Fractional excitation, -charge separation, spin incoherent and Luttinger liquids Xi-Wen Guan

Key collaborators:

Feng He, Yu-Zhu Jiang, Hai-Qing Lin, Han Pu, Randy Hulet

PCFT, University of Science and Technology of China, November 2020 Outlines

1 I. Fundamental concepts: Spin-charge separation

2 II. Yang-Gaudin model: Fractional and Luttinger liquids

3 III. Testing the theory of spin-charge separation

4 IV. Outlook Fundamental concepts: quantum liquids in 1D

I. Fundamental concepts: Spin-charge separation feature The zoo Quasiparticles are an extremely useful concept that provides a more intuitive understanding of complex phenomena in many-body physics. As such, they appear in various contexts, linking ideas across di !erent fields and supplying a common language.

LANDAU QUASIPARTICLES Even stranger quasiparticles showed A concept emerges up in the early 1980s, with the discovery One of the biggest surprises for of the fractional quantum Hall e $ect. experimental physicists of the early Robert Laughlin proposed that the twentieth century was the discovery of the elementary quasiparticle excitations super !uid behaviour of liquid helium. And involved carry just a fraction of an Moscow-based Pyotr Kapitza knew just the charge — as if, remarkably, are right theorist to explain it: Lev Landau 1. split up. Electronic transport measurements It was unfortunate then that the brilliant in 1997 detected quasiparticles carrying and outspoken Landau had got himself only a third of an electron charge and imprisoned by the KGB in 1938, the thereby proved — quite spectacularly — same year that super !uidity was publicly that Laughlin’s quasiparticles were more announced. Undeterred, Kapitza wrote than just mathematical entities. to Landau’s captors and miraculously Laughlin noted in his Nobel Prize managed to persuade them that it would be lecture 3 that both super !uidity and the a great loss to Soviet science if Landau was fractional quantum Hall e $ect are examples le " to perish in prison. He was promptly of emergent phenomena, which can only released, with Kapitza appointed as his be understood by considering the system Calculated (top) and measured (bottom) guarantor, and set to work without delay. as a whole and are impossible to derive dynamical structure factors for tin selenide, a Sure enough, two years later he published from microscopic principles. But one of highly thermoelectric material. Reproduced from a paper on helium super !uidity that, along the things an emergent phenomenon can ref. 30, NPG. the way, boldly introduced the notion do, Laughlin wrote, is create new particles. of quasiparticles. Landau’s discovery of quasiparticle Nature Physics 12, 1085 (2016)

! #$%& '()*+,,(- ./0,+12341 5+*+6378 9(46 :; <94+-=34 >(6/43? @,, 4+=261 43134A37? " " " " " " " " " " " Fundamental concepts: quantum liquids in 1D

(a)

(b)

Two-Spinon continuum Two-Spinon continuum

Many neutron scattering experiments present novel measurements of such fractional excitations through measuring the dynamic structure factor:

N ∞ aa¯ 1 −iq(j−j′ )+iωt a a¯ S (q,ω)= dte < Sj (t)Sj′ (0) >, a = z, , + N ′ −∞ − j,Xj =1 Z

Faddeev & Takhtajian, Phys. Lett. A 85, 375 (1981)

Lake et al. PRL 111, 137205 (2013); Stone et al, PRL 91, 037205 (2003) Fundamental concepts: quantum liquids in 1D

a b c d e f g h k 20 +– 20 !(3) D RPAP D zz

10 10

0 0 0 1 2 0 1 2 i l 20 20 +– 1 !(2) D zz RPAP D

10 10

0 0 0 0 1 2 0 1 2 Energy transfer (meV) j m 20 +– 20 –+ !(2) D RPP D B !(3) !(2) !(2) PAP PAP PP 10 10 GS R R R D+– D zz D+– D zz D+– D–+ 0 0 !Sz = –10 –1 0 –1 +1 0 1 2 0 1 2 Momentum transfer, q ( ") Momentum transfer, q ( ")

Spin strings and excitation spectra of spin-1/2 XXZ model SrCo2V2O8

Wang, et al. Nature 554, 219 (2018); A K Bera, et al. Nat. Phys. 16, 625 (2020)

Wu, et al. PNAS, 111, 39 (2014), Wu, et al. Nature Comm. 10:698 (2019) Fundamental concepts: quantum liquids in 1D Spin-charge separation

For interacting fermions in higher dimensions, an excited electron is dressed by interaction and fluctuation and forms a quasiparticle with a dispersion ω(k)= E(k )+ kF (k k ). The spectral F m∗ − F function has a Lorenzen peak. Whereas in 1D , the spectral function has two singular features corresponding to the spinon and . Fundamental concepts: quantum liquids in 1D

Challenges: a definitive observation of the SC separation is still missing 1) Observation of the separate collective excitation spectra of charge and spin 2) Determination of independent spin and charge sound velocities and their Luttinger parameters 3) Confirmation of the power-law decay with distance in correlation functions or of the power law of dynamical response functions

Kim, et. al. PRL, 77(19), 4054 (1996); Nat. Phys., 2(6):397 (2006)

Auslaender, et. al. Science, 308(5718):88 (2005)

Jompol, et. al. Science, 325 597 (2009) Fundamental concepts: quantum liquids in 1D

FIG. 1. Probing spin-charge deconfinement with cold atoms . A, Cartoon depicting of a fermionic excitation into quasiparticles. The dynamics is initiated by removing a fermion from the Hubbard chain. This quench creates a spin (spinon) and a charge (holon) excitation, which propagate along the chain at di fferent velocities vJ and vt. B, Using quantum gas microscopy, we simultaneously detect the spin and density on every site of the chain after a variable time after the quench. C, Average number of holes in the chain as a function of time. Error bars denote 1 s.e.m. The quench, performed at 0 ms creates a hole with a probability of ∼ 78% in the central site of the chain (bottom).

I. Bloch’s group, Vijayan et. al., Science 367, 186-189 (2020) Fundamental concepts: quantum liquids in 1D

Quantum criticality of spinons (Cooper pyrazine Quantum criticality of bosons dinitrate (CuPzN))

κ ~ Luttinger liquid : RW = K = vs/vN = πn/(mvs ) momentum distribution n(k) 1/k (1−1/2K ) ∼ d/z+1−2/νz µ µc Scaling function : cv /T = c(µ, T )+ T − K T 1/νz Critical exponents : z = 2, ν = 1/2, d = 1   −ν −z Correlation length : ξ µ µc , ∆ ξ ∼| − | ∼ νz Critical temperatures : T = a µ µc , T = a µ µc 1 − 1| − | 2 2| − | Yang, Chen, Zhang, Sun, Dai, Guan, Yuan, Pan, Phys. Rev. Lett. 119, 165701 (2017)

He, Jiang, Yu, Lin, Guan, PRB 96, 220401(R) (2017); Breunig, et al., Sci. Adv. 2017; 3:eaao3773 Fundamental concepts: quantum liquids in 1D PHYSICAL REVIEW LETTERS 125, 190401 (2020)

Emergence and Disruption of Spin-Charge Separation in One-Dimensional Repulsive Fermions

Feng He , 1,2 Yu-Zhu Jiang, 1 Hai-Qing Lin, 3,4 ,* Randall G. Hulet , 5 Han Pu, 5 and Xi-Wen Guan 1,6,7 ,† 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, APM, Chinese Academy of Sciences, Wuhan 430071, China 2University of Chinese Academy of Sciences, Beijing 100049, China 3Beijing Computational Science Research Center, Beijing 100193, China 4Department of Physics, Beijing Normal University, Beijing 100875, China 5Department of Physics and Astronomy, and Rice Center for Quantum Materials, Rice University, Houston, Texas 77251-1892, USA 6Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China 7Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia

(Received 27 April 2020; accepted 17 September 2020; published 2 November 2020)

At low temperature, collective excitations of one-dimensional (1D) interacting fermions exhibit spin-charge separation, a unique feature predicted by the Tomonaga-Luttinger liquid (TLL) theory, but a rigorous understanding remains challenging. Using the thermodynamic Bethe ansatz (TBA) formalism, we analytically derive universal properties of a 1D repulsive spin- 1=2 Fermi gas with arbitrary interaction strength. We show how spin-charge separation emerges from the exact TBA formalism, and how it is disrupted by the interplay between the two degrees of freedom that brings us beyond the TLL paradigm. Based on the exact low-lying excitation spectra, we further evaluate the spin and charge dynamical structure factors (DSFs). The peaks of the DSFs exhibit distinguishable propagating velocities of spin and charge as functions of interaction strength, which can be observed by Bragg spectroscopy with ultracold atoms.

DOI: 10.1103/PhysRevLett.125.190401

He, Jiang, Lin, Hulet, Pu, Guan, Phys. Rev. Lett. 125, 190401 (2020) Fundamental concepts: quantum liquids in 1D

Outlines

1 I. Fundamental concepts: Spin-charge separation

2 II. Yang-Gaudin model: Fractional quasiparticles and Luttinger liquids

3 III. Testing the theory of spin-charge separation

4 IV. Outlook Fundamental concepts: quantum liquids in 1D

II. Yang-Gaudin model: Fractional quasiparticles and Luttigner liquids

L ~2 d2 = φ†(x) φ (x)dx j 2 j H 0 − 2m dx Yang-Gaudin model(1967) j=X↓,↑ Z   L g1D > 0: antiferromagnetic † † +g1D φ↓(x)φ↑(x)φ↑(x)φ↓(x)dx 0 g1D < 0: ferromagnetic Z L H † † φ↑(x)φ↑(x) φ↓(x)φ↓(x) dx − 2 0 − Z  

H: effective magnetic field

2 2 ~ c a⊥ g1D = , c = 2/a1D, a1D = + Aa⊥ − m − − a3D Yang Phys. Rev. Lett. 19, 1312 (1967) Gaudin, Phys. Lett. 24, 55 (1967) Guan, Batchelor and Lee, Rev. Mod. Phys. 85, 1633 (2013) Fundamental concepts: quantum liquids in 1D

Bethe wave function for (0 < xQ1 <...< xQi < xQj ...< xQN < L)

ψ = Aσ ...σ (P ,..., P Q ,..., Q ) exp i(k x + ... + k x ) 1 N 1 N | 1 N P1 Q1 PN QN XP Bethe Ansatz equations

M kj λℓ + i c/2 exp(ikj L)= − , j = 1,..., N kj λℓ i c/2 ℓY=1 − − N M λα k + i c/2 λα λβ + i c − ℓ = − , α = 1,..., M λα kℓ i c/2 − λα λβ i c ℓY=1 − − βY=1 − −

Energy

~2 N E = k 2 2m j Xj=1

CN Yang, Phys. Rev. Lett. 19, 1312 (1967) Fundamental concepts: quantum liquids in 1D Spin charge separation: beyond the free fermion theory

M 1 ki Λj + ic eiki L − 2 i 1 N = 1 , = ,..., ki Λj ic Bethe ansatz equations Yj=1 − − 2 N 1 N Λj ki + ic Λj Λ + ic − 2 − ℓ j 1 M 1 = , = ,..., Λj ki ic Λj Λℓ ic Yi=1 − − 2 ℓY=1 − − An effective antiferromagnetic Heisenberg spin chain forc > 0

J N N 4E H = Sˆ Sˆ h sz , J F 2 i • i+1 − i ≈ c Xi=1 Xi

M π2 nJ 1 E = N N2 1 + O(c−2) 2 2 3L − − 2 ri + 1/4   Xi=1 i N M ri + r r + i 2 i − j i = ri ! − ri rj i − 2 Yj=1 − − Oelkers, Bathchelor, Bortz and Guan, J. Phys. A 39, 1073 (2006) Lee, Guan, Sakai and Batchelor, PRB 85, 085414 (2012) Yang, Pu, Phys. Rev. A 95, 051602 (2017) Fundamental concepts: quantum liquids in 1D

n,a n 1 −δL Spin strings : λα = λ + i(n + 1 2a)+ O(e ), a = 1, , n α 2 − ···

∞ Mn n 2(kj λα) −1 k L = 2πI θ − , θy (x) = 2tan (x/y) j j − nc Xn=1 αX=1   N 2(k λn ) ∞ Mm 2(λn λm) j α n α − β θ − = 2πJα + Θmn nc c ! Xj=1   mX=1 βX=1 ∞ Mn N Mn 1 I + Z, Jn − + + Z j ∈ 2 α ∈ 2 2 Xn=1 n ∞ n n N Mn 1 J J = mMm n Mm + | α|≤ + 2 − − 2 − 2 mX=1 mX=n+1 Fundamental concepts: quantum liquids in 1D

(a) Particle-hole and two-spinon excitations; (b) Four-spinon excitation; (c) Particle-hole and two- spinon excitations at finite magnetic field; (d) Two-spinon excitation with one length-2 string

N N M N 2π n ∞ Momentum : K = k = I + Jn , Energy : E = k 2 j L  j α j Xj=1 Xj=1 αX=1 Xn=1 Xj=1  2 λh  2 j 0 0 h Two spinon spectrum : ∆K = nc π + 2π ρs (λ)dλ, ∆F = φs (λj ) − ± 0 − Xj=1 Z Xj=1

He, Jiang, Lin, Hulet, Pu and Guan, Phys. Rev. Lett. 125, 190401 (2020) Fundamental concepts: quantum liquids in 1D

m∗/m Numerical 1.3 m∗ = m(1 + 4 ln 2/γ) 1.30

1.25 1.2 1.20 /m ∗ 1.15 m

1.7 5 10 15 20 1.1

1.0 0 50 100 150 200 250 300 γ Particle-hole continuum spectra at γ = 5.03 Effective mass of quasiparticle

1 2 Particle hole spectrum : ∆E = vc ~ ∆K (~∆K ) − | |± 2m∗ ε′ (k ) m ε′′(k ) πρ′ (k )ε′ (k ) Charge velocity : v = c 0 , Effective mass : = c 0 c 0 c 0 c ∗ 2 3 2πρc(k0) m 2(2πρc (k0) − (2πρc (k0))

4ln2 ∗ 4ln2 Strong coupling : vc 2πnc 1 , m m 1 + ≈ − γ ≈ γ    

He, Jiang, Lin, Hulet, Pu and Guan, Phys. Rev. Lett. 125, 190401 (2020) Fundamental concepts: quantum liquids in 1D

Quantum statistics:

1 quantum man-body systems

2 microscopic state energy Ei

3 ∞ −Ei /(kBT ) partition function Z = i=1 Wi e

4 free energy F = kBTPln Z − 5 Energy of the quasiparticle: 1 ′ ′ ′ − ′ ′ ′ ǫk,σ = ǫk +V k σ f (k, k ; σ, σ )δnk σ 6 challenge: findingP new physics

H ∞ 2 −φn (k)/T ε(k) = k µ T an ln 1 + e − − 2 − ∗ Xn=1   ∞ −ε(λ)/T −φm(λ)/T φn(λ) = nH Tan ln 1 + e + T Tnm ln 1 + e − ∗ ∗   mX=1   T ∞ p = ln[1 + e−ε(k)/T ]dk, n = 1,..., 2π ∞ Z−∞ Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press) Guan, Batchelor and Lee, Rev. Mod. Phys. 85, 1633 (2013) Fundamental concepts: quantum liquids in 1D

1.2

0.8 Velocity 0.4

0 0.05 0.1 0.15 0.2

Wilson ratio Phase diagram at T˜ = 0.005 Spin and charge velocities at µ˜ = 0.3, c = 1

χ Cov(M,M) κ Cov(N,N) Wilson ratio–Essence of Luttinger liquid: RW ∝ Cov(E,E) (or RW ∝ Cov(E,E) )

2 χ 4 πkB χ 2vc R = = Ks W 3 µ g c /T v + v  B  v s c 2 κ π κ 2vs RW = = Kc 3 cV /T vs + vc πvc Ks = 2πvsχ0, Kc = κ 2 Fundamental concepts: quantum liquids in 1D

5

2vc 1.5 Ks vs+vc Ks RW (T˜ = 0.0005) 4

1 3 s W K R 2 0.5

1

0 0 0 0.04 0.08 0.12 0.16 0 0.04 0.08 0.12 0.16 0.2 H˜ H˜ Wilson ratio at µ˜ = 0.3, c = 1 and T˜ = 0.005 Spin Luttinger parameter at µ˜ = 0.3, c = 1

χ Cov(M,M) κ Cov(N,N) Wilson ratio–Essence of Luttinger liquid: RW ∝ Cov(E,E) (or RW ∝ Cov(E,E) )

2 4 πkB χ 2vc RW = = Ks 3 µ g c /T v + v  B  v s c 2 κ π κ 2vs RW = = Kc 3 cV /T vs + vc πvc Ks = 2πvsχ0, Kc = κ 2 Fundamental concepts: quantum liquids in 1D

Tomonaga-Luttinger liquid πT2 1 1 f = E0 − + 6 vs vc 1/2 FG: Free gas cv T  ∼ MP: mixed phase

1 < RW < 2 FP: Ferromagnetic phase RW = 1

Quantum criticality (QC) of spinons

0 d/z+1−2/νz µ µc cv /T = c + T − , z = 2, ν = 1/2 v K T 1/νz   P2 3 Coexistence of Liquid and gas near Hs (COR): ǫp = vαp + ∗ + O(p ) α 2mα P Liquid BG 1 +1 s0∆H p = p + p + T z 0 0 G T 1/νz   2 3/2 Liquid πT BG 2 H p = , p = µc + 0 6v 0 3π 2 c   Fundamental concepts: quantum liquids in 1D

Specific heat for µ˜ = 2.5, c = 1 Quantum phases at criticality

Critical exponents : z = 2, ν = 1/2, d = 1 −ν −z Correlation length : ξ µ µc , ∆ ξ ∼| − | ∼ νz Critical temperatures : T = α µ µc , T = α µ µc 2 2| − | 1 1| − | 3 cv 3 TLL phase boundary : TTLL1 = , TTLL2 = cv vc π 1/vc + 1/vs π Fundamental concepts: quantum liquids in 1D

Read off critical exponents from scaling functions: ν = 1/2, z = 2

0 0 0.08 0.08

2 −0.1 −0.1 / 2 1 / 0.06 0.06 1 ˜ T − / ˜ ) −0.2 −0.2 T 0 / ) ˜

M 0.04 0.04 0 ˜ χ Exact Ana. − −0.3 µ˜ = 0.3 −0.3 Exact Ana. z − T˜ = 0.001 ˜ ˜ T = 0.001 χ M µ˜ = 0.3 (˜ 0.02 0.02 ( −0.4 −0.4 T˜ = 0.002 T˜ = 0.002 ˜ T = 0.003 T˜ = 0.003 −0.5 −0.5 0 0 0.15 0.155 0.16 0.165 0.17 −5 0 5 0.15 0.155 0.16 0.165 0.17 −5 0 5 H˜ ∆H/˜ T˜ H˜ ∆H/˜ T˜ scaling of magnetization scaling of susceptibility

1 2 2 1/2 1 arctan k0 arctan k0 T s ∆H z − π c c 0 M M0 = Li 1 e T −   π3/2√a   2 −   2 1 1 arctan 2 k arctan 2 k T −1/2 π c 0 c 0 s0∆H − T χ χ0 = Li− 1 e − −   π3/2√a   2 −   Fundamental concepts: quantum liquids in 1D

Luttinger liquid theory: single-particle Green’s function

ikF x 1/√vsτ ix e −1 G (x, τ) − + c.c. γ = (Kc + K 2)/8 σ 2 2 2 γ kc c ∼ (x + v τ ) Kc √vcτ ix − c − Incoherent spin liquid: single-particle Green’s function (Espin K T Echarge) ≪ B ≪

i(2k x−ψ+ ) e−2kF |x|(ln 2/π) e F Kc Gσ(x, τ) + c.c. 2 ∆ ∼ (x2 + v τ 2) Kc √vc τ ix c − Bosonization Hamiltonian: † π α 2 α 2 H = vα q bˆ bˆα,q + v ∆N + v J | | α,q 2L N α J α α=c,s α=c,s X Xq6=0 X   2c † † + ψˆ ψˆ ψˆ↓,r,k +2rk +pψˆ↑,−r,k −2rk −p L ↑,r,k1 ↓,−r,k2 2 F 1 F k1X,k2,p rX=±1 Near by the right/left Fermi point, r = 1, ψˆ is the Fermi field operator. ±

G. A. Fiete, Rev. Mod. Phys., 79, 801 (2007)

Cheianov and Zvonarev, Physics Rev. Lett. 92, 176401 (2004) Fundamental concepts: quantum liquids in 1D

Elementary excitations (a) the quantum numbers for the ground state in charge and spin degrees

(b) adding a particle near right Fermi point (∆Nα or backward scattering 2∆Dα). (c) a hole excitation. The total number of particles is N 1 in either spin or charge degrees. − ± (d) a single particle-hole excitation (Nα ). Calculate the total momentum and elementary excitation energies Fundamental concepts: quantum liquids in 1D

Origin of spin charge separation

2π − + π −1 2 t 2 Bethe ansatz result: ∆E = vα(N +N )+ vα(Zˆ ∆N~ ) +vα[Zˆ (2D~ )] L α α 2L α α α=c,s α=c,s X Xn o Z (Q ) Z (Q ) ∆N D Dressed charge Zˆ = cc c cs s ; Quantum number: ∆N~ = c , D~ = c Zsc(Qc) Zss(Qs) ∆Ns Ds       The leading order of correlations:

e−2iDc (kF↑+kF↓)x e−2iDskF↓x ϕ(x, t)ϕ(0, 0) = 2∆+ 2∆− h i (x ivat) a (x + ivat) a a=c,s − + − 2(∆ +∆ ) −2iDc (k +k )x −2iDsk x Q (πT /va) a a e F↑ F↓ e F↓ ϕ(x, t)ϕ(0, 0) T = − h i 2∆+ πT 2∆ πT sinh a (x ivat) sinh a (x + ivat) a=c,s va − va Q h i h i Giamarchi, T. Quantum Physics in one dimension (Oxford University Press, 2004) Guan, Batchelor and Lee, Rev. Mod. Phys. 85, 1633 (2013) Lee, Guan, Sakai, Batchelor, Phys. Rev. B 85, 085414 (2012) Fundamental concepts: quantum liquids in 1D

Outlines

1 I. Fundamental concepts: Spin-charge separation

2 II. Yang-Gaudin model: Fractional quasiparticles and Luttinger liquids

3 III. Testing the theory of spin-charge separation

4 IV. Outlook III. Probing the theory of spin charge separation

III. Testing the theory of spin charge separation Linear response theory

H = H0 + f (t)B ∞ ′ ′ ′ hA(t)i− A0 = χ(t − t )f (t )dt Z−∞ Kubo formula gives 1 χ(t − t ′) = h[A(t), B(t ′)]iΘ(t − t ′) i~ 1 1 ~ ~ χ (t) = e−βEm eiEmnt/ hm|A|nihn|B|mi− eiEnmt/ hm|B|nihn|A|mi AB i~ Z mn X   1 −βEm AmnBnm BmnAnm χAB (ω) = e − Z ω − ωmn + iη ω + ωmn + iη mn X   III. Probing the theory of spin charge separation

The dynamic structure factor relative to the operator F 1 ∞ S (ω) = hF(t)F +ieiωt dt F 2π Z−∞ 1 = e−βEm |hm|F|ni|2δ(ω − w ) Z nm mn X A simple tuning of the frequency of both Bragg lasers, one can measure both the density and spin responses. The responses are the imaginary part of the charge and spin susceptibilities, respectively. That relates to the energy (momentum) transfer from the external field to the system. Here ∆σ is the detuning.

2 2 P(k,ω) = 2πΩBrS(k,ω), ΩBr = ~Γ /(4∆) IAIB/Isat 1 1 2 p ∆P = ∝ 2 + 2 S↑↑(k,ω)+ S↑↓(k,ω) ∆↑ ∆↓ ! ∆↓∆↑

SD(k,ω) = 2 [S↑↑(k,ω)+ S↑↓(k,ω)]

SS (k,ω) = 2 [S↑↑(k,ω) − S↑↓(k,ω)]

1 −iωt † S ′ (q,ω) = dte hρˆσ(q, t)ˆρ ′ (q, 0)i σ,σ 2πN σ Z III. Probing the theory of spin charge separation

Momentum transfer by Bragg perturbation

1 1 2 p(k,ω) ∝ 2 + 2 S↑↑(k,ω)+ S↑↓(k,ω) ∆↑ ∆↓ ! ∆↑∆↓

1 −iωt † S ′ (q,ω) = dte hρˆσ(q, t)ˆρ ′ (q, 0)i σ,σ 2πN σ Z charge and spin density sensitive measurement

Figure: charge spin

charge channel: ∆↑ ≈ ∆↓(≫ w↑↓) spin channel: ∆↑ = −∆↓ III. Probing the theory of spin charge separation

1.0 Homogeneous one tube 3D sample 80 100 a0 200 a0 300 a0 0.8 400 a0 60 0.6 40 0.4 20 Atorms Per Tube

0 0.2 0 5 10 15 20 25 30 r(µm)

Normonized Excitation(A.U.) 0.0 0 5 10 15 20 25 30 35 40 Bragg Frequency (kHz) Charge DSF of homogeneous Fermi gas

Imχ(q,ω, kF , T , N) S(q,ω, kF , T , N)= ~ π(1 e−( ω/kBT ) − Dynamic susceptibility Nπm Imχ = (n n ) 2 q− q+ 2~ qkF −

∗ 2 Nm /(2~ qkc) 1 1 = −β~ω  2 2  (1 e ) m∗ ~q2 m∗ 2 − m∗ ~q2 m∗ 2  β ω− ∗ − v β ω+ ∗ − v  −  2q2 2m 2 c 2q2 2m 2 c  e "   # + 1 e "   # + 1     He, Jiang, Lin, Hulet, Pu, Guan, Phys. Rev. Lett. 125, 190401 (2020) III. Probing the theory of spin charge separation

Bragg spectroscopy for the spin-1/2 Fermi gas Bragg spectroscopy is used to measure a response of the gas to density (charge) mode excitations at a momentum q and frequency ω, as a function of the interaction strength.

1 −iωt † S ′ (q,ω)= dte ρˆσ(q, t)ˆρ ′ (q, 0) σ,σ 2πN h σ i Z R. Hulet’s group, Phys. Rev. Lett. 121, 103001 (2018) III. Probing the theory of spin charge separation

Charge DSF of homogeneous Fermi gas(imagine part of the response function is related to the energy transfer from the external field to the system)

Imχ(q,ω, kF , T , N) S(q,ω, kF , T , N)= ~ π(1 e−( ω/kBT ) − Dynamic susceptibility Nπm ωm q Imχ = (n n ), q = 2 q− q+ ± 2~ qkF − ~q ± 2

T. L. Yang, Phys. Rev. Lett. 121 103001 (2018) III. Probing the theory of spin charge separation

(a) (b)

) 16.5 7.0 1.0 400 a0 500 a0 q, ω

( 0.8 600 a0 s) S 15.5 6.6 ) 700 a0 / 0.6 hω ¯ β (cm − 0.4 c e

14.5 6.2 v

− 0.2 (1

0.0 Peak Frequency(kHz) 13.5 5.8 0 5 10 15 20 25 30 2 3 4 5 γ (c) Bragg Frequency (kHz) (d)

) 400 a0 13 Peak spin velocity 5.6

k, ω 1.0 500 a0 BA spin velocity δ

( 600 a0

0.8 s)

S 11 4.7

700 a0 / ) 0.6 hω ¯ (cm β

9 3.8 s −

0.4 v e

− 0.2 7 3.0 (1

0.0 Peak Frequency(kHz) 0 5 10 15 20 25 30 2 3 4 5 Bragg Frequency (kHz) γ

Spin DSF (H = 0): Luttinger liquid(LL) theory gives the DSF around k = π + δk

1 ALL w + vF δk w vF δk SLL = ~ Im ρ ρ − 1 e−( w/kBT ) T 4πT 4πT −      ρ(x) = Γ(1/4 ix)/Γ(4/3 ix), A = c2 α/2 − − LL − ⊥

He, Jiang, Lin, Hulet, Pu, Guan, Phys. Rev. Lett. 125, 190401 (2020) III. Probing the theory of spin charge separation

IV. Outlook

Fractional excitations (anyon, spinon, magnon, ...) Quasi-long range order (correlation function, M-body reduced density matrices...) Quantum liquid (Luttinger liquid, spin charge separation...) Quantum criticality (scaling laws, field theory, dynamical criticality... ) Evolution dynamics (transport properties, hydrodynamic, thermalization... ) Strongly correlated matter (topological matters, disorder, dynamical phase transition...) Quantum technology (quantum information, quantum metrology, quantum precision measurement, quantum battery, quantum refrigeration and quantum heat engine ... ) Conformal Field Theory, gauge field, Yang-Mills theory, quantum gravity of black holes, ......

Thacker, Rev. Mod. Phys. 53, 253 (1981)

F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982)

Andrei, Furuya and Lowenstein, Rev. Mod. Phys. 55, 331 (1983)

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