Wilson ratio: universal nature of quantum fluids

Wilson ratio: universal nature of quantum fluids Xi-Wen Guan key collaborators: Yi-Cong Yu, Yang-Yang Chen, Hai-Qing Lin Washington University, April 205

Giamarchi, Quantum Physics in one dimension, 2004 A. J. Schofield, Contemporary Physics, 200

3D c v = 3 D c v = πk 2 B T 3 polarization of Fermi surface m k F k 2 B T, χ = m k F µ 2 F g2 π 2 +F0 a, R W = +F0 a ( + ), χ =, R W = 2vc v s v c πv s v c + v s 3 R W = 4 3 ( ) 2 πkb χ µ B g c v/t

Wilson ratios The Wilson ratios, defined as the ratios of the magnetic susceptibility/compressibility to specific heat divided by temperature are dimensionless constants at the renormalization fixed point of these systems. The values of the ratio indicate interaction effects and quantifies spin/particle number fluctuations. G = E Nµ MH TS δm 2 = D k B Tχ, δn 2 = D k B Tκ R s W = 4 3 ( ) 2 πkb χ µ B g c, v/t Rc W = π2 kb 2 κ 3 c v/t

Wilson ratio: essence of Fermi and Luttinger liquids κ = κ +κ 2 + +κ w, /κ r = πv r Nc /r 2 = + + +, /χ rl = πvns rl χ l χ l χ l χ /r 2 wl c v = c v + c 2 v + c w v, cv r = πkb 2 T/( 3 vs r )

{ R s,c quantum fluctuation const. for FL, SL, TLL, W = thermal fluctuation = QC for, Non FL. RW sl = ( ) v Ns wl v2l ( +... w 2 Ns 4 + v Ns l vs w ( ) w 2 4 RW c = v Nc w +... v Nc 2 + v Nc ( ), R c,s vs w +... + W vs 2 vs +... v 2 s ) + vs ( ) ( µ µc) = Fc,s t νz ( ) ( µ µc) +λ 0 t β G c,s t νz

I. Two-component Fermi gas II. III. Wilson ratios of SU(N) Fermi gases Conclusion

Group of Quantum Integralable System 2004-204, Lieb-Liniger Bose gas 2008, Yang-Yang thermodynamics of Bose gas 2009, super Tonks-Girardeau gas 200, Yang-Gaudin spin-/2 Fermi gas 200, quantum criticality of the D quantum Ising model 202, quantum phonon fluctuations in the D Bose gas 202, Wilson ratio in a spin-/2 Heisenberg ladder 202-203, testing D interacting fermions: from few to many 203, quantum impurity and magnons in D Hubbard model of cold atoms 204, D liquid of multi-component fermions with tunable spins 204, measuring excitations energies in D Bose gas 204, Experimental Observation of a Generalized Gibbs Ensemble

I. Experimental test I. Two-component Fermi gas Yang-Gaudin model H = L j=, 0 φ j (x) ( 2 2m d 2 ) dx 2 φ j (x)dx L +g D φ (x)φ (x)φ (x)φ (x)dx 0 H L ) (φ 2 (x)φ (x) φ (x)φ (x) dx 0 many-body problem H: effective magnetic field H = 2 2m N i= 2 x 2 i g D = 2 c m, c = 2/a D, a D = a2 a3d + Aa + g D i<j N C. N. Yang,967; M. Gaudin, 967 Olshanii M., Phys. Rev. Lett., 8, 938 (998) Guan, Batchelor, Lee, Rev. Mod. Phys., Rev. Mod. Phys. 85, 633 (203) many others... δ(x i x j ) Hm z

I. Experimental test a Partially polarized (FFLO) Chemical potential Low P P c Vacuum High P Fully polarized Effective magnetic field Experimental measurement of phase diagram of two-component ultracold 6 Li atoms trapped in an array of D tubes through the finite temperature density profiles. Liao et al, Nature 467, 567 (200) Orso, PRL 07; Hu, Liu & Drummond, PRL 07 Guan, Batchelor, Lee & Bortz, PRB 07 Zhao, Guan, Liu, Batchelor & Oshikawa, PRL 09

I. Experimental test χ = R s W = 4 3 +, c v = π2 kb 2T ( χ u χ b 3 πvs u ( πkb µ B g + ) πvs b ) 2 χ c = 4 ( v/t v b N + 4vN) ( ) u vs b + vs u H = 2(µ µ 2 )+ 2 c2, n = 2n 2 + n χ = ( ) H = µ µ 2 = µ + µ 2 2 m n n n 2 n 2 ( ) ( ) n n2 χ u =, χ b = 2 µ n µ 2 n Guan, Yin, Foerster, Batchelor, Lee, Lin, PRL, 3040(203)

I. Experimental test.0 b c 0.8 0.6 0.4 t=0.0000 t=0.000 t=0.0004 t=0.0008 t=0.002 t=0.002 0.2 0.0 0.95.00.05.0.5.20.25.30.35 H/ b F(T) = E 0 πct 2 ( + ) 6 v b v u

I. Experimental test c V b /(T c ) 24 22 20 8 6 4 2 0 8 6 4 2 analytical t=0.0000 t=0.000 t=0.0004 t=0.0008 t=0.002 t=0.002 0 0.0 0.2 0.4 0.6 0.8.0 P c v = πk 2 B T 3 ( v b + v u ),

I. Experimental test 0.005 0.0003 2.000 0.004 0.0002 t 0.003 0.002 0.00 PP CR TLL 0.000 0.0000 0.96 0.97 0.98 0.99 CR F R W 0.000 0 0.9.0..2.3.4 H/ b RW s = 4 ( v b N + 4vN) ( ) u + vs b vs u

I. Experimental test κ = κ b +κ u, κ b = πv Nc b 2 ( ) vnc b µb = π n b,, κ u δn 2 = D k B Tκ, vnc u = H,c π = πvnc u ( ) µu n u H,c κ b π2 n 2 4 { { κ u 2π 2 n + 2n 2 c v u v b s + 6n c + 4n 2 c + n2 2 + 24n2 2cn c 2 + 24n n 2 c 2 + 7n2 2 c 2 2n3 2 c 2 + n4 2 n 4c 2 n 2 } 2m 2πn (+2A / c +3A 2 /c2) 2m πn 2 (+2A 2 / c +3A 2 2 /c2) + 6n2 cn 2 + 96n2 2 c 2 + 384n2 c 2 8n n 2 c 2 96n3 c 2 n 2 + 256n4 c 2 n 2 }

I. Experimental test RW c = π2 kb 2 3 ( R c 2 W = v b Nc κ c v/t + v u Nc ) ( / vs b + v u s ) Yu, Chen, Lin, Guan, 204 in preparation

I. Experimental test The second Wilson ratio at t 0.00ǫ b R c W = π2 k 2 B 3 κ c v/t, Rc W = K, TTL F : R c W =

I. Experimental test R c,s W The second Wilson ratio at t 0.00ǫ b ( ) ( ) ( µ µc) ( µ µc) = F c,s +λ t νz 0 t β G c,s t νz

I. Experimental test R c w 0 8 6 4 2 t=0.00 t=0.002 t=0.003 π 2 (κ κ0)/(3cv/t) 0.8 0.6 0.4 0.2 t=0.00020 t=0.00025 t=0.00030 0 0.73 0.74 0.75 0.76 0.77 x/(n 2a) 0 0.26 0.28 0.3 x/(n 2a) R c,s W = F c,s ( ( µ µc) t νz ) ( ) ( µ µc) +λ 0 t β G c,s Na 2 d a 2 = 4 n(y)/cdy, Ma 2 D /a2 = 4 t νz m z(y)/cdy

I. Experimental test The second Wilson ratio in an harmonic trap with P = 0.47 ( ) RW c = 2 vnc b + ( vnc u / vs b + ) vs u

I. Experimental test Experimental tests C. Chin s group, Nature 460,995 (2009) Ketterle s group, PRL 79, 553 (997) R c W = π2 k 2 B 3 κ c v/t, κ = n/ µ = n (x)/(xmω 2 x)

II. Wilson ratios of SU(N) Fermi gases II. Wilson ratios of SU(N) Fermi gases / b -.30 -.32 -.34 V (A).2.4.6 H / b (C) H 2 =H -.30 -.32 -.34 V (B) 0.8 0.9.0 H / b (D) H 2 =2H Sutherland, PRL 20, 98 (968) Ho, Yi, PRL, 82, 247 (999) Wu, Hu, Zhang, PRL, 9 86402 (2003) Controzzi, Tsvelik, PRL 96, 097205 (2006) Guan, Batchelor, et al PRL 00, 20040 (2008) Guan, Lee, et al PRA 82, 02606(R) (200) Guan, Ma, Wilson, PRA 85, 033633 (202) Cazalilla, Rey, Rep. Prog. Phys. 77, 2440 (204)... Atom Species Mass (u) Nuclear Spin (I) Symmetry Group Scattering Length (nm) 7 Yb 70.93 2 SU(2) 0.5(9) [ 7 Yb], 30.6(3.2) [ 73 Yb] [57] 73 5 Yb 72.94 2 SU(6) 0.55() [ 73 Yb], 30.6(3.2) [ 7 Yb] [57] 87 9 Sr 86.9 2 SU(0) 5.09(0) [ 87 Sr] [58] H = 2 2m N f i= 2 x 2 i + g D i<j N f δ(x i x j )

II. Wilson ratios of SU(N) Fermi gases pairs : k i = Λ i ± ic/2; trions : k i = λ i ± ic, λ i µ = (ǫ Z +ǫ 2 Z +ǫ 3 Z)/3, H = µ ǫ Z, H 2 = ǫ 3 Z +µ Guan, Batchelor, et al PRL 00, 20040 (2008) Guan, Ma, Wilson, PRA 85, 033633 (202) He, Lee, et al JPA 44, 405005 (20)

II. Wilson ratios of SU(N) Fermi gases (a) (b) (c) n n H r/r = G µ, n = G, n 2 = G H H 2 = n + 2n 2 + 3n 3, χ = M, χ 2 = M H H 2 = (µ r µ w)+ wr Experimental realization of three-component 6 Li Fermi gases: Ottenstein et al, PRL (2008); Huckans et al, PRL (2009)

II. Wilson ratios of SU(N) Fermi gases ε (k) = k 2 µ H + Ta ln(+e ε2/t )(k)+ta 2 ln(+e ε3/t )(k) T a n ln(+ξn ) n= ε 2 (k) = 2k 2 2µ H 2 2 c2 + Ta ln(+e ε /T )(k)+ta 2 ln(+e ε 2/T )(k) T(a + a 3 ) ln( + e ε 3/T )(k) T n= a n ln(+ςn )(k) ε 3 (k) = 3k 2 3µ 2c 2 + Ta 2 ln(+e ε /T )(k)+t(a + a 3 ) ln(+e ε 2/T )(k) +T(a 2 + a 4 ) ln( + e ε3/t )(k) lnξ n(k) = n(2h H 2 ) + a n ln(+e ε/t )(k)+ T mn ln(+ξm )(k) T m= S mn ln(+ςm )(k) m= lnς n(k) = n(2h 2 H ) + a n ln(+e ε2/t )(k)+ T mn ln(+ςm T )(k) m= S mn ln(+ξm )(k) m=

II. Wilson ratios of SU(N) Fermi gases ε (r) η (r) w = V r A rs ε (s), (r =, 2,, w). s= w = Ṽ(r) ˆB rs η (s), (r =, 2,, w ) s=

II. Wilson ratios of SU(N) Fermi gases Mapping out quantum liquids of the SU(3) Fermi gas from the W c R v Nc = v 3 Nc L π = 3L π ( ) µ, vnc 2 n = 2L ( ) µ2 H,H 2 π n 2 ( ) µ3 n 3 H,H 2 dµ dn = µ n + µ n 2 dµ 2 dn 2 = µ 2 n dµ 3 dn 3 = µ 3 n (H H 2 ) (n 3 n ) + µ (H H 2 ) n 3 (n 2 n 3 ) (H H 2 ) (n 2 n 3 ) + µ 2 + µ 2 (H H 2 ) n 2 n 3 (n 3 n ) (H H 2 ) (n 2 n 3 ) + µ 3 (H H 2 ) n 2 (n n 2 ) H,H 2 (H H 2 ) (n n 2 ) (H H 2 ) (n 2 n 3 ) (H H 2 ) (n n 2 ) (H H 2 ) (n 3 n ) (H H 2 ) (n 3 n ) + µ 3 (H H 2 ) n 3 (n n 2 )

II. Wilson ratios of SU(N) Fermi gases Mapping out quantum liquids of the SU(3) Fermi gas from the W c R R c W = π2 k 2 B 3 Yu, Chen, Zhou, Lin, Guan, in preparation κ c v/t, unpaired : Rc w = K u, pair : R c w = K b, trion : R c w = K t

II. Wilson ratios of SU(N) Fermi gases W c R = ( 9 v Nc ) + 4 vnc 2 + ( vnc 3 / vs + v 2 s + ) vs 3

II. Wilson ratios of SU(N) Fermi gases T T+B+U B+U T T+B+U B+U = π ( λ χ rl r 2 v Ns rl = Ô + ) Ô 2 Ô l E r, Ô r = r λ+ λ+ n r 3 χ = χ + χ 2 + χ 3, χ 2 = χ 2 + χ 22 + χ 32 n 3

II. Wilson ratios of SU(N) Fermi gases κ = κ t +κ b +κ u κ t = ( ) µ3, κ = ( ) ( ) µ2, κ µ b u = 3 n 3 H,H 2,c 2 n 2 H,H 2,c n H,H 2,c

II. Wilson ratios of SU(N) Fermi gases c v c r v = c v + c 2 v + c w v = πk 2 B T 3 vs r, r =, 2, 3

II. Wilson ratios of SU(N) Fermi gases The Lieb-Linger Bose gas H = ( ) πvsk dx 2 Π2 + vs 2πK ( xφ)2 Luttinger liquid vs Fermi liquid κ = πv N, c v = πk 2 BT 3 v s R c w = K

II. Wilson ratios of SU(N) Fermi gases 6 4 R W 2 0 0 0. 0.2 0.3 m Thielemann et al, PRL, 2008; Ruegg et al. PRL, 2008 Thielemann Ninios et al, PRL, 202

IV. Conclusion IV. Conclusion Wilson ratio presents the essence of the quantum liquids at the renormalization fixed point. This work has been supported by Wuhan Institute of Physics and Mathematics, Chinese Academy of Science and the Australian Research Council