The same is true for cases (III)and (V).Case (I)is e?ectively a tetragonal lattice,while cases (II)through (V)are e?ectively orthorhombic lattices.Notice that even though the optical lattice spacings a x =a y in cases (II)and (III),these cases correspond e?ectively to

”orthorhombic”lattices because the transfer energies t x and t y are di?erent.

Experimentally it may be easier to use only one type of laser with speci?ed wavelength,and thus generate a lattice with a x=a y,and change the focus width in one direction(say along?x)while keeping the width in the other direction?xed(say along?y).The change in focus width along the?x direction will modify the transfer inte-gral t x by either reducing it or enlarging it with respect to t y.This type of experiment can cover cases(I),(II) and(III).A second type of experiment(a bit harder) could use two types of lasers with di?erent wavelengths and produce a lattice with a x=a y.Further control of the focal widths can produce the situations encountered in cases(IV)and(V).

We concentrate here on cases(I),(II)and(III),where we set a x=a y=a.For case(I)we choose t x=t y=t, and V0,x=V0,y=V0.For case(II)we choose t x= (1+α)t,t y=t,and V0,x=V0,y=V0.For case(III) we choose t x=(1+α)t,t y=t,and V0,x=(1+α)V0, V0,y=V0.Here we take V0=0.6E0with E0=2t. In these cases,the transfer energies and the parame-tersαcan be determined by using exponentially decay-ing on-site Wannier functions and the WKB approxima-tion.When the lattice spacings a x=a y=a are the same but the optical lattice potentials U0,x>U0,y=U0 are slightly di?erent,the transfer integral t x/t is propor-tional to exp[?π(U0,x?U0)/

U0/E R),for smallδ.In the following sec-tions,we discuss two possibilities whereδ=0.01cor-responding toα≈0.1andδ=0.05corresponding to α≈0.6for U0≈2.5E R.As we increase(decrease)the ratioδ,both interactions and tunneling rate increase(de-crease)in?x direction.

Notice that a di?erent choice of on-site Wannier func-tions does not change our general conclusions,the only qualitative di?erence is that t x and V0,x are di?erent functions ofα.

A.Density of Fermionic States

The critical temperature and super?uid properties of Fermi systems depend highly on the order parameter symmetry,as can be seen by rewriting Eq.(18)as 1= i=x,y V0,i t x+t?t x?t dεtanhε??μ2(ε??μ)D p,i(ε),(27) where we de?ne an e?ective density of states(EDOS) D p,i(ε)= kδ[ε?ε(k)][√

2sin(k i a i)is the symmetry factor related to the SHS p-wave order parameter.We transformed discrete summations over k-space to continuous integrations to obtain

D p,x(ε)= f i dk y a

t x|≤1in the?rst Brillouin zone(1BZ). Plots of EDOS are shown in Figs.1a and1b for three

cases:α=0(hollow squares),α=0.1(solid squares), andα=0.6(line).

In Fig.1a,we plot EDOS which is smooth and maxi-mal at half-?lling in square lattices(α=0).For the SHS p-wave symmetry discussed,V0,i D p,i(ε)plays the role of a dimensionless coupling parameter which controls the critical temperature.This is simply because it is much easier to form Cooper pairs with a small attractive inter-action(and lower the free energy of the Fermi system) when the density of single fermion states is high.Ad-ditionally,EDOS decreases and?nally vanishes at the band edges where a small ratio of T c/T F was predicted from continuum models.However,this is not the case around half-?lling,and we expect weak interactions to be su?cient for the observation of super?uidity.

In contrast,the symmetry factor is1in the s-wave case and EDOS becomes

D(ε)= kδ[ε?ε(k)],(30)

which is identical to the density of single fermion states (DOS)of normal fermions.A similar calculation yields

D(ε)= f i dk y a

t x|≤1in the1BZ.Plots of DOS are shown in Fig.1c and Fig.1d for three cases:α=0(hollow square),α=0.1(solid square),andα=0.6(line).

For instance,notice that in the tetragonal case of a two dimensional lattice,the DOS at half-?lling is very large

0.1 0.2 0.3 0.4 0.5

-1-0.5

0 0.5 1D p ,x (εr )εr (a)

0.1 0.2 0.3 0.4 0.5

-1-0.5

0 0.5 1

εr (b)

0.1 0.2 0.3 0.4 0.5

-1-0.5

0 0.5 1

D (εr )εr

(c)

0.1 0.2 0.3 0.4 0.5-1

-0.5

0 0.5

1

εr

(d)

FIG.1:Plots of EDOS versus reduced energy εr =ε/E 0for SHS p -wave symmetry when a)α=0(hollow square),and

b)α=0.1(solid square)and α=0.6(line).In addition plots of DOS for s -wave symmetry when c)α=0(hollow square),and d)α=0.1(solid square)and α=0.6(line)are shown.

and tends to in?nity logarithmically (Fig.

1c).However,away from half-?lling the DOS decreases rapidly to a con-stant at the minimum and maximum band edges.This constant DOS for very low or very high ?lling factors is equivalent to the DOS of the continuous (homogeneous)systems since at the bottom or at the top of the band,the single fermion dispersion energy can be approximated by a parabola in momentum space.Furthermore,for s -wave super?uids,V 0D (ε)is the dimensionless coupling param-eter which controls the value of critical temperature T c .This important quantity (T c )is discussed next.

B.

Critical Temperature

Plots of reduced critical temperature T r =T c /E 0as a function of the number of atoms per unit cell (N c )are shown in Fig.2for SHS p -wave and s -wave super?uids in cases (I),(II),and (III).Here,T c is the critical tem-perature calculated from Eqs.(18)or (27)and E 0=2t is the half-?lling Fermi energy for the square lattice with respect to the bottom of the band.

The square lattice case (I)is shown in Figs.2a and 2c (hollow squares)for SHS p -wave and s -wave super?uids,respectively.Notice that T r is maximal at half-?lling,and that it has a value 0.01,which is much higher than the theoretically predicted T c from a continuum model,

0 0.004

0.008 0.012 0.016

0.3

0.4

0.5 0.6

0.7

N c (a)

0 0.004

0.008 0.012 0.016 0.3

0.4

0.5 0.6 0.7

N c (b)

0 0.004

0.008 0.012 0.016

0.3

0.4

0.5 0.6 0.7N c

(c)

0 0.004

0.008 0.012 0.016

0.3

0.4

0.5 0.6 0.7

N c

(d)

FIG.2:Plots of reduced critical temperature T r =T c /E 0versus ?lling factor N c for SHS p -wave symmetry are shown for cases a)(I)and (II)and b)(I)and (III).In addition,plots for s -wave symmetry is shown for cases c)(I)and (II),and d)(I)and (III).Plots for α=0,0.1and 0.6are shown with hollow squares,solid squares and line,respectively.

and comparable to experimentally attainable tempera-tures T/T F ≈0.0440.This implies that the super?uid regime of SHS fermion gases may be observed experimen-tally in a lattice,even in the limit of weak interactions (BCS regime).The observability of a super?uid tran-sition in SHS p -wave Fermi systems is clearly enhanced when the system is driven through a Feshbach resonance (in a lattice or in the continuum),as T c is expected to increase further in this case,however our calculations indicate that the weak interaction (BCS)limit may be su?cient in the lattice case.This possibility for Fermi gases in optical lattices has a parallel in the experimental observation of super?uid-insulator transitions in weakly interacting Bose systems in optical lattices 2.

In case (II),while we keep interaction strengths V 0,x =V 0,y =V 0the same,we vary only the transfer energy along ?x direction t x =(1+α)t .This case is shown in Fig.2a and Fig.2c for SHS p -wave and s -wave super?u-ids,respectively.Here,solid squares (line)correspond to α=0.1(α=0.6).

For SHS p -wave super?uids,an in?nitesimally small anisotropy in the transfer energy ratio t x /t y causes a dis-continous jump in EDOS at half-?lling because of the symmetry factor enhancement,however T c at half-?lling is the same as in case (I),because the discontinuity at

EDOS is limited to a single point.As the anisotropy ra-

tio increases(α=0.1),the discontinuous point in EDOS at half-?lling expands into a region where EDOS is higher than in case(I).Thus over a narrow range around half-?lling,T c is larger than in case(I),but varies smoothly as a function ofαor N c.Further increase ofαreduces EDOS near half-?lling pushing T c down.This behavior is characteristic of the SHS p-wave state through its symme-try

factors.In contrast,for s-wave super?uids shown in

Fig.2c,T c decreases for?xed N c around half-?lling with

increasing anisotropy(at least for0<α<0.6).Fur-thermore,T c develops two maxima which are symmetric

around half-?lling.This behavior in T c is also related to the DOS of the normal fermions which is shown in

Fig.1.Notice here that,when there is an anisotropy in the transfer energy,T c near half-?lling corresponding to

an SHS p-wave pairing is considerably higher than the T c of an s-wave pairing for the same parameters of interest

(assuming that V0,s=V0,p=V0).

In case(III),we vary both the interaction strength

V0,x=(1+α)V0and the transfer energy t x=(1+α)t along?x direction.This case is shown in Figs.2b and2d

for SHS p-wave and s-wave super?uids,respectively.In these?gures it is assumed that V0,s=V0,p=V0,x.In

case(III),both SHS p-wave and s-wave super?uids have critical temperatures that are much larger than in case

(II),because of the increase in V0,x,in addition to the density of states e?ect,discussed above.Notice again

that T c near half-?lling corresponding to an SHS p-wave pairing is also considerably higher than T c for an s-wave pairing for the same interaction parameters.

In this section,we established that the critical temper-atures for SHS p-wave super?uids in optical lattices can

be comparable or larger to the critical temperatures of s-wave super?uids,depending on the lattice anisotropy

and interaction strength.Thus,we expect that SHS p-wave super?uids may be experimentally attainable.In

the next sections,we discuss some experimentally rel-evant observables which could be used to identify the

super?uid phase of our exotic SHS p-wave state.

C.Order Parameter and Chemical Potential

In this section,we discuss the low-temperature be-

havior of the order parameter?(k)=?0,x sin(k x a x)+?0,y sin(k y a y)de?ned in Eq.(14)for the SHS p-wave state.In case(I)?0,x=?0,y=0,while in cases(II) and(III)?0,x=0and?0,y=0.Here,we also discuss for comparison the singlet s-wave case where?(k)=?0,s,

with?0,s=0for cases(I),(II)and(III).The re-duced amplitudes of the SHS p-wave order parameter ?r=?0,x/E0versus N c are shown in Figs.3a and3b.In addition,we plot s-wave?r=?0,s/E0in Figs.3c and3d for cases(I),(II),and(III).Notice that,the qualitative behaviour of?r as a function of N c is very di?erent from that of the chemical potentialμr=?μ/E0,which is dis-cussed next.

0.01

0.02

0.03

0.04

0.3 0.4 0.5 0.6 0.7

N c

(a)

0.01

0.02

0.03

0.04

0.3 0.4 0.5 0.6 0.7

N c

(b)

0.01

0.02

0.03

0.04

0.3 0.4 0.5 0.6 0.7

N c

(c)

0.01

0.02

0.03

0.04

0.3 0.4 0.5 0.6 0.7

N c

(d)

FIG.3:Plots of zero temperature reduced order parameter amplitude?r=?0,x/E0versus?lling factor N c for SHS p-wave symmetry for cases a)(I)and(II)and b)(I)and (III).In addition,plots for s-wave symmetry?r=?0,s/E0 is shown for cases c)(I)and(II),and d)(I)and(III).Plots forα=0,0.1and0.6are shown with hollow squares,solid squares and line,respectively.

Plots of the low-temperatureμr versus N c are shown in Fig.4for SHS p-wave and s-wave super?uids,respec-tively.Here,we present only the corresponding chem-ical potentials for cases(I)(hollow squares)and(II) (solid squares),whereα=0andα=0.1,respec-tively.Chemical potentials for case(III)are very sim-ilar to those of case(II).Notice that?μis always within the limits of the energy dispersion of the optical lattice,?t x?t

Although the chemical potentials of SHS p-wave and the singlet s-wave cases look similar at?rst glance,they are not.The major qualitative di?erences between the chemical potentials of the SHS p-wave and the singlet s-wave cases are more clearly seen in the derivative ??μ/?N c.The appropriate thermodynamic quantity that is related to this derivative is the atomic compressibility to be discussed next.

D.Atomic Compressibility

The isothermal atomic compressibility at?nite tem-peratures is de?ned byκT(T)=?(?V/?P)T,N/V where

-0.2-0.1 0 0.1 0.2 0.3

0.4

0.5 0.6

0.7

N c

(a)

-0.2-0.1 0 0.1 0.2 0.3

0.4

0.5 0.6

0.7

N c

(b)

FIG.4:Plots of zero temperature reduced chemical potential μr =?μ/E 0versus ?lling factor N c for a)SHS p -wave,and b)s -wave super?uids.Plots for α=0and 0.1are shown with hollow and solid squares,respectively.

V is the volume and P is the pressure of the gas.This can be rewritten as

κT (T )=?1??μ2

T,V =

1

??μ T,V .(32)The partial derivative ?N c /??μis ?N c

2E 3(k )

tanh

βE (k )

E 2(k )

,(33)

where Y (k )=(β/4)sech 2[βE(k )/2]is the Yoshida func-tion.

Plots of the reduced

isothermal atomic compressibility κr =κT (T )/κ0are shown in Fig.5for both SHS p -wave and s -wave super?uids for cases (I)and (II).For case (III),κr is both qualitatively and quantitatively similar to case (II),and it is not shown here.The normalization constant κ0is the SHS p -wave isothermal compressibility evaluated at T =0and half-?lling.

In case (I),κr has a peak at half-?lling and low tem-peratures,and a hump at T c for SHS p -wave super?u-ids.Notice the strong temperature dependence of κr near half-?lling.In the s -wave case there is only a hump both at T c and at low temperatures,and a very weak temperature dependence for all ?lling factors shown.In case (II),κr has the additional feature that the central peak (hump)splits into two due to degeneracy lifting,for the SHS p -wave case.Notice again the drastic di?erence between the values of κr for N c ≈0.45and N c ≈0.55for T ≈0and T ≈T c .However,in the s -wave case the original hump also splits into two,but there is very weak temperature dependence of κr .

To understand the peaks of κr at T ≈0,and its humps at T =T c limits for the SHS p -wave case,we indicate that ?N c /??μ[see Eq.(33)]has two contributions,where the ?rst term dominates at T =0,and only the second term survives at T =T c .Therefore,the ?rst term is responsi-ble for the peaks,and the second term is responsible for the humps.

0.2 0.4 0.6 0.8

1

0.3

0.4

0.5 0.6

0.7

κr

N c (a)

0.2 0.4 0.6 0.8

1 0.3

0.4

0.5 0.6

0.7

N c (b)

0.2 0.4 0.6

0.8 1 0.3

0.4

0.5 0.6

0.7

κr

N c

(c)

0.2

0.4 0.6 0.8 1

0.3

0.4

0.5 0.6

0.7

N c

(d)

FIG.5:Plots of reduced compressibility κr =κT (T )/κ0versus ?lling factor N c for SHS p -wave symmetry in cases a)(I)and b)(II).In addition,plots for s -wave symmetry are shown in cases c)(I)and d)(II)at temperatures T =0(squares),and T =T c (line).Plots for α=0and 0.1are shown with hollow and solid squares,respectively.

The peaks at T ≈0,can be understood by noting that the ?rst contribution to κT (T )in Eq.(33)can be written as

κT (T =0)=

2

E (k )

,

(34)

where n (k )is the momentum distribution and de?ned in Eq.(21).Therefore,the peaks are due to non-vanishing n (k )[1?n (k )]in regions of k -space where E (k )vanishes and n (k )is rapidly changing.

In case (I),the integrand n (k )[1?n (k )]/E (k )has 4k -space points (0,±π),and (±π,0)in the ?rst Brillouin zone (1BZ)where it diverges only when the chemical po-tential ?μ=0(N c =0.5).Similarly in case (II),the in-tegrand diverges at 2k -space points (±π,0)in the 1BZ when ?μ=t x ?t ≈0.021(N c ≈0.55).Furthermore,the integrand diverges at 2k -space points (0,±π)when ?μ=?t x +t ≈?0.021(N c ≈0.45).For other values of ?μ(other ?lling factors),the integrand is well-behaved and does not produce additional peaks in κr .

In contrast,the compressibility peaks at T =0do not exist for s -wave super?uids (see Figs.5c and 5d),where the quasi-particle excitation spectrum E (k )is al-ways gapped.Generally speaking,we expect compress-ibility peaks for nodal super?uids or superconductors,

where quasi-particle energy spectrum vanishes in regions of k-space.

At T=T c,κT(T)is dominated by the second term of the integrand(see Eq.(33))

κT(T=T c)=1

2

.(35)

Therefore,the humps are not related to the order pa-rameter(since at T=T c,?(k)=0),but are due to the peaks appearing in the single fermion(normal)DOS(see Fig.1c and Fig.1d).This is simply becauseκT(T=T c) can be written in terms of DOS(see Eq.30)for both SHS p-wave and s-wave symmetries.Notice that,while DOS have only one peak at half-?lling in case(I)(lead-ing to one hump inκr),DOS has two peaks in cases(II) and(III)(leading to two humps inκr).Thus,at T=T c the humps correspond to a Van Hove phenomenon44,but the atomic compressibility is smooth.Furthermore,we expect humps at T=T c for all pairing symmetries,since these humps are related only to normal state properties but not to the symmetry of the pairing interaction. Theoretically,the calculation of isothermal atomic compressibilityκT(T)is easier than the isentropic atomic compressibilityκS(T).However,performing measure-ments ofκS(T)may be simpler in cold Fermi gases,since the gas expansion upon release from the trap is expected to be nearly isentropic.Fortunately,κS(T)is related to κT(T)via the thermodynamic relation

κS(T)=

C V(T)

V κT(T),(37) where n and V are average density of atoms and vol-ume of the condensate,respectively.From the measure-ment of density?uctuations the isothermal compressibil-ity can be extracted at any temperature T. Furthermore,the spin susceptibility tensor component χηη(T)can be written as a spin-spin response function to a magnetic?eld h=h?ηapplied along an arbitrary η-direction42.In the case of SHS p-wave super?uids,

χηη(T)=??2?

??μ

.(38)

Therefore,for the SHS p-wave case,χηη(T)is directly

related to the isothermal atomic compressibility and is

given by[N2c/(gμB)2]κT(T).Thus,we expect similar

e?ects in the magnetic spin susceptibility as in the case

of isothermal atomic compressibility discussed above.In

cold atoms,the measurement of the spin susceptibility

will be most likely achieved by using techniques similar

to nuclear magnetic resonance(NMR),whereχηη(T)can

be measured via the Knight shift42.

Speci?cally,while we expect only one peak at exactly

half-?lling in the tetragonal lattices at T=0,two peaks

will appear in the orthorhombic lattices for a SHS p-

wave super?uid.However at T=T c,these peaks dis-

appear and turn into humps.Notice that the?lling fac-

tor dependence ofχηη(T)will be qualitatively di?erent

fromκT(T).This is because whileχηη(T)is symmetric

around half-?lling(N c=0.5),κT(T)is not.The relation

between the particle compressibility and the spin suscep-

tibility given above is not valid for the s-wave case.Their

relationship is more complicated in this case,and we do

not discuss it here.Another important property is the

super?uid density to be discussed next.

E.Super?uid Density Tensor

The super?uid density tensor is de?ned as a response

function to phase twists of the order parameter45.In

the approximation used in this paper,the temperature

dependence of its components is given by

ρij=

1

0.5 1 1.5

2 0 0.1 0.2

0.3 0.4 0.5 0.6

ρ

i j

r

T/T

c

FIG.6:Plots of reduced diagonal (ρxx r =ρyy

r ;upper curves)

and o?-diagonal (ρxy r =ρyx

r ;lower curves)super?uid density

tensor components ρij r =ρij /ρmax

xy

versus temperature T /T c for SHS p -wave super?uids in tetragonal lattices (case (I)).N c =0.5,0.45and 0.4are shown with line,hollow and solid squares.

0 0.5 1 1.5

2

ρ

i j

r

N c

FIG.7:Plots of reduced diagonal (ρxx r =ρyy

r ;upper curves)

and o?-diagonal (ρxy r =ρyx

r ;lower curves)super?uid density

tensor components ρij r =ρij /ρmax

xy versus number of atoms per unit cell N c for SHS p -wave super?uids in tetragonal lattices (case (I)).T 1=0.002T 0,T 2=0.02T 0and T 3=0.1T 0are shown with line,solid and hollow squares.Here T 0is the critical temperature at half-?lling.

half-?lling.It is important to emphasize that square lat-tices [case (I)]have identical diagonal elements ρxx =ρyy due to tetragonal symmetry,but have nonzero ρxy com-ponent as a result of the absence of re?ection symmetry in the yz -plane (x →?x )or the xz -plane (y →?y )for the d -vector de?ned in Eq.(26).In cases (II)and (III),re?ection symmetry is restored and ρxy vanishes identi-cally in the orthorhombic case for any temperature T .In the case of tetragonal symmetry [case (I)],ρxy has a peak as a function of N c ,but the diagonal components have a dip of the same size at exactly half-?lling.In Fig.7,we plot ρij /ρmax xy as a function of N c for three temperatures.

Furthermore,notice that at T =0the super?uid den-

sity tensor is reduced to

ρij =

1

strength V 0,x and crosses the bottom of the energy band (?min =?t x ?t y )at some critical value of V 0,x .The decrease of ?μis associated with the formation of bound particle pairs which are pulled out of the two-particle con-tinuum.Similarly,for high ?lling factors 0.5

Notice that the situation in lattices is very di?erent from the continuum,because of particle-hole symmetry.In the continuum the chemical potential is only pulled down below the bottom of the band since there is no energy upper bound for a parabolic dispersion.In the lattice case,as discussed above,the chemical potential can move below (above)the bottom (top)of the band for ?lling factors below (above)half-?lling.Notice in ad-dition,that for optical lattices of cold fermion atoms at high ?lling factors,super?uidity is associated with corre-lated motion of holes (atom voids),rather than particles (atoms).

A.Phase Diagram

Depending on the behavior of ?μ,the T =0phase di-agram can be plotted as a function of ?lling factor N c and interaction strength V 0,x .The BCS region,where |?μ|t x +t y ,corre-ponds to the stronger interaction region for ?xed density as shown in Fig.8.In the BEC region there is no Fermi surface locus,since ξ(k )can not be zero.

3 6 9 12 15

0.25

0.5 0.75

1

N c

(a)

3 6 9 12

15 0

0.25

0.5 0.75

1

N c

(b)

FIG.8:Plots of critical interaction strength V r =V 0,x /E 0and V r =V 0,s /E 0versus number of atoms per unit cell N c for a)SHS p -wave,and b)s -wave super?uids,respectively at T =0.Plots for α=0,0.1and 0.6are shown with hollow squares,solid squares and line,respectively.

In Fig.8we also compare the phase diagrams of SHS p -wave and s -wave super?uids.In case (I)(hollow squares)of SHS p -wave super?uids,a wide region of ?lling factors around half-?lling requires very strong critical V 0,x to reach the BEC regime.For a small anisotropy α=0.1(solid squares)corresponding to case (II),this region nar-rows.However,further anisotropy α=0.6(line),this region widens again.For s -wave super?uids,in case (I)(hollow squares),the region around half-?lling expands very little for small anisotropy α=0.1(solid squares)corresponding to case (II).Further anisotropy α=0.6(line)expands the BCS region around half-?lling,thus making it more di?cult to reach the BEC regime at ?xed ?lling factor.

Notice in addition,that when the ?lling factor N c →0or N c →1a ?nite interaction strength is necessary to evolve from the BCS to the BEC regime in both SHS p -wave and s -wave super?uids.This indicates that two-particle (or two-hole)bound states in a two-dimensional lattice require ?nite energy for the symmetries discussed.This is in contrast with the situation found in two dimen-sional continuum models,where a two-body bound state is found at arbitrarily small attractive interaction for s -wave.However,in two-dimensional continuum mod-els for the SHS p -wave case,the creation of a two-body bound state requires a ?nite interaction strength.

Furthermore,there are two possible ways of investigat-ing the evolution from BCS to BEC regime in cold atom experiments.The ?rst way is by changing the interac-tion strength for a ?xed ?lling factor,while the second is by changing ?lling factor for a ?xed interaction strength.Probably both ways are possible in cold atom experi-ments,where interaction strength and atom ?lling factor could be tuned independently.

In all physical properties to be discussed in the next sections,?rst we ?x the ?lling factor to N c =0.25(quarter-?lling)and vary V r to cross the BCS-BEC boundary.But,in addition,we ?x the interaction strength to V r =6,and vary N c to cross the BCS-BEC boundary.We present results for tetragonal (case (I)with α=0)and orthorhombic (case (II)with α=0.1)lattices.Notice that cases (II)and (III)have similar be-havior,and di?er only by a scale factor.Thus we do not present case (III)here.

B.Quasi-particle Excitation Spectrum

The quasi-particle excitation spectrum of SHS p -wave and s -wave super?uids during the BCS to BEC evolu-tion are very di?erent because of the symmetry of the order parameter.As discussed in Sec.II,the quasi-particle spectrum of SHS p -wave super?uids is given by E (k )=

they are pushed to extremely high energies and are not accessible.For instance,when a THS p-wave super?uid is formed from a pseudo-spin1/2system,there are two ac-cessible quasi-particle branches when time reversal sym-

metry is broken.Thus,in the SHS p-wave there is only one quasi-particle energy branch.

For the SHS p-wave case,notice that E(k)=0when bothξ(k)=0and?(k)=0.This implies that

the momentum space region of zero E(k)occurs when

?t x cos(k x a x)?t y cos(k y a y)=?μand?0,x sin(k x a x)+?0,y sin(k y a y)=0for chemical potentials inside the BCS

region(|?μ|t x+t y),sinceξ(k)can never be zero.However,the order parameter?(k)can still be zero.This implies that in the BEC region the quasi-particle excitation spectrum is fully gapped.Notice that this change in minimum excitation energy accompanies the existence or non-existence of the Fermi surface lo-cusξ(k)=0.Since quasi-particle excitations are evolv-ing from gapless BCS to gapped BEC regime as a func-tion of interaction strength for?xed?lling factor,or as a function of?lling factor for?xed interaction strength, the evolution between the BCS and BEC regimes is not smooth and a

quantum phase transition occurs.As we shall see in Sec.IV D,where ground state properties like the isothermal compressibility are calculated,this quan-tum phase transition is characterized by the chemical po-tential crossing either the bottom(?μc=?t x?t y)or the top(?μc=t x+t y)of the energy band.

In contrast,for the s-wave case,E(k)never vanishes and is always gapped in both the BCS and BEC regimes. This is a major di?erence between SHS p-wave(or more generally any nodal super?uids)and s-wave super?uids. In s-wave super?uids quasi-particle excitations are al-ways gapped and the evolution from the BCS to the BEC regime is smooth,so that no quantum phase tran-sition occurs.This gapless to gapped quantum phase transitions were?rst considered in the context of He3by Volovik47,and later in the context of high-T c supercon-ductivity48,49and atomic Fermi gases11,13.

C.Chemical Potential

Plots of the reduced chemical potentialμr=?μ/E0 at low temperatures(T≈0)and quarter?lling factor (N c=0.25)are shown in Fig.9as a function of reduced interaction V r=V0,x/E0and V r=V0,s/E0for SHS p-wave and s-wave super?uids,respectively.The tetrago-nal case(I)withα=0and orthorhombic case(II)with α=0.1are shown with hollow and solid squares,respec-tively.In the case of SHS p-wave super?uids,μr is a continuous function of V r,but its slope with respect to V r has a cusp when?μcrosses the bottom of the energy band;μr=?1(α=0:hollow squares)andμr=?1.1 (α=0.1:solid squares).However in the s-wave case,bothμr and the slope ofμr with respect to V r are con-tinuous functions of V r.

-3

-2.5

-2

-1.5

-1

-0.5

0 3 6 9 12 15

V r

(a)

-3

-2.5

-2

-1.5

-1

-0.5

0 3 6 9 12 15

V r

(b)

FIG.9:Plots of reduced chemical potentialμr=?μ/E0, versus reduced interaction strength V r=V0,x/E0and V r= V0,s/E0for a)SHS p-wave,and b)s-wave super?uids,respec-tively at T=0and N c=0.25.Plots forα=0and0.1are shown with hollow and solid squares,respectively.

Plots of the reduced chemical potentialμr=?μ/E0 at low temperatures(T≈0)and constant interaction strength(V r=6)are shown in Fig.10as a function of ?lling factor N c for both SHS p-wave and s-wave super-?uids.Here again the tetragonal case(I)withα=0and orthorhombic case(II)withα=0.1are shown with hol-low and solid squares,respectively.In the case of SHS p-wave super?uids,μr is a continous function of N c,but its slope with respect to N c has a cusp when?μcrosses the bottom of the energy band;μr=?1(α=0:hol-low squares)andμr=?1.1(α=0.1:solid squares).In addition,μr changes curvature atμr=0(N c=0.5)at the same place,where the topology of the Fermi surface locusξ(k)=0changes from particle-like to hole-like. However,in the s-wave case,bothμr and the slope ofμr with respect to N c are continuous functions of N c,and there is no change of slope when the topology of Fermi surface locusξ(k)=0changes.Thus,the slope change of μr for the SHS p-wave case is directly related to a higher angular momentum pairing channel(?=0).

This di?erence between SHS p-wave and s-wave super-?uids have great importance in describing the evolution from BCS to BEC regime.As discussed above,the major qualitative di?erences between the chemical potentials of SHS p-wave and s-wave cases are more clearly seen in the derivative??μ/?N c,which is directly related to the atomic compressibility to be discussed next.

D.Atomic Compressibility

Plots of the reduced isothermal atomic compressibil-ityκr=κT(T)/κ0at low temperatures(T≈0)(see Eq.34)and quarter?lling factor(N c=0.25)are shown in Fig.11as a function of reduced interaction strength V r=V0,x/E0and V r=V0,s/E0for SHS p-wave and s-

-2-1.5-1-0.5 0 0.5 1 1.5 2 0

0.25

0.5 0.75

1

N c

(a)

-2-1.5-1-0.5 0 0.5 1 1.5 2 0

0.25

0.5 0.75

1

N c

(b)

FIG.10:Plots of reduced chemical potential μr =?μ/E 0,versus ?lling factor N c for a)SHS p -wave,and b)s -wave su-per?uids at T =0and V r =6.Plots for α=0and 0.1are shown with hollow and solid squares,respectively.

wave super?uids,respectively.The tetragonal case (I)with α=0,and the orthorhombic case (II)with α=0.1are shown with hollow and solid squares,

respectively.

0.2 0.4 0.6 0.8 1

0 3 6 9 12 15V r

(a)

0.2 0.4 0.6 0.8 1 0 3 6 9 12 15

V r

(b)

FIG.11:Plots of reduced isothermal atomic compressibility κr =κT /κ0versus reduced interaction strength V r =V 0,x /E 0and V r =V 0,s /E 0for a)SHS p -wave,and b)s -wave super?u-ids,respectively,at T =0and N c =0.25.Plots for α=0and 0.1are shown with hollow and solid squares,respectively.

In the case of SHS p -wave super?uids,the evolution of κr from BCS to BEC regime is not smooth,and a cusp occurs at critical interaction strengths corresponding to μr =?1in case (I)(hollow squares)and μr =?1.1in case (II)(solid squares).These cusps are associated with the appearance of a full gap in the quasi-particle excitation spectrum as the evolution from BCS to BEC takes place.In contrast,the evolution of κr is smooth for s -wave super?uids,since the quasi-particle spectrum is always gapped during the BCS to BEC evolution.

Plots of N 2

c κr =?N c /?μr at low temperatures (T ≈0)and

at ?xed interaction strength (V r =6)are shown in Fig.12as a function of ?lling factor N c for both SHS p -wave and s -wave super?uids.The tetragonal case (I)with α=0and orthorhombic case (II)with α=0.1are shown with hollow and solid squares,respectively.

In the case of SHS p -wave super?uids,the evolution of

0.02 0.04 0.06

0.08 0.1 0

0.25 0.5 0.75

1N c

(a)

0.02 0.04 0.06

0.08 0.1 0

0.25 0.5 0.75

1

N c

(b)

FIG.12:Plots of N 2

c κr =dN c /dμr versus ?lling factor N c for a)SHS p -wave,an

d b)s -wav

e super?uids at T =0and V r =6.Plots for α=0and 0.1are shown with hollow and solid squares,respectively.

N 2c κr =?N c /?μr from the BCS to the BEC regime is not smooth.In both cases (I)(hollow squares)and (II)(solid squares),cusps occur for corresponding ?lling fac-tors when μr =±1and μr =±1.1,respectively.These cusps occur again as a result of the gapless to gapped transition in the quasi-particle excitation spectrum dis-cussed above.Notice that the cusps appearing at lower ?lling factors are associated with particle BCS-BEC tran-sitions,while the cusps appearing at higher ?lling factors are associated with hole BCS-BEC transitions.Further-more,N 2

c κr has a sharp peak at half-?lling in case (I),an

d two small peaks in cas

e (II)which are symmetric around half-?lling.These peaks occur inside the BCS re-gion o

f the phase diagram shown in Fig.8,and they arise due to the same reasons discussed in Sec.III D,where the BCS regime is extensively analysed.In contrast,the evo-lution of N 2

c κr as a function of N c is again smooth for s -wave super?uids during the BCS to BEC evolution.In col

d Fermi gases th

e measurement o

f the isother-mal compressibility κT (T )at low temperatures is very hard,because most measurements are performed when traps are turned o?and the gas expands.However,the gas expansion process is probably close to bein

g isen-tropic and κS is most likely measurable.As discussed in Sec.III D,for T ?T c ,the isentropic compressibility and the isothermal compressibility are essentially propor-tional,and their measurements can serve as an indicator of the quantum phase transition discussed in this section.Another important quantity that re?ects suc

h a transi-tion is the Cooper pair size to be discussed.

E.Average Cooper Pair Size

The average Cooper pair size in the saddle point ap-proximation is given by

ξ2pair = ?(k )|(??2k )|?(k ) / ?(k )|?(k )

(41)

where?(k)=?(k)/2E(k)plays the role of the Cooper pair wave function.This quantity re?ects the average size of a Cooper pair and not the coherence lengthξcoh of the super?uid.Although these quantities are directly related in the BCS regime,they are very di?erent in the BEC regime39.Here,however,we concentrate only on the analysis ofξpair.

Plots of the reduced average Cooper pair sizeξr=ξpair/a at low temperatures(

T≈0)and at quarter?lling

(N c=0.25)are shown in Fig.13.The plots are shown as

a function of reduced interaction strength V r=V0,x/E0

and V r=V0,s/E0for SHS p-wave and s-wave super?uids,

respectively.The tetragonal case(I)withα=0,and

the orthorhombic case(II)withα=0.1are shown with

hollow and solid squares,respectively.

5

10

15

20

25

0 3 6 9 12 15

V r

(a)

5

10

15

20

25

0 3 6 9 12 15

V r

(b)

FIG.13:Plots of reduced average Cooper pair sizeξr=

ξpair/a versus reduced interaction strength V r=V0,x/E0and

V r=V0,s/E0for a)SHS p-wave,and b)s-wave super?uids,

respectively,at T=0and N c=0.25.Plots forα=0and0.1

are shown with hollow and solid squares,respectively.

In the case of SHS p-wave super?uids,the evolution of

ξr from BCS to BEC regime is not smooth,and singular

behaviors occur at critical interaction strengths corre-

sponding toμr=?1in case(I)(hollow squares)and

μr=?1.1in case(II)(solid squares).These singular be-

haviors are associated with the appearance of a full gap

in the quasi-particle excitation spectrum and with pair-

ing in a non-zero angular momentum as the evolution

from BCS to BEC takes place.

This singular behavior can be understood in terms

of the vanishing of the momentum distribution when

?μsmaller(larger)than the bottom(top)of the band

for?xed?lling factor N c.For instance,in the case of

N c=0.25shown in Fig.13,when?μfalls slightly below

the bottom of the band,suddenly n(k)vanishes in the

neighborhood of k=0which leads to unbound pairs of

atoms with(k1=k=0;↑)and(k2=?k=0;↑),and

thus a rapid increase in the average pair sizeξpair for

interaction strengths below or above the critical values.

Beyond the critical interaction strength,ξpair decreases

monotonically for increasing V r and converges asymptot-

ically(when V r→∞and N c<0.5)to a?nite value

which is larger but of the order of the lattice spacing

a.This is a manisfestation of higher angular momentum

pairing and of the Pauli exclusion principle.

In contrast,the evolution ofξr is smooth for s-wave

super?uids,since the momentum distribution n(k)never

vanishes at low k,and thus the distribution of pair sizes

is well behaved.This monotonic decrease ofξpair is also

a re?ection of a quasi-particle spectrum that is always

gapped during the BCS to BEC evolution and of an or-

der parameter for zero angular momentum pairing.No-

tice that,ξr is decreasing monotonically as a function of

interaction V r for N c=0.25.The limiting pair size for

large V r and?xed N c is small in comparison to the lattice

spacing a,since the Pauli principle does not forbid atoms

of opposite pseudo-spins to be on the same lattice site.

Next,we discuss the behavior ofξr as a function of?ll-

ing factor N c.Plots ofξr=ξpair/a at low temperatures

(T≈0)and at?xed interaction strength(V0,x=6E0)

are shown in Fig.14as a function of N c for both SHS

p-wave and s-wave super?uids.The tetragonal case(I)

withα=0and orthorhombic case(II)withα=0.1are

again shown with hollow and solid squares,respectively.

5

10

15

20

25

0 0.25 0.5 0.75 1

N c

(a)

0.1

0.2

0.3

0.4

0.5

0 0.25 0.5 0.75 1

N c

(b)

FIG.14:Plots of reduced average Cooper pair sizeξr=

ξpair/a versus?lling factor N c for a)SHS p-wave,and b)s-

wave super?uids at T=0and V r=6.Plots forα=0and

0.1are shown with hollow and solid squares,respectively.

In the case of SHS p-wave super?uids,the evolution of

ξr from BCS to BEC regime is not smooth.In both cases

(I)(hollow squares)and(II)(solid squares),singulari-

ties occur for corresponding?lling factors whenμr=±1

andμr=±1.1,respectively.These singularities occur

again as a result of the gapless to gapped transition in

the quasi-particle excitation spectrum discussed above,

as well as the symmetry of the order parameter,which

strongly modi?es the nature of the pair wave function

?(k)=?(k)/2E(k).Notice thatξr has a singularity at

half-?lling(BCS region)in case(I),and two other sin-

gularities symmetric around half-?lling(BCS region)in

case(II).As the interaction is further increased,the low

and high?lling singularities(BCS-BEC boundaries)shift

towards half-?lling and merge with the half-?lling singu-

larity in case(I)(see also phase diagram in Fig.8).This

situation is analogous to the d-wave lattice case50,51.As

the interaction is further increased,similar behavior oc-

curs in case(II).Thus,at large interaction strengths the BCS(BEC)region shrinks(expands),and the character-istic value ofξpair is of the order of the lattice spacing a except when N c is very close to half-?lling,where the BCS-BEC phase boundary is located.

In contrast,the evolution ofξr as a function of N c is again smooth for s

-wave super?uids during the BCS to BEC evolution(see Fig.14b).For instance,below half-?llingξr decreases as a function of increasing N c(increas-ing DOS),while the pair wave function?(k)continues having the same qualitative behavior at low momenta. Above half-?lling,analogous discussions hold for holes. Furthermore,ξpair decreases monotonically with increas-ing interaction for any?lling factor N c,indicating that for very large interaction strengthsξpair→0for any N c. Thus far,we have investigated only momentum aver-aged quantities,but next we discuss the momentum dis-tribution,which could be measured in cold Fermi gases.

F.Momentum Distribution

The zero temperature momentum distribution

n(k)=1

E(k) (42)

is discussed in this section in the weak coupling BCS and strong coupling BEC regimes for both SHS p-wave and s-wave super?uids in a lattice.We discuss here only the case of particle super?uidity with N c=0.25.In the BCS regime,we choose V r corresponding toμr=?0.5for case(I)(α=0)andμr=?0.55for case(II)(α=0.1). Similarly in the BEC regime,we choose V r corresponding toμr=?2for case(I)andμr=?2.2for case(II).

In the case of SHS p-wave super?uids,the momentum distribution has a major rearrangement in k-space with increasing interaction strength as can be seen in Fig.15. This rearrangement is very dramatic when?μcrosses the bottom of the energy band?μ=?μc.This is a direct consequence of the change of the quasi-particle excitation spectrum from gapless in the BCS regime(|?μ|<(2+α)t) to fully gapped in the BEC regime(|?μ|>(2+α)t).

For case(I),plots of n(k)(in the?rst Brillouin zone) for BCS and BEC regimes are shown in Fig.15a and Fig.15b,respectively.Here the interaction strengths cor-respond toμr=?0.5for the BCS regime,andμr=?2 for the BEC regime.The boundary line between the two regimes occurs whenμr=?1.Notice that n(k)is sym-metric around the lines k y a y=k x a x and k y a y=?k x a x in both regimes as a re?ection of the symmetry properties ofξ(k)and|?(k)|.

While the line k y a y=?k x a x has the highest mo-mentum distribution in the BCS regime,n(k)vanishes along k y a y=?k x a x in BEC regime.As the interac-tion strength increases,two-particle states with opposite momenta are taken out of the two-particle continuum into two-particle bound states with zero center of mass momentum.As more of these tightly bound states are

k

-3

-2

-1

1

2

3

(a)

x

a x

k y a

k

-3

-2

-1

1

2

3

(b)

x

a x

k y a y -3-2-1 0 1 2 3

-3

-2

-1

1

2

3

(c)

k x a x

k y a

-3-2-1 0 1 2 3

-3

-2

-1

1

2

3

(d)

k x a x

k y a y

FIG.15:(Color online)Plots of momentum distribution n(k) versus k x a x and k y a y for an SHS p-wave super?uid in case(I) in a)BCS(μr=?0.5)and b)BEC(μr=?2)regimes.In addition plots for case(II)in c)BCS(μr=?0.55)and d) BEC(μr=?2.2)regimes at T=0.Here the brighter the region,the higher the value of momentum distribution.

formed the large momentum distribution in the vicinity of k=0splits into two peaks around?nite momentum values re?ecting the?=1value of the angular momen-tum associated with these p-wave tightly bound states in the BEC regime.For instance,in the case of d-wave (?=2)super?uids the momentum distribution in the BCS regime is centered around k=0and splits into four peaks around?nite momentum values re?ecting the ?=2value of the angular momentum associated with these tightly bound states in the BEC regime.49

For case(II),plots of n(k)for BCS and BEC regimes are shown in Fig.15c and Fig.15d,respectively.The ?gures shown correspond to interaction strengths asso-ciated toμr=?0.55,andμr=?2.2in the BCS and BEC regimes,respectively.The boundary line between the two regimes occurs whenμr=?1.1.Notice that, n(k)is symmetric around k x a x=0and k y a y=0in both regimes.While the highest values of n(k)occur near k=0in the BCS regime,n(k)vanishes near k=0 and develop two maxima for?nite value of k x in the BEC regime.

In contrast,the evolution of the momentum distribu-tion from BCS to BEC regime is smooth for s-wave su-per?uids as shown in Fig.16for case(I).In case(II) the results are very similar to case(I),except for the expected anisotropy and we do not present them here. Notice that for s-wave(?=0)super?uids,the momen-tum distribution in the BCS regime is centered around k=0,and remains centered around k=0even in the BEC regime.This re?ects the?=0value of the an-

-3

-2-1 0 1 2 3(a)

k x a x

k y a -3

-2-1 0 1 2 3(b)

k x a x

k y a y FIG.16:(Color online)Plots of momentum distribution n (k )versus k x a x and k y a y for an s -wave super?uid in case (I)in a)BCS (μr =?0.5)and b)BEC (μr =?2)regimes at T =0.Here the brighter the region,the higher the value of meomentum distribution.

gular momentum associated with s -wave tightly bound states in the BEC regime.Here,the essential qualitative di?erence in the BCS to BEC regimes is that as the inter-action gets stronger n (k )broadens due to the formation of tightly bound states.

V.

CONCLUSIONS

In summary,we considered SHS p -wave pairing of sin-gle hyper?ne state and THS s -wave pairing of two hyper-?ne state Fermi gases in quasi-two-dimensional optical lattices.We focused mainly on super?uid SHS p -wave (triplet)states that break time-reversal,spin and orbital symmetries,but preserve total spin-orbit symmetry.Since the paper was divided into two parts,we present our conclusions for each of these parts separately.In the ?rst part,we analysed super?uid properties of SHS p -wave and THS s -wave symmetries in the strictly weak coupling BCS regime.There,we calculate the order pa-rameter,chemical potential,critical temperature,atomic compressibility,and super?uid density as a function of ?lling factor for tetragonal and orthorhombic optical lat-tices.

We found that,for SHS p -wave super?uids,the crit-ical temperatures in tetragonal and orthorhombic opti-cal lattices are considerably higher than in continuum model predictions,and therefore,experimentally attain-able.In particular,we showed that a small anisotropy in the transfer energies increases the e?ective density of fermionic states of an SHS p -wave super?uid considerably around half-?lling which leads to higher critical tempera-tures.In contrast,a small anisotropy decreases the den-sity of states around half-?lling,and therefore,the criti-cal temperatures for an THS s -wave super?uid.We also noticed that further anisotropy leads to a larger decrease in critical temperatures for THS s -wave than for SHS p -wave super?uids.Therefore,we concluded that a small anisotropy favours SHS p -wave pairing near half-?lling,and that the critical temperatures for SHS p -wave super-?uids are comparable and even higher than THS s -wave

case for similar interaction parameters.

For SHS p -wave super?uids,we found a peak in the isothermal atomic compressibility at low temperatures,exactly at half-?lling of tetragonal lattices.This peak splits into two smaller peaks in the orthorhombic case.These peaks re?ect the SHS p -wave structure of the order parameter at low temperatures and they are related to the nodes (zeros)of the quasi-particle energy spectrum.We also showed that the atomic compressibility peaks decrease in size as the critical temperature is approached from below.The peaks turn into humps at T =T c .In contrast,for THS s -wave super?uids,we showed that the atomic compressibility does not show a peak structure at low temperatures since the quasi-particle energy spec-trum is always gapped,and that the compressibility is largely temperature independent.

We also discussed the super?uid density tensor for SHS p -wave and THS s -wave systems.In the SHS p -wave case,we concluded that in orthorhombic lattices,the o?-diagonal component ρxy of the super?uid density tensor vanishes identically,while the diagonal components ρxx and ρyy are di?erent.However,in tetragonal lattices,we showed that ρxy =0,while ρxx =ρyy .In contrast,for THS s -wave super?uids,ρxy =0and ρxx =ρyy in the orthorhombic,while ρxy =0and ρxx =ρyy in the tetragonal lattices.Therefore,the presence of non-zero ρxy in the square lattices is a key signature of our exotic SHS p -wave triplet state.

In the second part of the manuscript,we analysed su-per?uid properties of SHS p -wave and THS s -wave su-per?uids in the evolution from BCS to BEC regime at low temperatures (T ≈0).We discussed the changes in the quasi-particle excitation spectrum,chemical poten-tial,atomic compressibility,momentum distribution and Cooper pair size as a function of ?lling factor and inter-action strength for tetragonal and orthorhombic optical lattices.We found major di?erences between SHS p -wave and THS s -wave super?uids in the evolution from BCS to BEC regime.

In the case of SHS p -wave super?uids,the quasi-particle excitation spectrum,chemical potential,atomic compressibility,Cooper pair size and momentum distri-bution are not smooth functions of ?lling factor or in-teraction strength.In particular,the singular behavior of the atomic compressibility suggests the existence of a quantum phase transition when the chemical potential crosses the bottom or the top of the energy band.This transition is associated with the change in quasi-particle excitation spectrum and the higher angular momentum nature (?=1)of SHS p -wave super?uids,which are gap-less in the BCS regime,but fully gapped in the BEC regime.

In contrast,for THS s -wave super?uids,the quasi-particle excitation spectrum,chemical potential,atomic compressibility,Cooper pair size and momentum distri-bution are smooth functions of ?lling factor or interac-tion strength.In particular,the smooth behavior of the atomic compressibility suggests only a crossover when

the chemical potential crosses the bottom or top of the energy band.This crossover is associated with a fully gapped quasi-particle excitation spectrum and pairing at the zero angular momentum channel in both BCS and BEC regimes.

These major di?erences between THS s-wave and SHS p-wave super?uids in an optical lattice suggest that SHS p-wave cold atoms are much richer than THS s-wave cold atoms,and thus these systems may provide a new exper-imental direction in this?eld.

VI.ACKNOWLEDGEMENT

We would like to thank NSF(DMR-0304380)for?nan-cial support.

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脐带干细胞综述

脐带间充质干细胞的研究进展 间充质干细胞（mesenchymal stem cells,MSC S ）是来源于发育早期中胚层 的一类多能干细胞[1-5]，MSC S 由于它的自我更新和多项分化潜能，而具有巨大的 治疗价值 ,日益受到关注。MSC S 有以下特点：(1)多向分化潜能，在适当的诱导条件下可分化为肌细胞[2]、成骨细胞[3、4]、脂肪细胞、神经细胞[9]、肝细胞[6]、心肌细胞[10]和表皮细胞[11, 12]；(2)通过分泌可溶性因子和转分化促进创面愈合；(3) 免疫调控功能，骨髓源（bone marrow ）MSC S 表达MHC-I类分子，不表达MHC-II 类分子，不表达CD80、CD86、CD40等协同刺激分子，体外抑制混合淋巴细胞反应，体内诱导免疫耐受[11, 15]，在预防和治疗移植物抗宿主病、诱导器官移植免疫耐受等领域有较好的应用前景；(4)连续传代培养和冷冻保存后仍具有多向分化潜能，可作为理想的种子细胞用于组织工程和细胞替代治疗。1974年Friedenstein [16] 首先证明了骨髓中存在MSC S ，以后的研究证明MSC S 不仅存在于骨髓中，也存在 于其他一些组织与器官的间质中：如外周血[17]，脐血[5]，松质骨[1, 18]，脂肪组织[1]，滑膜[18]和脐带。在所有这些来源中，脐血(umbilical cord blood)和脐带(umbilical cord)是MSC S 最理想的来源，因为它们可以通过非侵入性手段容易获 得，并且病毒污染的风险低，还可冷冻保存后行自体移植。然而，脐血MSC的培养成功率不高[19, 23-24]，Shetty 的研究认为只有6%，而脐带MSC的培养成功率可 达100%[25]。另外从脐血中分离MSC S ，就浪费了其中的造血干/祖细胞(hematopoietic stem cells/hematopoietic progenitor cells,HSCs/HPCs) [26, 27]，因此，脐带MSC S (umbilical cord mesenchymal stem cells, UC-MSC S )就成 为重要来源。 一．概述 人脐带约40 g, 它的长度约60–65 cm, 足月脐带的平均直径约1.5 cm[28, 29]。脐带被覆着鳞状上皮，叫脐带上皮，是单层或复层结构，这层上皮由羊膜延续过来[30, 31]。脐带的内部是两根动脉和一根静脉，血管之间是粘液样的结缔组织，叫做沃顿胶质，充当血管外膜的功能。脐带中无毛细血管和淋巴系统。沃顿胶质的网状系统是糖蛋白微纤维和胶原纤维。沃顿胶质中最多的葡萄糖胺聚糖是透明质酸，它是包绕在成纤维样细胞和胶原纤维周围的并维持脐带形状的水合凝胶，使脐带免受挤压。沃顿胶质的基质细胞是成纤维样细胞[32]，这种中间丝蛋白表达于间充质来源的细胞如成纤维细胞的，而不表达于平滑肌细胞。共表达波形蛋白和索蛋白提示这些细胞本质上肌纤维母细胞。 脐带基质细胞也是一种具有多能干细胞特点的细胞，具有多项分化潜能，其 形态和生物学特点与骨髓源性MSC S 相似[5, 20, 21, 38, 46]，但脐带MSC S 更原始，是介 于成体干细胞和胚胎干细胞之间的一种干细胞，表达Oct-4, Sox-2和Nanog等多

脐带血造血干细胞库管理办法(试行)

脐带血造血干细胞库管理办法（试行） 第一章总则 第一条为合理利用我国脐带血造血干细胞资源，促进脐带血造血干细胞移植高新技术的发展，确保脐带血 造血干细胞应用的安全性和有效性，特制定本管理办法。 第二条脐带血造血干细胞库是指以人体造血干细胞移植为目的，具有采集、处理、保存和提供造血干细胞 的能力，并具有相当研究实力的特殊血站。 任何单位和个人不得以营利为目的进行脐带血采供活动。 第三条本办法所指脐带血为与孕妇和新生儿血容量和血循环无关的，由新生儿脐带扎断后的远端所采集的 胎盘血。 第四条对脐带血造血干细胞库实行全国统一规划，统一布局，统一标准，统一规范和统一管理制度。 第二章设置审批 第五条国务院卫生行政部门根据我国人口分布、卫生资源、临床造血干细胞移植需要等实际情况，制订我 国脐带血造血干细胞库设置的总体布局和发展规划。 第六条脐带血造血干细胞库的设置必须经国务院卫生行政部门批准。 第七条国务院卫生行政部门成立由有关方面专家组成的脐带血造血干细胞库专家委员会（以下简称专家委

员会），负责对脐带血造血干细胞库设置的申请、验收和考评提出论证意见。专家委员会负责制订脐带血 造血干细胞库建设、操作、运行等技术标准。 第八条脐带血造血干细胞库设置的申请者除符合国家规划和布局要求，具备设置一般血站基本条件之外， 还需具备下列条件： （一）具有基本的血液学研究基础和造血干细胞研究能力； （二）具有符合储存不低于1 万份脐带血的高清洁度的空间和冷冻设备的设计规划； （三）具有血细胞生物学、HLA 配型、相关病原体检测、遗传学和冷冻生物学、专供脐带血处理等符合GMP、 GLP 标准的实验室、资料保存室； （四）具有流式细胞仪、程控冷冻仪、PCR 仪和细胞冷冻及相关检测及计算机网络管理等仪器设备； （五）具有独立开展实验血液学、免疫学、造血细胞培养、检测、HLA 配型、病原体检测、冷冻生物学、 管理、质量控制和监测、仪器操作、资料保管和共享等方面的技术、管理和服务人员； （六）具有安全可靠的脐带血来源保证； （七）具备多渠道筹集建设资金运转经费的能力。 第九条设置脐带血造血干细胞库应向所在地省级卫生行政部门提交设置可行性研究报告，内容包括：

卫生部办公厅关于印发《脐带血造血干细胞治疗技术管理规范(试行)

卫生部办公厅关于印发《脐带血造血干细胞治疗技术管理规 范(试行)》的通知 【法规类别】采供血机构和血液管理 【发文字号】卫办医政发[2009]189号 【失效依据】国家卫生计生委办公厅关于印发造血干细胞移植技术管理规范(2017年版)等15个“限制临床应用”医疗技术管理规范和质量控制指标的通知 【发布部门】卫生部(已撤销) 【发布日期】2009.11.13 【实施日期】2009.11.13 【时效性】失效 【效力级别】部门规范性文件 卫生部办公厅关于印发《脐带血造血干细胞治疗技术管理规范（试行）》的通知 （卫办医政发〔2009〕189号） 各省、自治区、直辖市卫生厅局，新疆生产建设兵团卫生局： 为贯彻落实《医疗技术临床应用管理办法》，做好脐带血造血干细胞治疗技术审核和临床应用管理，保障医疗质量和医疗安全，我部组织制定了《脐带血造血干细胞治疗技术管理规范（试行）》。现印发给你们，请遵照执行。 二〇〇九年十一月十三日

脐带血造血干细胞 治疗技术管理规范（试行） 为规范脐带血造血干细胞治疗技术的临床应用，保证医疗质量和医疗安全，制定本规范。本规范为技术审核机构对医疗机构申请临床应用脐带血造血干细胞治疗技术进行技术审核的依据，是医疗机构及其医师开展脐带血造血干细胞治疗技术的最低要求。 本治疗技术管理规范适用于脐带血造血干细胞移植技术。 一、医疗机构基本要求 （一）开展脐带血造血干细胞治疗技术的医疗机构应当与其功能、任务相适应，有合法脐带血造血干细胞来源。 （二）三级综合医院、血液病医院或儿童医院，具有卫生行政部门核准登记的血液内科或儿科专业诊疗科目。 1.三级综合医院血液内科开展成人脐带血造血干细胞治疗技术的，还应当具备以下条件： （1）近3年内独立开展脐带血造血干细胞和（或）同种异基因造血干细胞移植15例以上。 （2）有4张床位以上的百级层流病房，配备病人呼叫系统、心电监护仪、电动吸引器、供氧设施。 （3）开展儿童脐带血造血干细胞治疗技术的，还应至少有1名具有副主任医师以上专业技术职务任职资格的儿科医师。 2.三级综合医院儿科开展儿童脐带血造血干细胞治疗技术的，还应当具备以下条件：

卫生部关于印发《脐带血造血干细胞库设置管理规范(试行)》的通知

卫生部关于印发《脐带血造血干细胞库设置管理规范(试行)》的通知 发文机关：卫生部(已撤销) 发布日期： 2001.01.09 生效日期： 2001.02.01 时效性：现行有效 文号：卫医发(2001)10号 各省、自治区、直辖市卫生厅局： 为贯彻实施《脐带血造血干细胞库管理办法（试行）》，保证脐带血临床使用的安全、有效，我部制定了《脐带血造血干细胞库设计管理规范（试行）》。现印发给你们，请遵照执行。 附件：《脐带血造血干细胞库设置管理规范（试行）》 二○○一年一月九日 附件： 脐带血造血干细胞库设置管理规范（试行） 脐带血造血干细胞库的设置管理必须符合本规范的规定。 一、机构设置 （一）脐带血造血干细胞库（以下简称脐带血库）实行主任负责制。 （二）部门设置 脐带血库设置业务科室至少应涵盖以下功能：脐带血采运、处理、细胞培养、组织配型、微生物、深低温冻存及融化、脐带血档案资料及独立的质量管理部分。 二、人员要求

（一）脐带血库主任应具有医学高级职称。脐带血库可设副主任，应具有临床医学或生物学中、高级职称。 （二）各部门负责人员要求 1．负责脐带血采运的人员应具有医学中专以上学历，2年以上医护工作经验，经专业培训并考核合格者。 2．负责细胞培养、组织配型、微生物、深低温冻存及融化、质量保证的人员应具有医学或相关学科本科以上学历，4年以上专业工作经历，并具有丰富的相关专业技术经验和较高的业务指导水平。 3．负责档案资料的人员应具相关专业中专以上学历，具有计算机基础知识和一定的医学知识，熟悉脐带血库的生产全过程。 4．负责其它业务工作的人员应具有相关专业大学以上学历，熟悉相关业务，具有2年以上相关专业工作经验。 （三）各部门工作人员任职条件 1．脐带血采集人员为经过严格专业培训的护士或助产士职称以上卫生专业技术人员并经考核合格者。 2．脐带血处理技术人员为医学、生物学专业大专以上学历，经培训并考核合格者。 3．脐带血冻存技术人员为大专以上学历、经培训并考核合格者。 4．脐带血库实验室技术人员为相关专业大专以上学历，经培训并考核合格者。 三、建筑和设施 （一）脐带血库建筑选址应保证周围无污染源。 （二）脐带血库建筑设施应符合国家有关规定，总体结构与装修要符合抗震、消防、安全、合理、坚固的要求。 （三）脐带血库要布局合理，建筑面积应达到至少能够储存一万份脐带血的空间；并具有脐带血处理洁净室、深低温冻存室、组织配型室、细菌检测室、病毒检测室、造血干/祖细胞检测室、流式细胞仪室、档案资料室、收/发血室、消毒室等专业房。 （四）业务工作区域应与行政区域分开。

脐带血间充质干细胞的分离培养和鉴定

脐带血间充质干细胞的分离培养和鉴定 【摘要】目的分离培养脐带血间充质干细胞并检测其生物学特性。方法在无菌条件下用密度梯度离心的方法获得脐血单个核细胞，接种含10%胎牛血清的DMEM培养基中。单个核细胞行贴壁培养后，进行细胞形态学观察，绘制细胞生长曲线，分析细胞周期，检测细胞表面抗原。结果采用Percoll(1.073 g/mL)分离的脐血间充质干细胞大小较为均匀，梭形或星形的成纤维细胞样细胞。细胞生长曲线测定表明接后第5天细胞进入指数增生期，至第9天后数量减少;流式细胞检测表明50%～70%细胞为CD29和CD45阳性。结论体外分离培养脐血间充质干细胞生长稳定，可作为组织工程的种子细胞。 【关键词】脐血;间充质干细胞;细胞周期;免疫细胞化学 Abstract: Objective Isolation and cultivation of mesenchymal stem cells (MSCs) in human umbilical cord in vitro, and determine their biological properties. Methods The mononuclear cells were isolated by density gradient centrifugation from human umbilical cord blood in sterile condition, and cultured in DMEM medium containing 10% fetal bovine serum. After the adherent mononuclear cells were obtained, the shape of cells were observed by microscope, then the cell growth curve, the cell cycle and the cell surface antigens were obtained by immunocytochemistry and flow cytometry methods. Results MSCs obtained by Percoll (1.073 g/mL) were similar in size, spindle-shaped or star-shaped fibroblasts-liked cells. Cell growth curve analysis indicated that MSCs were in the exponential stage after 5d and in the stationary stages after 9d. Flow cytometry analysis showed that the CD29 and CD44 positive cells were about 50%～70%. Conclusions The human umbilical cord derived mesenchymal stem cells were grown stably in vitro and can be used as the seed-cells in tissue engineering. Key words:human umbilical cord blood; mesenchymal stem cells; cell cycle; immunocytochemistry 间充质干细胞(mesenchymal stem cells，MSCs)在一定条件下具有多向分化的潜能，是组织工程研究中重要的种子细胞来源。寻找来源丰富并不受伦理学制约的间充质干细胞成为近年来的研究热点[1]。脐血(umbilical cord blood, UCB)在胚胎娩出后，与胎盘一起存在的医疗废物。与骨髓相比，UCB来源更丰富，取材方便，具有肿瘤和微生物污染机会少等优点。有人认为脐血中也存在间充质干细胞(Umbilical cord blood-derived mesenchymal stem cells，UCB-MSCs)。如果从脐血中培养出MSCs，与胚胎干细胞相比，应用和研究则不受伦理的制约，蕴藏着巨大的临床应用价值[2,3]。本研究将探讨人UCB-MSCs体外培养的方法、细胞的生长曲线、增殖周期和细胞表面标志等方面，分析UCB-MSCs 作为间充质干细胞来源的可行性。

脐带血干细胞检测

脐带血干细胞检测 对每份脐血干细胞进行下列检测： ①母体血样做梅毒、HIV和CMV等病原体检测，这一检测使脐血干细胞适合于其它家庭成员应用。如任何一种病原体测试阳性，需重复测定。 ②每份脐血干细胞样本同时检测确定没有微生物污染。 ③细胞活性检测、有核细胞数、CD34+细胞数、集落形成试验等。CD34是分子量115KD 的糖蛋白分子，使用特定单克隆抗体（抗-CD34）确定，脐血祖细胞的大部分，包括体外培养产生造血集落的细胞都包含在表达CD34抗原的细胞群中。 ④HLA组织配型、ABO血型。 一、采血方式及其优点 再生缘生物科技公司采用最严谨的封闭式血袋收集法，避免在收集脐带血液时可能遭受微生物污染的发生，且以最少之操作步骤，收集最大量之脐带血液方式，在产房内即可完成。 二、脐带血处理与保存 脐带血收集于血袋，经专人运送至再生缘生物科技公司之无菌细胞分离实验室后，由专业的技术人员于完全无菌的环境下，依标准操作程序将血液进行分离，收集具有细胞核的细胞，其中含有丰富的血液干细胞，经加入冷冻保护剂和适当品管检测后，并进行以最适合

血液干细胞的冷冻降温程序方式，进行细胞冷冻程序，达到避免细胞受到冷冻过程之伤害。完成后，冷冻细胞立刻保存于摄氏零下196度的液态氮槽中。所有操作程序记录和细胞保存相关数据，均由计算机条形码系统追踪确认，完全符合国际脐带血库之标准操作程序和品管要求。 母亲血液之检测 为确保所操作和保存的脐带血液细胞，符合国际血液操作规范，并提供客户最大的保障，对于产妇血液必须同时进行一些病毒传染病的检测，以确保没有下列病毒，如艾滋病毒(HIV)、C型肝炎病毒(HCV)、人类T细胞淋巴病毒(HTLV)和梅毒(syphilis)，同时对于B型肝炎病毒(HBV)和巨细胞病毒(CMV)加以侦测和纪录，作为将来可能应用脐带血细胞时之必要参考数据并符合卫生医疗之要求。 脐带血细胞之品管 对于所保存之脐带血细胞均进行多项操作流程监控和品管检测，如微生物污染检测、血液细胞浓度、细胞存活率、细胞活性测定等，每一步骤均有详细之纪录，在操作方法和使用仪器方面均定期进行验证和校验，以符合国际医疗标准。 三、实验室、贮存处所介绍 再生缘生物科技公司拥有符合美国联邦标准(FED-STD-209E)和中华民国优良药品制造标准（一区、二区、三区）的生物安全实验室和无菌操作设备，在专业的技术人员依标准操作程序下进行血液分离和保存步骤，保障客户珍贵样品和权益。 分离后之细胞将依浓度分装入4-6个冷冻管，计算机降温冷冻完成后，即由食品工业发展研究所国家细胞库专业液态氮库房人员，将冷冻细胞分别存放于二个不同的脐带血细胞专属液态氮槽中保存，在安全机制上更有保障。液态氮库房拥有五吨的液态氮供应系统，每一液氮槽均有自动充填装置和异常警报系统，和每日值勤人员监控，确保冷冻细胞处于最佳的冷冻状态。 四、安全管制措施 脐带血液经快递送达无菌细胞分离实验室后，每一步骤均有专业技术人员操作和监督，并将所有分析数值详细填于具有条形码管制之分析表格和计算机数据表中，利用条形码和读码系统确认样品之专一性，避免人为失误，且便于追溯和数据品管。 在冷冻细胞保存上

胎盘干细胞与脐带血干细胞的区别

胎盘干细胞与脐带血干细胞的区别 干细胞是一类具有自我复制和多向分化能力的原始的未分化的细胞，它可以向多种类型细胞分化，并具有相应的功能，可以用来修复和替代受到损伤，病变的组织和器官。目前主要的成体干细胞来源有胎盘来源、脐带来源、脐带血来源和骨髓来源的干细胞，本文主要介绍一下胎盘来源的干细胞和脐带血来源的干细胞的区别： 1、分离部位不同：胎盘干细胞是从胎盘组织中分离提取的干细胞，脐带血干细胞是从脐带里面血液中分离提取的干细胞。 2、种类不同：胎盘中的干细胞主要指的是间充质干细胞，而脐带血干细胞主要指的是造血干细胞。 3、分化能力不同：1）胎盘干细胞分化能力强，在特定的诱导条件下可以分化成血管干细胞、神经干细胞、肝干细胞等多种类型的干细胞，从而修复受损和病变的组织和器官2）脐带血干细胞可以在体内向红细胞、血小板等各种血液细胞分化。 4、数量不同：1）胎盘体积大，从中提取的干细胞数量丰富，并可以在体外培养扩增，扩增培养的子细胞数量达十亿个，可以供成人多次使用2）脐带血干细胞的数量要根据抽取的脐带血多少而定，不可以在体外培养扩增，一份脐带血干细胞可以供40kg以下患者一次使用。 5、治疗使用：1）胎盘干细胞目前在治疗脑瘫、糖尿病、肝硬化、心血管疾病等诸多疾病都显示了良好的效果2）脐带血干细胞可以治疗白血病，再生障碍性贫血等血液系统疾病。不过对于儿童白血病多以先天性为主，所以对于这种情况，自己的脐带血干细胞是不能使用的，需要配型使用捐献的脐血干细胞。 6、配型方面：二者自体使用都不需要配型，如果异体使用，胎盘干细胞由于免疫源性低的特点，所以配型成功率非常高，有血缘关系的亲属都可以使用；脐带血干细胞与父母配型有1/2的几率，与兄弟姐妹有1/4的几率。

脐带血造血干细胞库技术规范

脐带血造血干细胞库技术规范(试行) I 脐带血造血干细胞库(简称脐带血库)的质量控制 1 规章制度和操作规程 脐带血库必须制定脐带血采集、制备、检测、库存、选择和发放的规章制度、操作规程。 1.1 脐带血供者筛选标准和咨询。 1.2 脐带血采集和运送。 1.3 脐带血制备、冷冻、库存。 1.4 标签。 1.5 传染性疾病、人类组织相容性抗原（HLA）分型、造血干细胞和其他检测。 1.6 库存脐带血和脐带血检测标本的确认。 1.7 脐带血的发放。 1.8 脐带血库与移植机构之间的运输。 1.9 数据管理、申请查询、供者与受者配型、脐带血的选择。 1.10 移植随访资料的收集和分析。 1.11 人员培训和继续教育。 1.12 材料、试剂和设备。 1.13 不合格产品、操作错误和事故的报告。 1.14 卫生清洁。 1.15 保密制度。 2 规章制度和操作规程的执行 2.1有脐带血库主任的签字及开始实施的日期。 2.2各项规章制度和操作规程修改，须由脐带血库主任或文件起草人进行审查、签字并标明日期。 2.3 规章制度和操作规程应置于方便工作人员随时取用的位置。 2.4 存档的各种规程和标准记录应长期保留。 2.5 如认为本技术规范不适应当前发展，允许各脐带血库根据情况适当调整，但应报脐带血造血干细胞库专家委员会备案，备案期为30天。专家委员会如不同意备案的规章制度和操作规程，应在备案期间内通知备案单位停止执行。 3 质量控制 3.1 应有专门的质量控制规程，以便对脐带血库工作人员在常规操作中所使用的规程、试剂、设备和材料进行质量控制。 3.2 脐带血库内部的质量控制 3.2.1应由脐带血库主任或指定专人进行质量控制。 3.2.2脐带血库内部的质量控制包括质量评估，改进和修正的措施，错误和事故的处理。 3.2.2.1 脐带血库必须有不合格产品的记录和报告。 3.2.2.2 脐带血库主任应定期召开质量评价会议，对错误和事故进行评价；对重大事故应及时处理。 3.2.2.3 脐带血库主任必须签发对规程的修正。

脐带血干细胞的基础与应用研究

生命科学 Chinese Bulletin of Life Sciences 第18卷 第4期2006年8月 Vol. 18, No. 4Aug., 2006 脐带血干细胞的基础与应用研究 顾东生, 刘 斌, 韩忠朝* （中国医学科学院中国协和医科大学血液学研究所实验血液学国家重点实验室，天津 300020） 摘　要：作为造血干/祖细胞(hematopoietic stem cells/hematopoietic progenitor cells, HSCs/HPCs)的另一 来源，脐带血已经应用于临床治疗多种恶性和非恶性疾病。脐带血中HSCs/HPCs 的质与量是决定其临床应用效果的最重要因素。同时，脐带血中还存在多种非造血的干细胞和前体细胞，如间充质干细胞(mesenchymal stem cells, MSCs)、内皮前体细胞(endothelial progenitor cells, EPCs)和非限制性体干细胞(unrestricted somatic stem cells, USSCs)等，这些细胞可能会在未来的细胞治疗和再生医学中发挥重要作用。本综述还讨论了脐带血的临床应用及HSCs/HPCs 的体外扩增、增加HSCs 归巢和再植能力等提高其临床应用能力的相关研究。关键词：脐带血；造血干细胞；移植 中图分类号：R322.2; R323.3; Q813 文献标识码：A The research and application of cord blood stem cells GU Dong-Sheng, LIU Bin, HAN Zhong-Chao* (State Key Laboratory of Experimental Hematology, Institute of Hematology, Chinese Academy of Medical Sciences and Peking Union Medical College, Tianjin 300020, China) Abstract: Umbilical cord blood (UCB), as an alternative source of hematopoietic/progenitor stem cells (HSCs/HPCs), has been used clinically for a large number of malignant and non-malignant disorders. The quality and quantity of HSC and HPC may be the most important factors on which the capacity of UCB to perform clinical function depends. Other non-HSCs/HPCs, such as mesenchymal stem cells (MSCs), endothelial progenitor cells (EPCs) and unrestricted somatic stem cells (USSCs), also present in cord blood, which may play a future role in cell therapy and regenerative medicine. This review also covers the efforts to expand HSCs and HPCs ex vivo and recent studies on attempts to enhance the homing and engrafting capability of HSCs as means to enhance the clinical utility of UCB. Key words: umbilical cord blood; hematopoietic stem cells; transplantation 收稿日期：2006-04-25 基金项目：“863”计划(2003AA205060)；“973”项目子项(2001C B5101) 作者简介：顾东生(1981—)，男，硕士研究生；刘 斌(1973—)，男，硕士，助理研究员；韩忠朝(1953—)，男，博士，教授，博士生导师，*通讯作者。 文章编号 ：1004-0374(2006)04-0323-05 1988年，Broxmeyer 首先以实验证明脐带血(umbilical cord blood, UCB)中富含造血干细胞(hematopoietic stem cells, HSCs)。法国Gluckman 等[1]在巴黎圣路易斯医院为一位患有先天性再生不良性贫血的儿童实施了世界上首例脐带血移植术，并取得成功。从此，人们对于一直被当成废弃物丢掉 的胎盘和脐带血有了全新的评价和认识，至今，各国学者对脐带血的基础研究和临床应用进行了大量工作并取得很大成绩。本文对脐带血中存在的多种干/祖细胞的生物学特性及临床应用研究进行综述。1　脐带血干细胞 在现阶段，脐带血之所以能够应用于临床治疗

胎盘干细胞和脐带血干细胞是否需要都保存

胎盘干细胞和脐带血干细胞是否需要都保存？ 近来,保存胎盘干细胞热潮在京城悄然兴起，引起了人们对于保存胎盘干细胞和脐带血干细胞的一番讨论，即究竟该保存哪个更有价值呢？是否需要都保存呢？针对这一问题，记者特意查找了大量专业性新闻报导及相关信息，并就该问题咨询了北京博雅干细胞库技术人员，以求寻获一个明确的答案。 记者了解到，在保存胎盘干细胞热潮兴起之前，保存脐带血干细胞或捐献 脐带血干细胞的观念早已深入人心，而且已经占据了一定的市场。并且大多数人都已知晓利用脐带血干细胞能够治愈血液系统疾病等等诸多好处。但是对于胎盘干细胞在生物医学领域的重大用途却知之甚少。 现如今，胎盘干细胞同脐带血干细胞一样被人们所熟知，可是面临的问题 却使得人们不知该如何选择，那就是胎盘干细胞和脐带血干细胞是否应该都保存呢? 针对此问题,记者经过一番周折终于可以比较专业的为大家解答这个问题了。其实，保存胎盘主要是保存从胎盘中提取的间充质干细胞，而保存脐带血是保存脐带血中的造血干细胞。那么，胎盘间充质干细胞和脐带血造血干细胞有何区别呢？首先，胎盘中含有的间充质干细胞非常丰富，这种间充质干细胞属于多能干细胞，具有强大的增殖能力和多向分化潜能，能培养发育成人体的各个组织器官与神经系统。而脐带血中的造血干细胞含量较少，这种造血干细胞属于单能干细胞，不能进行体外扩增，不能分化成为其他的干细胞。其次，胎盘间充质干细胞治疗疾病范围比较广泛，如对于治疗心、脑血管疾病、神经系统疾病、肝脏疾病、骨组织病、角膜损伤、烧伤烫伤、肌病等多种疾病都有不错的疗效。造血干细胞主要针对治疗血液系统疾病和免疫系统疾病具有较好的疗效。最后，记者想告诉大家的是，胎盘间充质干细胞和脐带血造血干细胞两者同样都具有很高的医学价值，都能治疗多种疾病，虽然造血干细胞治疗疾病范围有限且不能进行体外扩增，导致干细胞数量仅可供30公斤以下儿童一次使用，但是胎盘间充质干细胞可以和造血干细胞共移植治疗血液系统疾病，并可供成人使用。另外，两者在采集的时候都不会给产妇和新生儿带来任何不适的感觉和产生任何不良的影响，并且在未来生物医学研究发展过程中，都将起到不可或缺的作用，并将推动再生医学的发展。那么，人们更应该保存哪个呢？记者建议，如果经济条件许可的话，建议人们将两者一起保存起来，以备将来不时之需。

脐血干细胞移植实施方案

附件3 脐带血造血干细胞移植实施方案

(一）准备 患者准备 1、身体准备全面体检和实验室检查； 2、心理准备移植病人大多数对治疗方法及过程缺乏了解，又因长期接受化疗，造成很大的痛苦，病人对移植既抱有希望，又有焦虑和恐惧的心理。因此，在移植前护理人员应主动与病人及家属进行交谈，尽可能做好心理。 物品准备： 病人入舱前，舱内所有物品包括药品、被服、纸张、卫生材料、医疗器械都要经过灭菌处理后，由传递窗送入无菌舱内。 病人在舱内的生活用品，经灭菌处理后入舱。 环境准备： 无菌层流舱: ?患者舱：100级 ?护士站、治疗室等：1000级 ?手消毒间、备无菌餐间：10，000级 ?更衣间、药浴间：100，000级 ------舱内压力递减 患者入住前环境准备 1、彻底卫生清洁： 2、熏蒸24小时：每立方米用高锰酸钾5mg+40%甲醛10ml，熏蒸24小时，通风24小时。 3、入住前的全面消毒液擦拭。 4、空气培养：达标。目前选用平皿沉降法检测； 5、入室物品一律消毒灭菌：可以高压灭菌或适合环氧乙烷消毒的物品，一律灭菌后进舱，须浸泡消毒的物品要确保浸泡消毒的效果可靠。 患者入住后无菌全环境的保持 （一）入住后患者要求： 1、每日以KL-98消毒液洗头、洗脸、擦身、洗脚，早晚各一次（20分钟）。 2、每日以KL-98消毒液于晨起、睡前、便后坐浴一次（20分钟）。

3、睡前、饭前、饭后（进食任何饮食后）认真漱口。 4、3%双氧水擦洗鼻前庭、外耳道每日三次，然后用碘伏消毒液擦拭，再涂以红霉素软膏等。 5、抗菌及抗病毒的眼药水交替点眼，每日三次。 6、经常以含KL-98消毒液棉球擦手（代替洗手）。 （二）入住后环境要求： 1、净化舱内地面、所有物品表面每日消毒液擦拭一次，发现有污染随时擦拭消毒。 2、室内墙壁隔天消毒液擦拭一次。 3、被服高压消毒更换每日一次。 4、空气喷雾消毒每日一次。 5、坐便桶、污水桶每日更换消毒一次。 （三）无菌饮食要求： 1、食物新鲜，彻底洗净、煮熟、微波炉消毒7分钟。 2、水时须做成水果羹后微波炉消毒，或须经消毒后用无菌刀削皮后方可食用。 3、饼干、馒头放微波炉隔水蒸7分钟。 4、饮水均须用开水经舱内电热水瓶二次沸腾后方可饮用。 5、餐具严格消毒。 工作人员入室要求： 严格控制入室人员。医护人员入室前先淋浴，更换清洁衣裤，戴清洁帽子。在缓冲间用肥皂洗手，清水冲净后，再用手快速消毒剂擦手，然后更换无菌拖鞋进入更衣间。戴一次性无菌手套，按无菌操作要求穿无菌分体式隔离衣，戴无菌口罩，进入消毒间再次消毒手，更换无菌拖鞋方可进入护士站。如果进入病人所在的百级层流病房，还需戴无菌手套，穿无菌隔离衣，更换无菌拖鞋方可进入。(二）预处理 定义：是指在输注造血干细胞前对病人进行的大剂量化疗或放疗。 目的：尽可能杀灭病人体内的异常细胞或肿瘤细胞，最大限度减少复发；破坏病人免疫系统，为造血干细胞的植入提供条件，防止移植物被排斥；为造血干细胞的植入、生长提供必要的空间。