Vladimir A. Yurovsky Tel Aviv University

Abandoned symmetry

Vladimir A. Yurovsky

Tel Aviv University

PERMUTATION SYMMETRY

Each physical object consists of indistinguishable particles of several kindsand is invariant over their permutations.



Symmetry =⇒ conservation law(Noether’s theorem)



Symmetry =⇒ conservation law(Noether’s theorem)What are the conservedquantities for the permuta-tion symmetry?




What are the conservedquantities for the permuta-tion symmetry?Dirac[Proc. R. Soc. A (1929)]— characters of irreduciblerepresentations of the sym-metric group




What are the conservedquantities for the permuta-tion symmetry?Dirac[Proc. R. Soc. A (1929)]— characters of irreduciblerepresentations of the sym-metric group

ABANDONED SYMMETRY

SELECTION RULES

Symmetric system + non-symmetric perturbation

S

S

D

FORBIDDEN

P

ALLOWED

SELECTION RULES


S

S

D

FORBIDDEN

P

ALLOWED

PijΨ(. . . ri . . . rj) ≡ Ψ(. . . rj . . . ri) = +Ψ(. . . ri . . . rj) — bosons

PijΨ(. . . ri . . . rj) ≡ Ψ(. . . rj . . . ri) = −Ψ(. . . ri . . . rj) — fermions

— the Pauli exclusion principle.

SELECTION RULES


S

S

D

FORBIDDEN

P

ALLOWED

PijΨ(. . . ri . . . rj) ≡ Ψ(. . . rj . . . ri) = +Ψ(. . . ri . . . rj) — bosons

PijΨ(. . . ri . . . rj) ≡ Ψ(. . . rj . . . ri) = −Ψ(. . . ri . . . rj) — fermions

— the Pauli exclusion principle. WHAT TO SELECT?

Outline

Permutation-invariant eigenstates for spin-independent interactions:— SU(M ) vs. permutation symmetry— spatially-chaotic states with defined spins— stability in a presense of spin-dependent interactions

Selection rules, related to the permutation symmetry.

Correlation rules— physical sense of Young diagrams.

MULTIDIMENSIONAL REPRESENTATIONS

Rotation group: Lx (Lz = 1) = (Lz = 0)+ (Lz = 2)



Symmetric groupTwo spin 1/2 atoms, Sz = 0

| ↑↓〉, | ↓↑〉P(| ↑↓〉 ± | ↓↑〉)/

√2 = ±(| ↑↓〉 ± | ↓↑〉)/

√2




| ↑↓〉, | ↓↑〉P(| ↑↓〉 ± | ↓↑〉)/

√2 = ±(| ↑↓〉 ± | ↓↑〉)/

√2

Three spin 1/2 atoms, Sz = 1/2

| ↑↑↓〉, | ↑↓↑〉, | ↓↑↑〉P(| ↑↑↓〉 + | ↑↓↑〉 + | ↓↑↑〉)/

√3 = (| ↑↑↓〉 + | ↑↓↑〉 + | ↓↑↑〉)/

√3




| ↑↓〉, | ↓↑〉P(| ↑↓〉 ± | ↓↑〉)/

√2 = ±(| ↑↓〉 ± | ↓↑〉)/

√2

Three spin 1/2 atoms, Sz = 1/2

| ↑↑↓〉, | ↑↓↑〉, | ↓↑↑〉P(| ↑↑↓〉 + | ↑↓↑〉 + | ↓↑↑〉)/

√3 = (| ↑↑↓〉 + | ↑↓↑〉 + | ↓↑↑〉)/

√3

Ξ1 = (−| ↑↓↑〉 + | ↓↑↑〉)/√2

Ξ2 = (2| ↑↑↓〉 − | ↑↓↑〉 − | ↓↑↑〉)/√3

PΞt =∑

t′Dt′t(P)Ξt′



Symmetric group: multicomponent wavefunction Ξ[λ]t :

PΞ[λ]t =

∑

t′D[λ]t′t (P)Ξ

[λ]t′

Hypothetical particles (intermedions, Gentileons) do not exist in nature.



Symmetric group: multicomponent wavefunction Ξ[λ]t :

PΞ[λ]t =

∑


[λ]t′

Hypothetical particles (intermedions, Gentileons) do not exist in nature.

SPINOR GASES: SPIN AND SPATIAL DEGREES OF FREEDOM

Experiments: Myatt, Burt, Ghrist, Cornell & Wieman, (1997)Stamper-Kurn, Andrews, Chikkatur, Inouye, Miesner, Stenger & Ketterle(1998)Theory: Tin-Lun Ho (1998), Ohmi & Machida (1998)

SPINOR GAS WITH SPIN-INDEPENDENT INTERACTIONS

H = Hspat + Hspin

spin-independent Hspat, coordinate-independent Hspin

permutation invariance: P−1HspatP = Hspat, P−1HspinP = Hspin


H = Hspat + Hspin


permutation invariance: P−1HspatP = Hspat, P−1HspinP = HspinMulticomponent spatial and spin wavefunctions:

HspatΦ[λ]t = EspatΦ

[λ]t , HspinΞ

[λ]t = EspinΞ

[λ]t

PΦ[λ]t =

∑

t′D[λ]t′t (P)Φ

[λ]t′ , PΞ

[λ]t =

∑


[λ]t′


H = Hspat + Hspin




[λ]t , HspinΞ

[λ]t = EspinΞ

[λ]t

PΦ[λ]t =

∑


[λ]t′ , PΞ

[λ]t =

∑


[λ]t′

Total wavefunction: Ψ[λ] =∑

tΦ[λ]t Ξ

[λ]t

HΨ[λ] = (Espat + Espin)Ψ[λ], PΨ[λ] = ±Ψ[λ]


H = Hspat + Hspin




[λ]t , HspinΞ

[λ]t = EspinΞ

[λ]t

PΦ[λ]t =

∑


[λ]t′ , PΞ

[λ]t =

∑


[λ]t′

Total wavefunction: Ψ[λ] =∑

tΦ[λ]t Ξ

[λ]t

HΨ[λ] = (Espat + Espin)Ψ[λ], PΨ[λ] = ±Ψ[λ]

Spin-free quantum chemistry1D gases — Yang-Gauden model [Yang (1967), Sutherland (1968)]

QUANTUM GASES WITH SU(M ) SYMMETRY

Hspin invariance over spin rotations =⇒ SU(M ) symmetry(M = 2s + 1 — multiplicity, s — spin of the atom).


Hspin invariance over spin rotations =⇒ SU(M ) symmetry(M = 2s + 1 — multiplicity, s — spin of the atom).Theory:C. Wu, J.-p. Hu, and S.-c. Zhang, PRL 91, 186402 (2003);A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P.Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6, 289 (2010);M. A. Cazalilla, A. F. Ho, and M Ueda, NJP 11, 103033 (2009).— Atoms with closed electron shell.— Interactions are independent of the nuclear spin.


Hspin invariance over spin rotations =⇒ SU(M ) symmetry(M = 2s + 1 — multiplicity, s — spin of the atom).Theory:C. Wu, J.-p. Hu, and S.-c. Zhang, PRL 91, 186402 (2003);A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P.Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6, 289 (2010);M. A. Cazalilla, A. F. Ho, and M Ueda, NJP 11, 103033 (2009).— Atoms with closed electron shell.— Interactions are independent of the nuclear spin.Observation of SU(M ) symmetry:87Sr: s = 9/2 — SU(10)X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus,M. S. Safronova, P. Zoller, A. M. Rey, J. Ye, (2014);173Yb: s = 5/2 — SU(6) F. Scazza, C. Hofrichter, M. Hofer, P. C. De Groot,I. Bloch, and S. Folling, (2014);G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, Florian Schafer, HuiHu, Xia-Ji Liu, J. Catani, C. Sias, M. Inguscio, and L. Fallani (2014).



Classification of SU(M ) invariant states — Young diagrams.

M

λ

λ1

M

λ=[λ ,...,λ ]1 M

∑

m

λm = N

— symmetrization over rows— antisymmetrization over columns




M

λ

λ1

M

λ=[λ ,...,λ ]1 M

∑

m

λm = N


s = 1/2:2S

S — the total many-body spin




M

λ

λ1

M

λ=[λ ,...,λ ]1 M

∑

m

λm = N


s = 1/2:2S

S — the total many-body spin

s > 1/2: the total spin is undefined, Smax = (s + 1)N −∑Mm=1mλm

SU(M ) AND PERMUTATION SYMMETRIES

N=4, s=1, M=3

transformations in the spin space [SU(M)]


perm

utat

ions

SN

N=4, s=1, M=3



perm

utat

ions

SN

N=4, s=1, M=3


SU(M ) — for the spin and total wavefunctions

Permutation symmetry — for the spin and spatial wavefunctions— can persist if SU(M ) is violated(Hspin is coordinate-independent but is not SU(M ) invariant)

Consequences of the SU(M ) and permutation symmetries do not coincide(and do not contradict).

EIGENSTATES OF QUANTUM GASES: INDIVIDUAL SPINS

Ψ ∝=∑

P

∏

j

ϕpj(rPj)|σj(Pj)〉P1

P4

P2

P3


Ψ ∝=∑

P

∏

j


P4

P2

P3

Excited states of interacting gases Thermal chaotic eigenstate[Srednicki, PRE 50, 888 (1994)]


Ψ ∝=∑

P

∏

j


P4

P2

P3

Excited states of interacting gases Thermal chaotic eigenstate[Srednicki, PRE 50, 888 (1994)]

COLLECTIVE SPIN AND SPATIAL WAVEFUNCTIONS

Individual spins of interacting elec-trons are undefned

singlet

triplet

MANY-BODY COLLECTIVE SPIN AND SPATIAL WAVEFUNCTIONSs = 1/2

Degenerate stateswith collective spinwavefunctionsUnitary transforma-tion to degeneratestates with individualspins

non−interacting atom

s

}

}S=0

S=N/2−1

S=N/2E

interactionsspin−

independent

Non-degenerate states withdefined total spins S[ Heitler, Z. Phys. 46, 47(1927)]S — good quantum num-berNo thermal equilibrium be-tween states with differentS

How long can the states with defined S survive spin-dependent interactions?

SPATIAL CHAOS FOR DEFINED TOTAL SPINS[VY, ArXiv 1509.01264]

Atoms with spin-independent interactions, the total spin S is conserved.Conditions of chaos — high energy-density of states [Deutch (1991)],

—delocalization in the Fock space[Altshuler et al (1997),Jacquod & Shepelyansky (1997), Silvestrov (1998)]

a > 2π~/(√mTN2) (3D), a > 2

√

2π~/(mωconf )/N2 (2D)




a > 2π~/(√mTN2) (3D), a > 2

√


Berry conjecture [Berry (1977)]: an egenfunction is a superposition ofplane waves with random phases and Gaussian random amplitudes, but withfixed wavelength.




a > 2π~/(√mTN2) (3D), a > 2

√



random phase and amplitude ⇓ ⇓ fixed wavelength

Wavefunctions ofinteracting atoms

Ψ(S)nSz

= N∑

r,{p}A(S)n (r, {p})δ({p}2 − 2mE

(S)n ) Ψ

(S)r{p}Sz⇑

wavefunctions of non-interacting atoms — symmetrized plane waves


random phase and amplitude ⇓ ⇓ fixed wavelength

Wavefunctions ofinteracting atoms

Ψ(S)nSz

= N∑

r,{p}A(S)n (r, {p})δ({p}2 − 2mE

(S)n ) Ψ

(S)r{p}Sz⇑

wavefunctions of non-interacting atoms — symmetrized plane waves

A(S)n

∗(r′, {p′})A(S)

n (r, {p}) =δr′rδ{p′}{p}

δ({p′}2 − {p}2)(generalization of[Srednicki (1994)])

DECAY OF STATES WITH DEFINED TOTAL SPINS

Weisskopf-Wigner estimate for the decay rate

Γ = 2π|〈Ψ(S′)n′S′

z|Vspin|Ψ(S)

nSz〉|2 dn

dE(S′)n′

|E(S′)n′ =E

(S)n

|〈Ψ(S′)n′S′

z|Vspin|Ψ(S)

nSz〉|2 = (NN ′)2

∑

{p}{p′}δ({p}2−2mE

(S)n )δ({p′}2−2mE

(S′)n′ )

calculated with sum rules −→ ×∑

r,r′|〈Ψ(S′)

r′{p′}S′z|Vspin|Ψ(S)

r{p}Sz〉|2

[VY, PRA 91, 053601 (2015); ArXiv 1506.01268]

Spin-dependent two-body interactions

Vspin =g⇈Dd

2V⇈+

g�Dd

2V�+g

↑↓DdV↑↓, g3d = 4π~2

a

m, g2d = g3d

√

mωconf2π~

Γ(S)Sz

= ΓDd

{

[(S − 1)2 − S2z ](S

2 − S2z)

2S(S − 1)(2S − 1)(2S + 1)N(N + 2S)(N + 2S + 2)α2+

+(S2 − S2

z)(N + 2S + 2)

S(2S + 1)N

[

α2+S2z(N + 2)

S2 − 1+ α2−(N − 2) + 4α+α−Sz

]

+[(S + 1)2 − S2

z ](N − 2S)

(S + 1)(2S + 1)N

[

α2+S2z(N + 2)

S(S + 2)+ α2−(N − 2) + 4α+α−Sz

]

+[(S + 1)2 − S2

z ][(S + 2)2 − S2z ]

2(S + 1)(S + 2)(2S + 1)(2S + 3)N(N − 2S)(N − 2S − 2)α2+

}

α+ = (a⇈ + a� − 2a↑↓)/a ,α− = (a⇈ − a�)/a, a = (a⇈ + a� + a↑↓)/3

Γ3d = 2√

πT/ma2n3d, Γ2d =π

2a2ωconfn2d

THE DECAY MINIMIZATION BY FESHBACH RESONANCE

10-4

10-3

10-2

10-1

100

101

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Sz=S-200

Sz=S-2

N=10000 atoms 87Rb,

Γ Sz

(S) /Γ

Dd

S

no tuningFeshbach tuning of a↓ ↓ (B0≈1007.4G, ∆≈0.21G)Feshbach tuning of a↑ ↓ (B0≈9.13G, ∆≈15mG)

10-4

10-3

10-2

10-1

100

101

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

1

2

3

4

5

6

7

B-B0

Feshbach tuning of a↑ ↓

Γ Sz

(S) /Γ

Dd

B-B

0, G

S

Sz=S-100Sz=S-20Sz=S-5

87Rb, n3d = 1012cm−3, T = 1µK: Γ3d ≈ 0.9s−1

a↑↓ tuning at Sz ≈ ±S, N ≫ 1: α+ ≈ ∓α−

Γ(S)Sz

∼ α2+ΓDd

N

TWO-BODY CORRELATIONS

Local correlations of atoms in different spin-states— calculated with sum rules.

Averaged over states with defined total spin

ρ(S,Sz)↑↓ =

〈ρ2(0)〉N (N − 1)

(

1

4N (N − 2) + S(S + 1)− 2S2

z

)

〈ρ2(0)〉 — the average two-body density, independent of S and Sz

Averaged over the thermal equilibrium with given total spin projection(occupations of the individual spin-states)

ρ(N↑,N↓)↑↓ =

〈ρ2(0)〉N (N − 1)

(

1

4N2 − S2

z

)

= ρ(S,Sz)↑↓ − 〈ρ2(0)〉

N (N − 1)

(

S2 − S2z + S − N

4

)

SELECTION RULES (arbitrary s)

perm

utat

ions

SN

N=4, s=1, M=3


allowed?

H = Hspat + Hspin +∑

1≤j1<j2<···<jk≤N

Wk(j1, . . . , jk)

Wk({j}) =∑

{m},{m′}〈{m′}|W (rj1, . . . , rjk)|{m}〉

k∏

i=1

|m′i(ji)〉〈mi(ji)|,

— k-body interaction, depends on the spin states |m(j) >and coordinates rj of k atoms j1, . . . , jk

SELECTION RULES (arbitrary s)


1≤j1<j2<···<jk≤N

Wk(j1, . . . , jk)

Wk({j}) =∑

{m},{m′}〈{m′}|W (rj1, . . . , rjk)|{m}〉

k∏

i=1



〈Ψ[λ′]|Wk({j})|Ψ[λ]〉 = 〈Ψ[λ′]|Wk(N − k, . . . , N )|Ψ[λ]〉(both Ψ[λ] and Ψ[λ′] are either symmetric or antisymmetric)Invariance over permutations of the first N − k atoms

— reduction to subgroup SN ⇒ SN−kOrthogonality relations.

SELECTION RULES


1≤j1<j2<···<jk≤N

Wk(j1, . . . , jk)

Wk({j}) =∑

{m},{m′}〈{m′}|W (rj1, . . . , rjk)|{m}〉

k∏

i=1



〈Ψ[λ′]|Wk({j})|Ψ[λ]〉 = 〈Ψ[λ′]|Wk(N − k, . . . , N )|Ψ[λ]〉(both Ψ[λ] and Ψ[λ′] are either symmetric or antisymmetric)Invariance over permutations of the first N − k atoms

— reduction to subgroup SN ⇒ SN−kOrthogonality relations.

λ and λ′ differ by relocation of no more than k boxes between their rows

SELECTION RULES

Wk({j}) =∑

{m},{m′}〈{m′}|W (rj1, . . . , rjk)|{m}〉

k∏

i=1


— k-body interaction, depends on the spin states |m(j)〉and coordinates rj of k atoms j1, . . . , jk

〈Ψ[λ′]|Wk({j})|Ψ[λ]〉λ and λ′ differ by relocation of no more than k boxes between their rows

Two-bodyinteraction(k = 2) λ λ’

SELECTION RULES

Wk({j}) =∑

{m},{m′}〈{m′}|W (rj1, . . . , rjk)|{m}〉

k∏

i=1



〈Ψ[λ′]|Wk({j})|Ψ[λ]〉λ and λ′ differ by relocation of no more than k boxes between their rows

Two-bodyinteraction(k = 2) λ λ’

M∑

m=1

|λm − λ′m| ≤ 2k

DIPOLE TRANSITIONS

s = 1/2, k = 1

2(S+1)2S

2(S−1)

Wk = C0 + CxSx + CySy + CzSz

HIGH SPIN

0 2 4 6 8 10λ2

0Smax=λ1-λ3=2

43

5

76

8

910

11

12

λ

λ’

s = 1, k = 1

CORRELATION RULES

Eigenstate-averaged local correlations of k particles vanish if k exceeds thenumber of columns (for bosons) or rows (for fermions) in the associated Youngdiagram

correlations (fermions)spatial

(fermions)spin

correlations (bosons)

CORRELATION RULES

Eigenstate-averaged local correlations of k particles vanish if k exceeds thenumber of columns (for bosons) or rows (for fermions) in the associated Youngdiagram

correlations (fermions)spatial

(fermions)spin

correlations (bosons)

Valid for both spatial 〈Ψ[λ]|∏ki=2 δ(r1 − ri)|Ψ[λ]〉

and momentum 〈Ψ[λ]|∏ki=2 δ(p1 − pi)|Ψ[λ]〉 correlations

POPULATION OF MANY-BODY STATES

Spatially-homogeneous spin-changing (conserves the Young diagram)Whom(t) =

∑

m6=m′Wmm′(t)|m〉〈m′| (example — the π/2 pulse)

Spatially-inhomogeneous spin-conserving (moves one box)Winh(r, t) =

∑

mWm(r, t)|m〉〈m|



∑



∑

mWm(r, t)|m〉〈m|

1 spin state



∑



∑

mWm(r, t)|m〉〈m|

fermion correlations

Whom1 spin state

2 spin states



∑



∑

mWm(r, t)|m〉〈m|

Whom Winh1 spin state

2 spin states




∑



∑

mWm(r, t)|m〉〈m|

Whom Winh Whom1 spin state

2 spin states

3 spin states


CONSERVATION LAWS

Integrals of moton — the character operators

[Dirac, Proc. R. Soc. A (1929)]

χ(CN ) =∑

P∈CN

P/g(CN )

(The sum is over all g(CN ) permutations in a conjugate class CN )Their eigenvalues χλ(CN ) — the normalized characters

Each spatial orbital is occupied only by one particle

Average spatial and momentum local correlations

ρ[λ]k ({0}) = ρ

[λ]k 〈ρk({0})〉 , g

[λ]k ({0}) = ρ

[λ]k 〈gk({0})〉

〈ρk({0})〉 and 〈gk({0})〉 λ— multiplet independentMultiplet dependence — the universal factor

ρ[λ]k =

∑

Ck

sig(Ck)g(Ck)χλ(Ck)

Vladimir A. Yurovsky Tel Aviv University

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