Recent advancesfor

integrable¥÷¥.long-range spin chains

.¥;÷!;!,;n÷by

Jules LamersUniversity of Melbourne

based on

TL,PRB C'781214416 TL

,V Pasquier, D Serban RKlabbers.TL

arXiv : 1807.05728 arXiv : 2004.73270 arXiv : 2009.74573

and ongoing

CMT seminar,UVA -

' 27 . 06 7/75

Motivation:#

yintegrab-eng-r-angespinchainsspi.nohains model magnetic materials

based on quantum mechanics

a. are one . .im.... .

i¥¥i¥÷÷¥( nature : e.g . Kcufz ; laboratory)

'

inner:c::::c::::÷ii: an .on.ca. ...... :i÷:÷÷:÷:applications in string theory

condensed -matter physicsquantum computing

quantum- integrable cases allow one to ¥,study the effect of long-range interactions

in an exactly - solvable setting using quantum algebra2/75

Overviewandyscaieofongrang Pinchas

.gg/;.4jtp.!.J&g&.K-→interaction range

nearest neighbour intermediate long rangeµ degree of rangespin symmetry

? 7anisotropic Heisenberg XYZ C- .

→ .

Sutherland '70

Baxter '

73 µ It

partially q -deformed HSHeisenberg XXZ ← ? →c.uglov -Lamers )isotropic Orbach '58

Yang-Yang '

66 Uglov'

95 , TL'

78

µ JL Pasquier Serban'20

t t

isotropic Heisenberg XXX c- lnozemtsev → Haldane- ShastryHeisenberg '28 lnozemtsev '90 Haldane ' 88

, Shastry' 88

Bethe '31 lnotemtsev '90 -

'

OO,

Haldane '91 , Bernard et al '

93tslabbers TL '20

3/75

Isotropic level : overview-#

Consider N sites with spina think : viz) !fg!,¥Iqtq&§!q!! "p sin.

H=iEniino;µ !"" "s

Tinh He is

Heisenberg XXX 're lnotemtsev '90 Haldane '88 - Shastry -88

potential VHCiijt-odci.jg.ie c- Vztiij ) lit) → Vash;jj= =Lsin'#Ci-JD r2

nearest neighbour k→• (Minas)ElNzzxilR>o K→0

exact up to solving up to solving → in closed form

solubility "

Bethe equations" ← "

Bethe equations" via connection to Haldane 'm

(spectrum) Bethe '31 via connection totrig Calogero -Sutherland

ell Calogero -Sutherland

Haig Bethe ansate Inozemtsev'

go ,'95,100 It I {Yemzimngtheotb

Klabbers 7.L'20

quantum Yangian structure degenerate affine Hecke algintegrals Faddeev et al , late '7os-805

unknown Yahgian symmetryBernard et al '93

I transfer matrix µcentre

V t freezinghigher known conjectured Bernard et al '93Ham's partial proof known

talstra Haldane 'es

Dittrich Inozemtsev'08 7/75

Isotropic level . symmetri--

.

'

es-

H ' chinaz.io?Efunctionofr=aistci.i , •¥!µ!rp&&÷fp!f÷I '-

j

isotropy global Mz : Csa,H

?- o S III.of Cx=x.az)

g[5. SB ]=zi MS'

N

Mz - Huda 5- = =Eo± Sz CSIS 's]=±S±Ii '

[5,51=25allows us to organise N=@2)

④"as

#1=14 ,←ITT . . -M

O 90,0 o o o o

^ : ⑧ so it remains to findSt l l S-

2 . . ⑧ ⑧ ⑧ 000

:& :÷÷:÷÷÷:÷÷÷:÷÷÷:÷;:÷ :÷:÷:.÷:.÷:.÷:÷! :÷! :÷! ! ! ! ! ! se.- nighest weight vectors

. ⑧ ⑧ ⑧SZ -- Nz - M Ny : . at 147*2N o

"

i---

E.g .(1/2)×06=3 to 2+05 to y④9 ⑤ 0+05

-

Gt = G- I

homogeneity translations : GHG- '

=H G=Pn→n . . - 133172 { gN=zfixes 14=7 : magnohs eipho-llf.it )

h

G- eigenvalue eip , momentum p=YfI (KIEN-7 )

H - eigenvalue Elp) dispersion5/75

Isotropic level spectrum : overview pI

--L-- sin'

N LN→ asHeera .net.in ÷. :i÷:÷÷÷:i÷isj

Heisenberg XXX lnozemtsev Haldane- Shastry471N )2

potential VHCiijt-odci.js.ie c- VICKI) lit) → VHS"'t )=sinzyza =Lnearest neighbour k→• CN.in/k)ElNzzxilRsoK30

dispersion EHCpl=4 sin-

E c- Efp ) - Elp ) - (51Mt- est - p'T→ EHsCp)=Zp( 217 - p)-

y 'tE Cznsink) s 't .5ypl= -Jlp) I}f ICN -I) p= I

HSHe is antiferro

HS spectrum is

highly regular. I "

ni;¥e: rateHaldane '88

ferromagn

6/75

Isotropic level spectrum : M=2 Inozemtsev'

so--L- Klabbers TL -20

p sin.

Heisenberg XXX lnozemtsev.tn. . Haldane- ShastryTinh

method (e.g. coord ) Bethe ansate extended CBA connection to trig Cal -sat

ansatz for ftp.pzeilkhitkk ) ← Ip, ,pz(n , ,n, )eikmtkh2)

#Chainz ) + ftp.pzeicpihztpzh; ) on shell → yI(Wh;wh2 ) WEE""N

- ftp.pzchn.nz/eilhh2tkhn ) on shell qCOO rd off's 9' '[

accounts plane 't nparamscfm.fm Pm- rm symmetric polynomialspace of for (contact) wave doubly quasi periodic deg < N in each argumentsolutions interactions simple poles at equal args =o at equal arguments

if identify Fm=Xlpm) , fanfiction~V

Schrodinger ftp..pe Ip, ,pdx, ,xz) must be eigenfn of Ice"",ei×2 ) must be eigehfh Of

eqh fixes fp=e"

In ~*,

Laine ~* µ,µpOf

Hees- - -212×9+2×22 ) -1284. - Xa) → Ht = - Its.tk/t2sIqxzTwhich are known (Hermite) which are known f- Gegenbauer)

as well asn

energy Eµ=EHCPnltEµCPd,EI=EICpnltEICPz) - Uhh , Pz) → EHs=EHsCPnItEHsCPz)

RE- ftp.hp:)

recall EHCP )=4 sin-

Iz C- EICP ) - g-Cpl - (5lplt-cst.PT → Ehs (p)=zIp( 217 - p)additivity functional '

quasi' strict

periodicity values of Pn ,Pz : Bethe values of Pn ,pz : Bethe eqs nothing - periodicity is built-in

fixes egsPETTI , -1 ftp..pe) PF I ,-1¥ p, YIM,{Pz=2Iz- ftp.pz )

← { pz=7fIz- ¥ 4 91=-92 ' ' ' ' ' ' {pz=2fgµzquantum hrs Bethe integers Bethe integers motif 7/75

Isotropic level spectrum : M=2 dictionary of States klabbers 7.L'20--t---

Heisenberg XXX lnozemtsev Haldane - Shastryenergy EH=EHCPnltEHCPa)

,EI=EICpnltEICPz) - UTR ,Pz)→ EHS -_ Etiscpnltetiscpz)

dispersion EHCP )=4 sin-

Iz c- EICP ) - g-Cpl - (5lplt-cst.pl/--EHsCpl=zIpC2n - p)PETTI , -101Pa ,pz) Pn= I ,-1¥ p, YIM,{pz=2Iz- ftp.pz )

← { pz= Iz- I ' ' ' ' ' ' ' ' ' {pz=7Iµzclassics - ) 0=IyEIzfN - I 0=IyEIzfN - I µ=0 empty , M=CI ) , IEIEN- Isee - desc pn=o , pz= Is , 0=0 = pn=O , Pz= Is , 4=0 = py=pz=o p,=o, pz=2NIIse , - hw

EH=EHCP2 ) ← EI=EICPz ) -→ Et,s= Etiscpz )

class II 1StMEN -1, Iz > Intl nsImfN - 1, Iz> Intl TEMMFN - 1, Nz>Matt

Pm= Im,a real ← Pma Im , cereal -→ pm=FfMm

Of

→ No → o

scattering scattering Yangianhwclass III IIIMEN -1, Iz=In ,Intl nsImfN - 1, Iz=I , ,Iit7 M=(I ) , ZEIEN- 2

Rep.m= III , -0 cpx F. pm cpx , 9 cpx gift,

PEO , PETTI I=IntIzbound state affine descendent§,=EEHCPntPz) EHs= Etiscpz )

8/75

Isotropic level spectrum : hidden symmetry of HSNaint

--L---- H = E -14 - Pij )Hs

- sins# Ci-JDisotropy enhanced to Yang iah symmetry ( Sd, H 3=105,H 7=0

"tHS HS

N

Mz - Huda : SIZEof [ sgsfs ]=zi Post 5- =S×±it (x x. Y, z) Z

n

ysez : OYE. cotfzci-iDEMofg.ES?QI=ziEeYaffInQt- I Ha et al

'

ez(co - dim) d & Serre - like Bernard etat

'

93

allows us to organise N=@2)④"as

ITT . . - T) [ either-

ftp..ly by homogeneity#t = M mod -

n

°a did

^ : hi y y⑨onion onion on , one cnn.sn,Stl IS

-

2 % : x xx x⑧a , a 0%

i :*:÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷:÷÷:÷÷÷÷÷÷÷:÷ : anti.noSZ -_ Nz - M N -y g g l l ra.gs, G.3.si ""

µ £ I 4,3) (1,41 G.S) ( 2,4) (2,5) (3,5)

& Gl Cz) ( 3) (4) ( s )( 1,413,5)

o( 2,47 Ciel (2,5)

so it remains to find"""

G.3) (3,5) . Cys)

Yslz- highest weight vectors "" "s) is ,

Giles 951(21141

at 1472 ↳(Y!s,"

mist

labelled by motifs. 4,2)

'EMMEN-IMmt , >Mmt l alls,

FerroMo o

He is HS9/75

Isotropic level . algebraic structure of HS Bernard et al '93--2--- talstra Haldane '95

N

dj= axjt Eat th - si;) HI = -zjEpijtjittj) 4

- I gnli-f) sij trig spin - Cal-Sat model a

-

# • abelian symm : Pcs , tttcs . . . . IDunkl operators N bosons

"

→

degenerate with spin z• hohabeliansymm : Yglz

affine Hecke alg{ freezing k→- (semiclass lim) y

L N bosons = j (class equilib) ✓

Haldane - Shastry spin chain

trig cat - suit model , ✓• abelian symm : G

, Hus , . . .• abelian symm : Pcs ,Htcs , . . . • honabeliansymm : Yglz of of d t• exact eigenfunctions : Jacks- • exact eigenvectors

NNEM

~ * N

PCE - ifpxj k=2H = [ a- Pij )

µ µ9 x*m= nm

Hs isjsihfffci-JDKCK- 1)-HI = -Effi.t§xxHj t

it Ilm,. . . ,nm)= Cwm

,. . .,wM) , WEE

"""

→ K : reduced

€÷h coupling Infzi , . . . ,zm ) = .nl/m!zm-Zmt2xPfTffzn.....Zm)-Jastrow Jack polynomial(Slater det) (zonal spherical )

10/75

Isotropic level spectrum general descriptions--t-:-p

sin.

Heisenberg XXX '28 lnozemtsev '90 lynn Haldane 88 - Shastry '88Tinh

method (e.g. coord ) Bethe ansate extended CBA connection to trig Cal -sat→ Conn to ell Cal -Sat

ayins .at?nfn9rEAp.neiPin" nfsm-Ip.cn.ie"-Ft "" Ecw"; . . . ,wnm )

-"ESM '

f'[ Plwanaeves 't nparamscem.fm 't

symmetric polynomial WEE"""

cord 0ft's account doubly quasi periodicfor (contact) simple poles at equal arg,

deg LN in each argumentinteractions =o at equal arguments

if identify pm=XCpm) , rapidity✓ function

Schrodinger ftp.n as fu of p Ifk, , . . . ,xm) must be eigenfn of Ice"",. . . ,ei×m) must be eigehfh Of of

eqh fixes (upto normalisation) * M* y 2 a 2 (NIV)

'

Fees- -zmE.pxmt2.in?Emplxm-xmt-tTtcs=--mEpxmt2E sixmnTHMsm'

which are known (complicated) which are known (simple : -Jack)

periodicity values of Pn , . . 's Pm : values of Pn , . . -spm : nothing - periodicity is built infixes Bethe ansatzegs Bethe ansatzegs

Pm= Im -h§OCpm , pm .) Pm=FYImtf9m IEIMEN - I Pm= Mm motif ismmslv-I⇐m) Im+ ,> Im Mmti >Mmt '

dispersion EH 1=4 sin-

E c- Etnocp ) - g-Cpl - (51Mt- est -pf → EHSCP)=Ep( 217 - p)M

energy EH EHCPM) c- EI=§=,EICpm) - VT → EHs=EEHsCPm)'I m=L

functionally additive '

quasi additive'E- TERM

strictly additive17/15

Isotropic level : summary---

consider N sites with spina. &f!%¥Iqtq&q!I "

H " in

⇒is

Heisenberg XXX 're lnotemtsev '90 Haldane '88 - Shastry -88

potential VHCiijl-odci.jg.ie c- Vztiij ) lit) → Vtiscicj )= =Lsin'#Ci-JD r2

nearest neighbour k→• (N, ink)ElNzzxilR>o K→0

exact up to solving up to solving → in closed form

solubility Bethe equations←

Bethe equationsvia connection to Haldane 'M

(spectrum) Bethe '31 via connection totrig Calogero -Sutherland

ell Calogero -Sutherland

Haig Bethe ansate Inozemtsev'

go ,'95,100 It rfofnrselfmzimngtheotb

Klabbers 7.L'20

quantum Yangian structure degenerate affine Hecke algintegrals Faddeevetal , late '7os-805

unknown Yahgian symmetryBernard et al '93

I transfer matrix µcentre

V t freezinghigher known conjectured Bernard et al '93Ham's partial proof known

talstra Haldane 'es

Dittrich Inozemtsev'08 12/75

Partially isotropic level'- --

"

partially isotropic csz,HI=o :I!µ!÷.!&&i!÷I 'H= [ Vlt , 't) Sci .jo not quite homogeneousisjsaint:I¥

.

it.

Heisenberg XXZ ← unknown → Uglov '

es - JL 're

Orbach 'so

potential VHeislisjl-odci.jg.ie conjecture V (icj )= -1Klabbers 7.L'20

Uh rt knearest neighbour

exact up to solving unknownin closed form Thet al '20

Solubility Bethe equations via connection to

(spectrum) Yang -Yang'66 trig Ruijsenaars-Macdonald

µ nonsymm theoryItaly Bethe ansate t freezing

quantum quantum - loop structureunknown

affine Hecke algintegrals Faddeev et al , late '7os quantum - loop symmetry

Bernard et al '93I transfer matrix / centrev y t freezing

higher knownHam's

unknown known TL et al'

20

73/75

Outlookandyscaieofngrang Pinchas qq.gg?jj4,.&.Jtg.K-→interaction range

nearest neighbour intermediate long rangef degree of contact potential range

trig potentialspin symmetry elliptic potential

? 7anisotropic Heisenberg XYZ c- .

→ .

elliptic quantum alg Sutherland '70

Baxter '

73 µ It

partially g -deformed HSHeisenberg XXZ ← ? →c.uglov -Lamers )isotropic Orbach '58

trig quantum alg : Yang-Yang '

66 Uglov'

95 , TL'

18

quantum loop alg , f IL Pasquier Serban'20

affine Hecke alg t t

isotropic Heisenberg XXX c- lnozemtsev → Haldane- Shastryrational quant alg : Heisenberg 128 lnozemtsev '90 Haldane ' 88

, Shastry' 88

Yang ian, Bethe '31 lnotemtsev '90 -

'

00,

Haldane '91 , Bernard et al '

93

degenerate AHA klabbers TL '20

74/75

outlook-andopeofquan-tummany-bdysystemsq.IQ-→interaction range

nearest neighbour intermediate long rangecontact ( positions) range

elliptic (positions) trig (positions)

C ? ) 2 C-'DELL

' → ?elliptic (momenta)

'

t t trelativistic 2 C- ell Ruijsenaars → trig Ruijsenaars-

. Macdonaldtrig (momenta)affine Hecke alg J J Jnon- rft n Lieb-Li niger ? c- ell Cal - Sort → trig Cal - suitrational (momenta)

degenerate AHAtowards grand unified theory for

quantum- integrable long - range spin chains ?15/75