Scaling behavior near the critical point grand canonical ensemble hl/2,4c- /2 Near critical point, density of particle number satisfies scaling law np,h,T)-m(,h,T)=rd+1-129 d/z+1-1/Uz=1/2,1/Uz=1, (x)=F-12(-4c+)(2x12) Focus on this model dimension d=1 critical exponent z=2, correlation length exponent v= 1/2
dimension d=1, critical exponent z= 2, correlation length exponent v= 1/2 Grand canonical ensemble Scaling behavior near the critical point Near critical point, density of particle number satisfies scaling law Focus on this model
Anisotropic spin-1 Bose gas N N 月=∑。+c∑0+P2 +p 2,0 )δ(xz-x 2 ∑a2+∑ 2+32s3)2 i≠ 6(x7-x) k IC sab(k) 0,0,D2,2⊥D2 0 k+ici订111订 )+ ab ab ab M2 E=∑k2+∑k2 S=M-2M
II. Anisotropic spin-1 Bose gas
1-(1)_1(1) +1C e 1≠ km):,j=1,2…M, IC M2k-k+ic MK-n/-ic/2 e k2-k2-1c12-1(+ic2 2)-ic/2 ni-ni-ic 关)1-A1+in 1,2,…,M 1-k2+ic/2 repulsive interaction k),k)∈R;k2),k2)∈R; 2=k2,k22ER; Anzj=Anz+(n+1-2jic/2,j=1,2…,n,n=1,2
repulsive interaction
Ground state 15 h=0, m|=n/2. half of the atoms occupy the state with sz=0 and the rest stay on the state with 05 s2=1(or-1) fully polarized state Cl s2=l and m=n 0.5 h<-hhcl, S=-land m=-n partially polarized states Partially polarized states with sz=1(or-1 n=l and c=1 and sz=0 In the case of strong repulsive interaction the critical field hc is he|=n2m21-16n/3y+O(1/y2 r=C/n>1
Ground state In the case of strong repulsive interaction, the critical field hc is fully polarized state partially polarized states half of the atoms occupy the state with s z=0 and the rest stay on the state with s z=1 (or -1). Partially polarized states with sz =1 (or -1) and sz=0
The compressibility at the ground state KT=(OH)-pIhT/n Compressibility can also be use to quantify the phase transition 2.5 At the critical point, the compressibility is divergent 0.5 0.5 1.5 n=l andc=1
The compressibility at the ground state At the critical point, the compressibility is divergent. Compressibility can also be use to quantify the phase transition