Fermi liquids and correlations in 1D attractive Hubbard model Xi-Wen Guan key collaborators: Song Cheng, Yi-Cong Yu, Yu-Zhu Jiang, Murray T Batchelor

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1 Fermi liquids and correlations in 1D attractive Hubbard model Xi-Wen Guan key collaborators: Song Cheng, Yi-Cong Yu, Yu-Zhu Jiang, Murray T Batchelor University of Melbourne, June 2017

exp [ (ǫ j µ)/t ] + 1 ) 1 Quantum statistics: 1 quantum man-body systems 2 microscopic

2 Maxwell-Boltzmann distribution: < n j > 1/ ( exp ( (ǫ j µ ) /T) ) Bose-Einstein distribution: < n j >= ( exp [ (ǫ j µ)/t ] 1 ) 1 Fermi-Dirac distribution: < n j >= ( exp [ (ǫ j µ)/t ] + 1 ) 1 Quantum statistics: 1 quantum man-body systems 2 microscopic state energy E i 3 partition function Z = i=1 W ie E i /(k B T) 4 free energy F = k B T ln Z

3 !"!!!!!#!! "!"!!!!#!! '!"!! '!!! " #!"!! $ # $! % & " #!!! # $! % & More is different: emergent physics 1 1+x = 1 x + x2 x 3 ln 2 = Continuous phase transition: ñ(t, µ) = ñ 0 ( µ)+t d z +1 1 νz F ) ( µ µc T 1 νz ( κ = κ 0 + T d/z+1 2/νz µ µc ) K T 1/νz ξ µ µ c ν, ξ z µ µ c zν

4 "#!$%&'!()&*!+%,-.'/!0,1'& "#!2'345!()&*!&-56!541).)6+' $%&%672'345!4580,3'&99!!! Cazalilla, M. A., Citro, R., Giamarchi, T., Orignac, E. & Rigol, M., Rev. Mod. Phys. 83, 1405 (2011) One dimensional bosons: From condensed matter systems to ultracold gases Guan, X.-W., Batchelor, M. T. & Lee, C.-H., Rev. Mod. Phys. 85, 1633 (2013) Fermi gases in one dimension:from Bethe ansatz to experiments

one band Hubbard Hamiltonian Ĥ L = (c j,a c j+1,a + c j+1,a c j,a) j=1 a=, L +4u n j n j 2uˆN + ul j=1 c j,a and c are the annihilation and creation j,a operators of electrons with spin a at site j n

5 one band Hubbard Hamiltonian Ĥ L = (c j,a c j+1,a + c j+1,a c j,a) j=1 a=, L +4u n j n j 2uˆN + ul j=1 c j,a and c are the annihilation and creation j,a operators of electrons with spin a at site j n j,a = c j,a c j,a is the particle number operator Lieb, Wu, PRL 20 (1968) 1445 V. Korepin, A. A. Ovchinnikov, F. Essler, H. Frham, P. Schlottmann, N. Kawakami, J. M. Carmelo, N.M. Bogoliubov, F. Woynarovich, B.S. Shastry... ˆN = L j=1 n j + n j is the total particle number operator u < 0 (u > 0) represents the on-site attractive (repulsive) interaction between particle L is the lattice number, N/L 1, the model reduces to the Yang-Gaudin model The Hubbard model has long been important in solid state physics for understanding strongly interacting electrons.

$Magnetism, transport properties, transition from metallic to a Mott insulator, fractional excitations, quantum walks, quantum dynamics, quantum criticality, etc.$

6 Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations. Magnetism, transport properties, transition from metallic to a Mott insulator, fractional excitations, quantum walks, quantum dynamics, quantum criticality, etc. recent new experiments with the Hubbard model: Hart, et al. Nature 519, 211 (2015) Boll et al. Science 353, 1257 (2016) Parsons et al. Science 353, 1253 (2016) Cheuk, et al. Science 353, 1260 (2016) Cheuk, et al. PRL 116, (2016) Hilker, et al. arxiv

7 1d Hubbard chains 1.15 µm Split 1 y x 1d chains (spin unresolved) (a.u.) 0 Split 1d chains (spin resolved) Reconstruction Schematic of the spin- and density-resolved imaging with Li-6 atoms Boll et al. Science 353, 1257 (2016)

8 I. Bethe ansatz solution for the 1D Hubbard model II. III. Pair Correlation and FLLO IV. Quantum criticality: From a Bethe ansatz perspective

9 Bethe ansatz solution for the 1D Hubbard model Bethe ansatz equations: Lieb, Wu, PRL 20 (1968) 1445 exp(i k j L) = N j=1 M α=1 sin k j Λ α + i u, j = 1, 2, 3,, N sin k j Λ α i u M sin k j Λ β + i u sin k j Λ β i u = Λ α Λ β + 2 i u, β = 1, 2, 3,, M Λ α Λ β 2 i u α=1 string hypothesis: Takahashi, Prog. Theor. Phys. 47 (1972) 69 Essler, Korepin, NPB 426, 503 (1994) single real k s; α-th Λ-Λ string of length m Λ m,j α = Λ m α + i (m + 1 2j) u, j = 1, 2, 3,, m. α-th k-λ string of length m, these are 2m k s k 1 α = arcsin(λ αm + i m u ), k 2 α = arcsin(λ αm + i (m 2) u ), k 3 α = π k2 α,. k 2m 2 α = arcsin(λ αm i (m 2) u ), k 2m α = arcsin(λ αm i m u )

10 Bethe ansatz solution for the 1D Hubbard model ε(k) = g 0 (k) a n ( F[ε n ] F[εn]) (k) n=1 ε n(λ) = 2nB a t n F[ε](Λ) A nm F[ε m](λ) m=1 ε n (Λ) = gn(λ) at n F[ε](Λ) A nm F[ε m ](Λ) m=1 F[x](y) = T ln[1+exp( x(y) π T )], at n F[x](Λ) = dy cos y a n(sin y Λ)F[x(y)] π a n F[x](k) = dy a n(k y)f[x(y)], g 0 (y) = 2 cos y µ 2u B, g n(y) = 4Re 1 (y + i n u ) 2 n(2µ+4u) - π dk f = u + π 2π F[ε](k)+ dλ n=1 2π ξn(λ)f[ε n](λ) π ξ n(λ) = dk a n(λ sin k) π Song Cheng, Yi-Cong, Murray T Batchelor, Xi-Wen Guan, preprint

11 Bethe ansatz solution for the 1D Hubbard model Ground Sate A ε u (k) = 2 cos k µ 2u B π ε b (Λ) = 2µ 2 π A dk cos 2 k a 1 (sin k Λ) A dλ ( a 2 Λ Λ ) ε b (Λ ) A dλ a 1 (sin k Λ)ε b (Λ), Q Q dk cos k a 1 (sin k Λ)ε u (k)

12 Bethe ansatz solution for the 1D Hubbard model The phase boundaries in 1-D Hubbard model at zero temperature could be obtained analytically: µ c = 2 u u 2 ; B c2 = u ; B c1 d J 1 (ω)e u ω = u ω cosh(uω) IV-V phase boundary: A >> 1 µ 4 ( 1) n G ( i α n) I 1 (α n) exp( α n A) (2n + 1) 2 n=1 B ( 1) n 2 2u + ỹ + (i α n) exp( α na) u n=0 dω J 1 (ω) exp( u ω ) +2 0 ω cosh(u ω) I :Vacuum II :Fully Polarized III :Half Filling IV :Mixed V :Paired IV-V phase boundary: A << 1 [ ] B = 2 2u + 2 π u a 2 (A) 1 µ 2 π + dk cos 2 k a 1 (sin k A) π u a 2 (A) π

13 Bethe ansatz solution for the 1D Hubbard model I. Bethe ansatz solution for the 1D Hubbard model II. III. Pair Correlation and FLLO IV. Quantum criticality: From a Bethe ansatz perspective

14 κ(k) = 2 cos k µ 2u B + T dλ a n(sin k Λ) ln[1+exp( ε n(λ) T )] T n=1 lnη n(λ) = 2nB T + + T n=1 π π dλ a n(sin k Λ) ln[1+ηn 1 (Λ)] dk cos k a n(sin k Λ) ln[1+exp( κ(k) T )] ˆT nm ln[1+exp( ε m T )](Λ) m=1 ε n(λ) = 2nµ 2 + T + T π π m=1 π π dk cos 2 (k) a n(λ sin k) dk cos k a n(sin k Λ) ln[1+exp( κ(k) T )] ˆT nm ln[1+exp( ε m T )](Λ)

15 Wilson ratio 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 quantify 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 Rκ W = π2 kb 2 κ 3 c v/t Fermi Liquid : R χ W = 1 1+F0 a,, RW κ 1 = 1+F0 s

16 Interacting Fermions with spins and charges in 1D: critical phenomena Yang-Gaudin model H: effective magnetic field H = L j=, 0 φ j (x) ( 2 2m d 2 ) dx 2 φ j (x)dx L +g 1D φ (x)φ (x)φ (x)φ (x)dx 0 H L ) (φ 2 (x)φ (x) φ (x)φ (x) dx 0 g 1D = 2 c m, c = 2/a 1D, a 1D = a2 a3d + Aa Yang Phys. Rev. Lett. 19, 1312 (1967) Gaudin, Phys. Lett. 24, 55 (1967) Guan, Batchelor and Lee, Rev. Mod. Phys. 85, 1633 (2013)

17 The second Wilson ratio at t 0.001ǫ b R c W = π2 k 2 B 3 ( κ c = v/t 4 v b Nc ) + 1 ( 1 vnc u / vs b + 1 ) vs u Guan, Yin, Foerster, Batchelor, Lee, Lin, PRL 111, (2013) Yu, Chen, Roemer, Lin, Guan, Phys Rev B 94, (2016)

18 energy transfer relation H/2 = (µ 1 µ 2 )+ε 2 /2, n 1 + 2n 2 = n χ 1 2 = 1 4 (µbg) 2 πd χ 2, χ 1 1 = (µ Bg) 2 πd χ 1 hidden simplicity: free particle nature κ 1 2 = 1 4 πdκ 2, κ 1 1 = πd1 κ ( ) µr D κ r = r π c 1 V = π2 k 2 BT 3 n r 1 πv 1 s H, D χ r = r π, c 2 V = π2 k 2 BT 3 ( ) µr n r 1 πv 2 s n Low energy physics of the spin-1/2 Fermi gas displays additivity rules 1 χ = 1 χ χ 2 κ = κ 1 +κ 2 c V = c 1 v + c 2 V

19 Wilson ratios: conceptual simplicity, essence of quantum liquids!"#$$% R χ W = 4 3 ( ) πkb 2 χ at T=0.01 and u = 1. µ B g L C v/t R κ W = π2 k 2 B 3 κ at T=0.01 and u = 1. C v/t

20 R W c V IV II µ = 0.08 T= T= T= T= T= T= III B for u = 7 and µ = 0.08 R κ,χ W [ ] [ ] (η ηc) (η ηc) = F κ,χ +λ T νz 1 0 T β G κ,χ T νz 1

21 1D strongly attractive Hubbard model: Low temperature B/T 1: effective chemical potentials: κ(k) = k 2 µ 1 a 1 F[ε 1 ] ε 1 (Λ) = α(λ2 µ 2 ) a 1 F[κ] a 2 F[ε 1 ] π dk f Hubbard = u + π 2π F[κ(k)]+ dλ 2π βf[ε 1 (Λ)] µ 1 = µ+b + 2u + 2, µ 2 = 1 α [2µ+4( u u )] Yang-Gaudin model: Low temperature B/T 1: ε 1 (k) = k 2 µ 1 2 H a 1 F[ε 2 ] ε 2 (k) = 2k 2 2µ 1 2 c2 a 1 F[ε 1 ] a 2 F[ε 2 ] f SU(2) = dk 2π F[ε1 (k)]+ dk 2π 2F[ε2 (k)]

22 The explicit form of the function α( u ) and β( u ) are: π α = dk 1 2 u cos 2 k(u 2 3 sin 2 k) π π π (u 2 + sin 2 k) 3, β = dk 1 π 2π 2 u u 2 + sin 2 k α( u ) 1 β( u ) u κ = κ α κ 2, u 1 χ = 1 χ 1 + α 2 1 χ 2 κ 1 = n 1, κ 2 = 2 n 2, χ 1 = n 1, χ 2 = 2 n 2 µ 1 B µ 2 B µ 1 n µ 2 n

23 µ 1 = µ+b + 2u + 2, µ 2 = 1 α [2µ+4( u u )] µ 1 = πn1 2 A π2 α 1 3β1 3 n2 3 A2 2 µ 2 = π 2 n2 2 β1 2 A π2 A 1 = 1+ 2n 2 u + n1 3 3α A π2 1 3β1 3 n2 3 A3 2 ( 2n2 u ) 2, A 2 = 1+ 2n 1 + n 2 β 1 u + ( ) 2n1 + n 2 2 β 1 u dµ = dµ 1 = α 2 dµ 2 κ = n = dn 1 + 2dn 2 µ H dµ = dn 1 + 2dn 2 α dµ 1 dµ = κ u α κ b κ u = n 1 µ 1 B κ b = 2 n 2 µ 2 B

24 µ 1 = µ+b + 2u + 2, µ 2 = 1 α [2µ+4( u u )] µ 1 = πn1 2 A π2 α 1 3β1 3 n2 3 A2 2 µ 2 = π 2 n2 2 β1 2 A π2 A 1 = 1+ 2n 2 u + n1 3 3α A π2 1 3β1 3 n2 3 A3 2 ( 2n2 u ) 2, A 2 = 1+ 2n 1 + n 2 β 1 u + ( ) 2n1 + n 2 2 β 1 u dn 1 + 2dn 2 = 0, db = dµ 1 α 2 dµ 2, χ = m B χ 1 = dµ 1 α dµ 2 2 dn 1 = dµ 1 dn 1 + α 2 1dµ 2 = χ 1 u + α 2 dn 2 2 χ 1 b χ u = n 1 µ 1 n χ b = 2 n 2 µ 2 n n

25 T= T= T= T= T= T= Ground state χ V IV B susceptibility for u = 1, µ = χ 1 = χ 1 u + α 2 χ 1 b χ 1 u χ 1 b = π 2 2n 1 (1 2n 1 + 4n 2 + n2 1 u u u 2 12n 1 n 2 u 2 = π 2 α 2β 2 n 2 (1 + 4n 1 β u + 12n 2 2 u 2 ) 4n n2 1 β u β 2 u 2 24n 1 n 2 β 2 u 2 6n 2 2 β 2 u 2 )

26 T= (a) T= T= T= GS analytical V IV II κ χ (b) V IV T= T= T= T= GS analytical II χ B (c) B B= B= B= B= B= B c B= B= B= B= B= T x 10 3 Compressibility for u = 1, µ = κ = κ α κ 2

27 Universal feature of the susceptibility in the gapped phase χ s Susceptibility for the gapped phase at µ = 0.08, u = 7 = T 1/2 ( 4 π Li 1/2 e /T) 1 4 πt e /T, as T = q 2 4(2π q3 /3) 3 u π 2 q = Re µ+2u + B + 2 ( 1+ 2 u πµ ) 3/2 2π q 3 /3

28 Quantum Criticality The critical exponents ν = 1/2 and z = 2 Free Fermi liquids ( ) n(µ, B, T) =n 0 (µ, B, T)+T d/z+1 (1/νz) µ µc G T 1/νz ( ) κ(µ, B, T) =κ 0 (µ, B, T)+T d/z+1 (2/νz) µ µc F T 1/νz G(x) =Li 1/2 (x) F(x) =Li 1/2 (x) κ = κ α κ 2, ( 1 1 χ = 1 + α 1 χ 1 2 χ 2 ) c v = πt v 1 v 2 ( α v 2 = β πn u β (2n 1 + n 2 ) ), v 1 = ( 2πn ) u n 2

29 phase diagram at zero temperature

30 Quantum Criticality Between Phase II and IV ( )) n = n b2 +λ 1 T 1/2 2(µ µc4 ) Li 1/2 ( exp T ( )) m = m b2 +λ 2 T 1/2 Li 1/2 ( exp 2(µ µc4 ) T ( )) κ c = κ cb2 +λ 3 T 1/2 2(µ µc4 ) Li 1/2 ( exp T ( )) χ s = χ s b2 +λ 4 T 1/2 2(µ µc4 ) Li 1/2 ( exp T

31 I. Bethe ansatz solution for the 1D Hubbard model II. III. Pair Correlation and FLLO IV. Quantum criticality: From a Bethe ansatz perspective

32 n k pair 200 (a) U=-8t, L=80, n=0.5 (b) Q=πnP k max P 50 increasing P=0,0.1,0.2,..., k max 0 0 π/2 π momentum k 1D attractive Hubbard model (Bogoliubov and Korepin, 1988, 1889): the single particle Green s function ψ n,sψ 1,s e n/ξ, ξ = v F / the singlet pair correlation function ψ n, ψ n, ψ 1, ψ 1, n θ Fulde & Ferrell: Cooper pairs form with finite centre-of-mass momentum. Larkin & Ovchinnikov: the order parameter oscillates in space. the 1D FFLO state (Feiguin & Heidrich-Meisner, 2007; M. Tezuka & M. Ueda 2008): numerical evidence of the power-law decay n pair cos(k FFLO x )/ x α

33 FFLO pairing correlations I α 0 particle-hole excitation I + α I α,j I α 0 adding particles I + α I α,j I α 0 backscattering process I + α I α,j I α 0 ground state I + α I α,j

34 finite-size correction E = ε 0 π v 6L 2 α α=u,b v α = ± dεα(kα) dp α(k α) = ± ε α(±q α) kα=±qα p α(q α) particle-hole excitations E = 2π L v α(n α + + Nα) α=u,b adding particles and backscattering E = 2π ( ) 1 t ( N) t (Z 1 )VZ 1 N + t ( D)ZV t Z D L 4 ( ) ( ) ( ) Nu Du vu 0 N =, D =, V = N b D b 0 v b ( ) Zuu(Q u Q ± ) Z Z = ub (Q b Q ± ) Z bu (Q u Q ± ) Z bb (Q b Q ± )

35 2.4 Dressed Charge Matrix in the Low Density Regime In the ground state, the integral equation for the dressed charge eq. (2.102) could be much simplified if we only focus the physics in the regime where the particle density is very low, which implies that the integration boundaries ( ) are small. We explicitly write down the four integral equations for the dressed charges, d d d d d d (2.157)

36 The conformal dimensions are expressed as, (2.167) (2.168) which in the strong coupling regime could be approximately expressed as, (2.169) (2.170) The typical contributions to the long distance asymptotics of correlation functions is written as, (2.171)

37 Pair correlation function ψ (x, t)ψ (x, t)ψ (0, 0)ψ (0, 0) A p,1 cos(π(n n )x) x + iv ut θ 1 x + ivb t θ 2 θ 1 1 2, θ n b u α G z(x, t) = S z(x, t) S z(0, 0) θ 1 m 2 z + A z,1 cos(2π(n n )x) x + i v u t 2θ 1 +A z,2 cos(2πn x) x + i v b t 2θ 2 2, θ nb u α The spatial modulations are characteristic of a Fulde-Ferrell-Larkin- Ovchinnikov (FFLO) state. The backscattering among the Fermi points of bound pairs and unpaired fermions results in a 1D analog of the FFLO state and displays a microscopic origin of the FFLO nature.

38 I. Bethe ansatz solution for the 1D Hubbard model II. III. Pair Correlation and FLLO IV. Quantum criticality: From a Bethe ansatz perspective

39 Quantum criticality and Luttinger liquid of 1D Bose gas E H Lieb & W Liniger 1963: δ-function Bose gas Continuum field theory problem of bosons with δ-function interaction L [ ] H = dx xψ (x) xψ(x)+cψ (x)ψ (x)ψ(x)ψ(x) 0 [ ] Ψ(x, t),ψ (y, t) N-particle eigenstate Lieb-Liniger equations [ ] = δ(x y), [Ψ(x, t),ψ(y, t)] = Ψ (x, t),ψ (y, t) = 0 L Φ >= dx N χ(x 1,, x N )Ψ (x 1 ) Ψ (x N ) 0 > 0 E = 2 2m N j=1 k 2 j, exp(ik j L) = N k j k l + i c k l=1 j k l i c

40 !"#$%!"#$&'()*+,-.#"+/01& Group of Quantum Integralable System!"#$%&'()*! Grand Canonical ensemble Entropy Minimization

41 t Classical Gas QC z=2, =1/2 TLL z=1, = / S/ Lc Boltzmann Statistics p = m 2π 2 T 3 2 e µ T, p = p 0 + T 3 2 Z 2 p 2π 2 for T Fermi Statistics p = m 2π T Li 3 ( e µ T ), for c 2 Bose Statistics p = m 2π T Li 3 (e µ T ), for c 0, 2 Fractional Statistics (1+ω i ) ( ωj ) αji j = e (ǫ i µ i )/T for c, T 0 1+ω j Haldane, Phys. Rev. Lett. (1991) Batchelor, Guan, Olkers, Rev. Rev. Lett. (2006) Jiang, Chen, Guan, Chin. Phys. B (2015)

42 atoms: ω x = 2π 22.2(1)Hz; ω = ω yω z = 2π 7.99(1)k Hz, T = 18 74nK in collaboration with Zhen-Sheng Yuan s group at USTC, arxiv:

43 ñ(t, µ) ñ 0 ( µ, t)+t d z +1 1 νzf ) ( µ µc t 1 νz ξ µ µ c ν, ξ z µ µ c zν scaling functions read off z = and ν = equation of state, compressibility, specific heat, speed of sound Wilson ratio determines the Luttinger parameter

44 regular parts for n r(t), p r(t), S r(t) scaling: p = p/[ 2 c 3 /(2m)], S = S/(k B c) ( ) p(µ, T) = p r(µ, T)+T z 1+1 µ µc G T 1/νz ( ) S(µ, T) = S r(µ, T)+T z 1 µ µc H T 1/νz! scaling functions read off z = and ν = equation of state, compressibility, specific heat, speed of sound Wilson ratio determines the Luttinger parameter

45 κ = 1 (πv N ), cv = πk2 B 3 T v s, for Luttinger liquid

46 R κ W = π2 k 2 B 3 κ c V /T, Rκ W = K = v s/v N, for Luttinger liquid

47 The central atoms can be transfer from F = 1, m F = 1 to F = 2, m F = 2, 1, 0, then removed the atoms in F = 2 to create a density dip. We finally extrapolated v s(η 0) from the finite perturbation ratio η base on the relation v s(η) = v s(0) 1 η/2. R κ W = π2 k 2 B 3 κ c V /T, Rκ W = K = vs/v N, for Luttinger liquid

48 The magnetic field compensates the gravity along x; after switching off the optical confinements; the cloud expands in a weak magnetic potential along y, the trapping frequency ω y = 2π 10.0(2)Hz. After a quarter period of oscillation, the momentum distribution is mapped to the spatial density profile k = mω yy/2π by focusing technique. n(k) exhibits a power-law decay at intermediate momenta (1/l φ k 20/l φ ), where l φ = v sk/(πk B T). n(k) A(K)Re[Γ(1/4K + ikl φ /2K)/Γ(1 1/4K + ikl φ /2K)] ( ) 1 ( ) 2K 2K ρ0 L φ (T) 1 1 2K A(K) = Γ (1 1 ), l φ = ρ 0 Λ 2 T π K 2K M. A. Cazalilla, J. Phys. B 37, S1 (2004)

49 For an harmonic trap, U = 1 2 kx2 with a frequency ω = k/m and the period T = 2π/ω. The motion of a particle X(t) = A cos(ωt +φ). Assuming the particle has a position x and velocity v at t = 0, then we have Thus we obtain x = A cos(φ), v = Aω cosφ, A = x/ cos(φ),φ = arctan v ωx, Finally after a quarter of the period t = T/4 = π/(2ω), X T4 = A cos( π 2 +φ) = x tanφ = v ω which means that the positions of atoms at t = T/4 only depend on the initial speeds and are independent of their initial positions. Shvarchuk, et. al. PRL 89, (2002)

50 Conclusions!"!!!!!#!! "!"!!!!#!! '!"!! '!!! " # $ # " #!"!! $! % &!!! # $! % & Quantum liquids: Luttinger parameter, Wilson ratios (R χ W, Rκ ), additivity rules, correlation W functions, universal relations, etc. Quantum criticality: scaling functions, critical exponents, specific peaks

51 Maxwell relations for Tan s Contact: dp = ndµ+sdt + MdH ρs 2 dw 2 + cd(a 1 ( ) ( ) C µ ( = T,H,a ( 1D C ) = H µ,t,a 1D ( C ) T µ,h,a 1D = ( n ( a 3D ) m z ( a 3D ) s ( a 3D ) ( c = w )µ,t,h,a D ( ) ) µ,t,h µ,t,h µ,t,h ρ s a 1 3D ) µ,t,w,h 3D) Tan s Contact at quantum criticality: C c 2 ( p ) c µ,h,t ( ) p(t, h, µ) = P 0 + t d/z+1 µ µc P t, h hc 1/νz t 1/νz Chen, Jiang, Guan, Zhou, Nature Communications 5: 6140 (2014)

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