Genralised Floquet Time Crystal Systems I. INTRODUCTION

Transcription

1 Gerlsed Floquet Tme Crystl ystems Gutm Kmlr N Physcs Deprtmet, Id Isttute of Techology (Brs Hdu Uversty), Vrs - 005, Id Ktsur Lbortory, Deprtmet of Physcs, Uversty of Toyo, Hogo, Buyo-u, Toyo , Jp (Dted: 9 August 07) Tme trsltol symmetry breg s possblty eplored oly the recet pst. I ths report, we frst revew pper by Else, Buer d Ny whch defes the oto of tme trsltol symmetry breg qutum systems d gves the lytcl d umercl study of sp ½ model tht shows ths behvor. ecodly we eted ths de to lytclly d umerclly show tht t s possble to hve tme trsltol symmetry breg hgher sp systems. I. INTRODUCTION A. poteous ymmetry Breg (B) poteous symmetry breg s process where symmetrc stte of system spoteously chges to symmetrc stte. Ths occurs whe the uderlyg lws of system obeys cert symmetry but the groud sttes or the low eergy sttes of the system do ot obey the sme symmetry. The smplest emple for ths pheomeo s tht of crystls. A volume of gs uform potetl hs cotuous trsltol symmetry. If ths system cools dow d suppose the gs ow crystlles, the, the system o more hs cotuous trsltol symmetry but ow hs dscrete trsltol symmetry. Cotuous trsltol symmetry s sd to be broe ths cse. Aother emple s tht of the Isg model. The Isg Hmlto (for ch of sp ½ prtcles) H Isg J s symmetrc uder the smulteous flp of ll the sps of the system. I the ferromgetc cse ( J 0 ), there re low eergy sttes wth o-ero mgetto. Uder the cto of the symmetry (.e. uder sp-flp), we obt other low eergy sttes whch hve opposte mgetto. For fte chs, symmetrc d tsymmetrc superpostos of such sttes re the egesttes. These superpostos, lso ow s ct sttes, cot be dstgushed by y locl mesuremets d re etremely sestve to eterl felds. Eve ftesml felds cuse the system to decohere d gve stte wth mgetto log the eterl feld. B ths system my be defed terms of the low eergy egesttes beg ct sttes or to the developmet of mgetto o pplcto of ftesml eterl feld. There re my other emples of B prtcle physcs d codesed mtter physcs - guge symmetry breg, electromgetc guge symmetry breg, chrl symmetry breg d Euclde symmetry breg to me few. My emples we hve for B seems to suggest tht for every symmetry we c th of, we c come up wth system whch bres tht symmetry. I ths report we re terested tme trsltol symmetry breg. B. Tme Trsltol ymmetry Breg (TTB) The cocept of TTB ws frst descrbed by Frc Wlce 0 [] [3]. ystems whch bre tme trsltol symmetry hve bee clled Tme Crystls logy wth ordry crystls whch bre cotuous trsltol symmetry. The defto of TTB s ot strghtforwrd. As dscussed by Else, Buer d Ny (EBN) [], the most obvous defto d ts modfctos, ll bsed o the epectto vlues d correlto

2 fuctos of the therml equlbrum stte e H must be ruled out due to o-go theorems [4]. These theorems show tht t s mpossble to hve TTB equlbrum sttes d hece suggest the eed to eplore the possblty of hvg TTB o-equlbrum sttes. Eve the cse of symmetres other th tht of tme trslto, whe we cosder spoteously broe symmetres, we loo t oequlbrum systems whch show ergodcty breg d hve lfe tmes whch dverge wth the system se. It should be possble to hve systems whch show logous behvor for TTB. Tg these to cosderto the EBN pper gves two equvlet deftos of TTB. Cosderg Hmlto H ( t ) perodc wth perod T wth ts correspodg utry s U ( t, t ) for tme evoluto from tme t to tme t, TTB s defed s: TTB-: TTB occurs f for ech t, d for every stte ( t) wth short-rged correltos, there ests opertor such tht ( t T ) ( t T) ( t ) ( t), where ( t T ) U ( t T, t ) ( t ). TTB-: TTB occurs f the egesttes of the Floquet opertor U U( T, 0) cot be short-rge correlted. Here, stte s sd to hve short rge correltos f, for y locl opertor ( ), ( ) ( ) ( ) ( ) 0 s,.e. f cluster decomposto holds. Notce tht the Hmlto cosdered hs dscrete tme trsltol symmetry d ot cotuous tme trsltol symmetry. f II. MODEL OF TTB A. p ½ system The EBN pper gves my body locled system of oe dmesol sp ½ prtcles wth Floquet utry U ept H ep ept H f 0 j 0 j j j whch hs perod T t0. The utry hs two Hmltos ppled oe fter the other, the frst beg the whch whe ppled for perod effectvely flps ll the sps the ch bout j j the -drecto. The secod s my body locled Hmlto H whch gves Isg tercto coupled wth eterl mgetc feld log the -drecto wth perturbto cosdered log the - drecto,.e. 3 H J h h where J, h d h re uformly chose from J,, h 0, d h 0, h. Oly slghtly perturbed systems wth smll vlues of h re cosdered.

3 For o perturbto (.e. h 0 ), ths system c be lytclly solved to show tht TTB occurs. uppose we cosder tht {s } wth s re the egesttes of the opertors σ so tht { s} s { s}. As H commutes wth whe h 0, sttes {s } re lso egesttes of H wth the egevlues s H { s} E { s} E { s} { s} where we defe E { s } J s s d { } E s h s. Wth these ottos the egesttes of the Floquet opertor c be wrtte s: s ep t0e { s} { s} ep t0e { s} { s} The correspodg Floquet egevlues re ep { } t0e s. Notce tht the egesttes stsfy TTB-. To see tht TTB- s stsfed s well, we study the vrto of epectto vlues of wth tme: d U ep j j j j Let U ep t0 J h Cosderg s the tl stte, we hve U U U U UU tt f f t0, so tht U f UU. s U, 0 d, 0 for s, 0 Ths shows tht the epectto vlues of show osclltory behvor re perodc wth perod T hece verfyg tht TTB- s stsfed. The EBN pper goes o to rgue tht the bove behvor s see eve whe we perturbtos to the Floquet utry such s smll devtos the legth of pplcto of the flppg Hmlto from π d o-ero vlues of h do ot destroy TTB. The behvor of the system uder o ero vlues of h c be studed by smultg the systems. Ths system s smulted by usg tme-evolvg bloc decmto scheme (TEBD) [6] whch s dscussed the pped. Fg.. shows the dsorder verged epectto vlues of d over 00 dsorders for system se of L=0, h=0.3 d t 0 wth the sp flps doe stteously. The TEBD clcultos were doe wth Trotter step of 0.T d bod dmeso of 6. The lfetmes of the TTB observed the perturbed systems c be studed by smultg the system usg ect dgolto (ED). We defe mgetto Z(t) bsolute vlue of t. Fg.b. shows the vrto of the dsorder verged mgetto over 500 dsorders wth tme for system wth

4 h=0.3, t 0 d tl product stte polred log the -drecto for smll system ses. We c see tht the lfetmes dverge epoetlly wth the system ses. Fg.. Tme depedece of dsorder verged d for system of 0 sp ½ stes shows tht the osclltos of the former perssts whle tht of the ltter des dow N=4 N=5 N=6 N= Z(t) ^ 0^ 0^3 0^4 0^5 0^6 0^7 0^8 t/t Fg.b. The decy of dsorder verged mgetto Z(t) sp ½ system wth tme. I ths sp ½ system, we observed tht the epectto vlue of hd perod twce tht of the Hmlto. I ttempt to observe TTB wth the epectto vlue of some opertor hvg perod three tmes tht of the Hmlto, we study sp system.

5 B. p system We cosder oe-dmesol sp system wth Floquet opertor smlr to the oe cosdered the sp ½ system. We use the followg otto for the opertors. The mtr form of the opertors re epressed by cosderg the stdrd bss ep The Floquet utry s 0 0 ep where e 3 U ept H ep ept H f Alogous to the p ½ cse, the Floquet utry hs two Hmltos ppled oe fter the other wth totl perod of T t 0 3. Frst the Hmlto ppled for perod 3 cycles every sp the ch betwee the sttes wth mgetto log the -drecto beg, 0 d - (referred to s cyclg of sps). Ths s wy the log of sp flp sp ½ systems. Ths s followed by the Hmlto H gve by: H J h h where J, h d h re uformly chose from,, d J h 0, h 0, h 3. The frst prt of the Hmlto s the tercto term d the secod prt s tht of the eterl mgetc feld log the -drecto wth perturbto log the -drecto. Ths specfc tercto term s chose out of the my possbltes of the tercto terms sp systems becuse ths tercto term s vrt uder the cyclg of sps whch s vtl for the d of behvor we wsh to hve the system. Just s the sp ½ cse, ths system c be solved lytclly for o perturbtos.e. h=0 d t c be proved tht TTB occurs. Cosder the egesttes of s { } s,0, such tht s wth. For h=0, we hve H { s } E { s } E { s } { s } { s } s { s } where s s { } Re d E { s} h s E s J. We defe the cyclg opertor wth

6 the followg set of equtos: 0, 0, d s s for d. Usg the otto {,,3} d E s E { s } E { s} 3, the Floquet egesttes re s ep t E { s} { s} ep t E { s } { s } ep t E { s } { s } wth the egevlues ep { } epectto vlues of t0e s. These egesttes stsfy TTB-. The vrto of wth tme c be foud s follows: Let U ep t0 J U f d U U U. Cosderg s the tl stte, we hve U U U U UU tt f f t 0 ep 3, so tht s U, 0 d, 0 for s ce, we get the result 3 whch shows tht the epectto vlues of Hece TTB- s stsfed for ths system. Im 3 tt t 0 re perodc wth perod thrce tht of the Hmlto. Whe perturbtos to the system re cosdered, the behvor of the system my be studed by smultos. The system s smulted by usg TEBD. Fg.. shows the dsorder verged epectto vlues of d over 67 dsorders for system se of L=50, h=0.3 d t 0 wth the cyclg of sps doe stteously. The TEBD clcultos were doe wth Trotter step of 0.T d bod dmeso of 0. We see tht the TTB behvor s stll see the system. To study the lfetme of TTB the system, we smulte t usg ED. Here we defe mgetto Z(t) s the mmum vlue of the bsolute vlue of t the tme tervl t to t+t. Fg.b. shows the vrto of the dsorder verged mgetto over 500 dsorders, wth tme for system wth h=0.3, t 0 d tl product stte polred log the -drecto for smll system ses. Notce tht just s the cse of the sp ½ systems, the lfetmes dverge epoetlly wth the system se but the decy of mgetto s ot s steep.

7 Fg.. Tme depedece of dsorder verged d for system of 50 sp stes N=4 N=5 N=6 N=7 Z(t) ^ 0^ 0^3 0^4 0^5 0^6 0^7 0^8 t/t Fg.b. The decy of the dsorder verged mgetto Z(t) sp system wth tme.

8 C. p system The TTB model of sp system proposed c be turlly eteded to sp systems. We use the followg ottos ( ) ( ) d where d d e d The Floquet utry d the my body loclsed Hmlto cosdered re smlr structure to tht cosdered the sp cse. U ep t H f 0 where H J h h To show tht the system s fct model of TTB, we fd the vrto of epectto vlue of tme., b d ( b ) b d d U f U f U U UU t T s t0 wth s U, 0 d, 0 for Hece we hve d b tt d, b ( ) b to

9 Here g we c see tht the epectto vlues of re perodc (by substtutg =d) wth perod d tmes the perod of the Hmlto hece stsfyg TTB-. Fg.. shows the epectto vlues of d for system wth =3/ d = respectvely d hvg system se of L=50, h=0 d t 0 wth the cyclg of sps doe stteously. The TEBD clcultos were doe wth Trotter step of 0.T d bod dmeso of 4 d 5 respectvely. Fg.3. Tme depedece of d (bottom pel) stes wth o perturbtos. for system of 50 sp 3 (top pel) d sp

10 III. ACKNOWLEDGEMENT Frst d foremost I would le to th Professor Hosho Ktsur for gvg me the opportuty to wor hs lb, for hs ptet gudce d hs vluble sght durg dscussos. I would lso le to th hot Tmur for hs support d ssstce wth the project, Yut Ag, Nobuyu Yosho d the other members of the lb for ther hosptlty, fellow UTRIP studet Ze Ross for the my frutful dscussos d for shrg the project, d Tsh Mor for hs commets d suggestos regrdg the project. I would lso le to eted my grttude to Uversty of Toyo's Grdute chool of cece for ther geerous scholrshp d the ILO for orgg d helpg me throughout the progrm. Apped: Tme Evolvg Bloc Decmto (TEBD) TEBD s lgorthm used to effcetly smulte the tme evoluto of qutum systems whch hve low log rge etglemet throughout the perod for whch t s evolved. Geerlly low eergy sttes of qutum systems follow re lw of etglemet etropy (.e. the etglemet etropy scles lerly wth the se of boudry of the system) [7] whle hgher eergy sttes follow the volume lw of etglemet. But my body loclsed systems, eve hgher eergy sttes follow re lw of etglemet whch mples tht ll sttes my body loclsed system hve oly short rge correltos. Ths llows for my body loclsed systems to be effcetly smulted by TEBD. Vdl hs gve comprehesve method of mplemetg TEBD [6]. The tl stte epressed the mtr product stte (MP) form s:......,...,,..., [] [] [] [] [ ], where {,,..., } d... represets the sp orthoorml bss of sp system. For short-rge correlted sttes, the compoets of vectors fll epoetlly [5]. Eplotg ths fct c led to sgfct decrese the storge spce d provde speed up the clcultos requred to smulte these systems. The umber of sgfct vlues of cosdered s te to be. For smultg the system wth ero error, the vlue of tht eeds to be cosdered s the chmdt r of the system. The evoluto of the system hvg Hmltos wth sgle sp tercto or eghborg sp terctos s reltvely smple ths method. gle sp terctos re crred out by updtg just the pproprte tesor. Cosderg U s the utry correspodg to the Hmlto ctg o the lth [ l ] [ l ] sp, the updted tesor s gve by. The computtol cost for the opertos volved s Ø( ) bsc opertos. U j The evoluto of the system wth eghborg sp terctos c be crred out by updtg the correspodg two tesors d the correspodg vector. The computtol cost for crryg out ll 3 the clcultos volved s Ø( ).

11 Hece usg TEBD, computto whose complety would hve scled epoetlly wth the umber of sps f doe covetolly, c be doe wth complety of polyoml of whch tur grows s polyoml of the umber of sps f there s low log rge etglemet. Refereces [] Else, D. V., Buer, B. & Ny, C. Floquet tme crystls. Phys. Rev. Lett. 7, (06). [] F. Wlce, Phys. Rev. Lett. 09, 6040 (0). [3] A. hpere d F. Wlce, Phys. Rev. Lett. 09, 6040 (0). [4] H. Wtbe d M. Oshw, Phys. Rev. Lett. 4, 5603 (05). [5] G. Vdl, Physcl Revew Letters 93, (004). [6] G. Vdl, Phys. Rev. Lett. 9, 4790 (003). [7] Esert, J., M. Crmer, d M. B. Pleo, Rev. Mod. Phys. 8, 77 (00).