Quantum Annealing for Heisenberg Spin Chains

Transcription

1 LA-UR # - Quatum Aealig for Heiseberg Spi Chais G.P. Berma, V.N. Gorshkov,, ad V.I.Tsifriovich Theoretical Divisio, Los Alamos Natioal Laboratory, Los Alamos, NM Istitute of Physics, Natioal Academy of Scieces, Kiev, Ukraie Physics Departmet, Polytechic Uiversity, Brookly, New York, Abstract We suggest usig the method of quatum aealig for computig the groud state of the Heiseberg spi chais. Our iitial Hamiltoia describes a spi system i a highly ouiform magetic field. The iitial Hamiltoia gradually trasforms ito the Heiseberg Hamiltoia, which describes the exchage iteractio ad the Zeema iteractio with the uiform magetic field. We demostrate quatum aealig for -, -, ad -spi systems. I particular, we cosider the alterate (ferro- ad atiferromagetic) exchage itegral. By chagig the sig of the exchage itegrals we switch from the frustrated to the correspodig o-frustrated spi system. We discuss the similarity ad differece betwee the frustrated ad o-frustrated spi systems.. Itroductio Recetly, the quatum aealig method has bee widely used to compute the groud state of Isig spi chais []. Quatum aealig assumes that the iitial Hamiltoia of the system has a simple groud state ad does ot commute with the Isig Hamiltoia. The iitial Hamiltoia smoothly (adiabatically) trasforms ito the Isig Hamiltoia. I this case a simple iitial groud state smoothly trasforms ito the complicated groud state of the actual Isig Hamiltoia. Typically, the iitial Hamiltoia describes the iteractio of the spi system with the trasverse magetic field. The Isig spi chai has umerous applicatios to mathematically itractable problems. From the other side the Heiseberg spi chais may be used to create atomic-scale magetic structures, i particular, for the data storage devices. Heiseberg spi chais are actively studied ow for applicatios i aotechology []. Cosequetly, we cosider the applicatio of the quatum aealig to the Heiseberg spi chai. The uiform trasverse magetic field caot be used ow because the isotropic Heiseberg Hamiltoia iteracts with ay compoet of the total spi. Thus, we have to cosider iteractio with ouiform magetic field, which does ot commute with the Heiseberg Hamiltoia.

2 . Two-spi Heiseberg system I order to uderstad the properties of the quatum aealig computatio we start with a trivial two-spi Heiseberg system. We cosider the followig timedepedet Hamiltoia of the system: ˆ () ( z z t ) ( x x H t = JSS + γ B S + S + γ B S S )( t / τ ), () τ where < t < τ, J is the exchage iteractio costat, γ is the absolute value of the electro gyromagetic ratio, BB is the uiform magetic field which poits i the positive z - directio, B is the magitude of the o-uiform magetic field which poits i the positive x - directio at the locatio of the first spi ad poits i the egative x - directio at the locatio of the secod spi. A o-uiform magetic field is itroduced i order to fid the groud state of the Heiseberg system usig the quatum aealig approach. At time t = the Hamiltoia () describes two idepedet spis iteractig with the o-uiform magetic field. Thus, the eigestate of the Hamiltoia () at t = represets the first spi poitig i the egative x -directio ad the secod spi poitig i the positive x -directio. We choose this state as the iitial state for the quatum aealig. At t = τ the Hamiltoia () describes the coupled Heiseberg spis iteractig with the uiform magetic field BB, which poits i the positive z - directio. If τ is large eough oe ca expect that the groud state of the iitial Hamiltoia Ĥ(t = ) will trasfer smoothly to the groud state of the fial Hamiltoia Ĥ(t = τ ). Below we describe our computer simulatios for the twospi system. We use BB as the uit of the magetic field, γb B as the uit of frequecy, γħbb as the uit of eergy, ad /(γħb B ) as the uit of time. I these uits we have chose the followig parameters of our Hamiltoia: J =, B =, ad τ =. () Positive value of J correspods to the simplest ferromagetic groud state. We take z the eigestates of the operators, S k, as the basis states ad represet the wave fuctio as a superpositio of all basis states with the amplitudes C. The iitial values of the amplitudes are C = - ½, C = - ½, C = ½, C = ½. () Here ad below we assume that for a sigle spi state represets the dow directio, ad state represets the up directio. For two or more spis we use the same rules to trasfer from the biary to the decimal otatio. Thus, for the two

3 spis state represets, state represets, state represets, ad state represets. Note that we cout spis from the left to the right. For example, the state idicates that the first spi poits dow while the secod spi poits up. Fig. a shows evolutio of the probabilities p = C. Oe ca see that the probability p gradually approaches to uity, while all the other probabilities approach to zero. Cosequetly, the spi system approaches to the ferromagetic groud state with spi poitig i the egative z - directio. Fig. a shows probabilities p for the atiferomagetic exchage iteractio J =. I this case probabilities p ad p approach the value. while the two other probabilities vaish. This reflects evolutio to the atiferromagetic state S =, which is a superpositio of the two states ad with equal probabilities ad phase differece π. (Phase is ot show i Figs. a ad a.) I Figs. b ad b, we show also the chage of the eergy levels for J = ad J =, correspodigly. These graphs do ot have a direct coectio to the quatum aealig. They are preseted to better uderstad the spi systems uder cosideratio.. The Heiseberg system of three ad more spis For three or more spis we use a Hamiltoia similar to () ˆ () k t ( / ) ( ) k+ Ht = JkmSS k m + γ B Sz + γ B tτ S x k, () km, k τ k At time t = the spis poit i the directio of their local magetic field (i.e. i the positive x - directio for eve k, ad the egative x - directio for odd k ). We assume a closed chai of spis. If the product of the exchage itegrals is a odd umber the the spi system is frustrated. Fig. a demostrates quatum aealig to the ferromagetic groud state for the - spi system with J km =. Fig. b shows the evolutio of the eergy levels for the same system. Note that the probability p does ot icrease mootoically as it did for the -spi system. Fig. a shows quatum aealig for the frustrated -spi system: J = J =, J = -, ad Fig. b shows the chage of the eergy levels. I this case, the groud state is a superpositio of the states = ad = with equal probabilities ad phase shift π. Oe ca see that the probabilities p ad p mootoically icrease durig the process of quatum aealig. Surprisigly, quatum aealig for the frustrated system appears simpler tha for the ferromagetic system.

4 Next, we cosider a more complicated system of spis. The iitial eergy levels for t = are show i Fig.. All eergy levels except for the lowest ad the highest oes are degeerate. Fig. shows the fial eergy levels at t = τ for the case of the ferromagetic exchage iteractio, J km =. I the scale of the picture we observe a almost cotiuous eergy spectrum except for the highest levels. The groud state of the system is the ferromagetic state with the total spi S = / ad the z- z compoet of the total spi S = /. The first excited state is the same z ferromagetic state with S = / ad S = /. Correspodigly, the spacig betwee the groud ad the first excited state is exactly oe uit, which is ot visible i the scale of Fig.. Quatum aealig drives the spi system to the ferromagetic groud state. Now, we cosider a system of spis with alterate exchage itegral. This spi system is show i Fig.. The five double lies i the Fig. correspod to positive values of the exchage itegral J =, ad the four solid lies correspod to egative values, J =. The fial eergy spectrum is show i Fig.. The two lowest eergy levels have a Zeema spacig of oe uit, the same as for the ferromagetic case. I the scale of Fig. these two levels are represeted by a sigle dot. These two levels are separated by a gap of approximately. uits from the followig quasi-cotiuous spectrum. The secod gap appears ear the highest levels. The groud state obtaied by the quatum aealig is a complicated superpositio of the basis states. (See Fig..) The maximum probability correspods to the basis state : p.. Fig. demostrates the structure of the state : the circles at locatios,,, ad show the spis, which poit up. Fially we discuss the same spi system with the opposite sig of the exchage itegrals J km. This spi system is frustrated because the umber of egative exchage itegrals is odd. The eergy spectrum of the frustrated spi chai is very similar to that of the o-frustrated oe. (See Fig..) At the same time, the structure of the groud state, which is obtaied usig quatum aealig, is much more complicated. Fig. shows the values of the amplitudes C. Oe ca see that the groud state of the frustrated spi system has four domiat states istead of a sigle oe for the o-frustrated system. These domiat states are show i Fig.. The correspodig values of the amplitudes are: C C =., C = -., C =., C = -.. () Thus, we have the two pairs of the domiat states. The two states i a pair have approximately equal probabilities ad a phase shift of π.

5 . Coclusio We have demostrated the successful applicatio of the quatum aealig method to Heiseberg spi chais. I order to use quatum aealig we apply a highly o-uiform magetic field. The correspodig Hamiltoia does ot commute with both the Zeema ad the exchage terms of the basic Hamiltoia. First, we demostrated fast quatum aealig for the simple two- ad three-spi systems. The we cosidered a more complicated -spi system. We have show that the frustrated spi system has almost the same eergy spectrum as the correspodig o-frustrated system but the structure of the groud state for the two systems is sigificatly differet. The o-frustrated system has a sigle domiat state while the correspodig frustrated system has the two pairs of the domiat states. The states i each pair have approximately equal probability ad a phase shift of π. This work was carried out uder the auspices of the Natioal Nuclear Security Admiistratio of the U.S. Departmet of Eergy at Los Alamos Natioal Laboratory uder Cotract No. DE-AC-NA. Refereces. Arab Das, Bikas Chakrabarti (Eds), Quatum Aealig ad Other Optimizatio Methods, Lecture Notes i Physics, Publisher: Spriger Berli / Heidelberg, V. /.. Mark Wilso, Physics Today,, ().

6 . p a. = E (t) b , Fig.. a-probabilities of the basis states, which cotribute to the groud state of the -spi system with the ferromagetic exchage iteractio, as a fuctio of time i the process of the quatum aealig; b- the eergy levels of the spi system as a fuctio of time.. p a, E (t) b.... = Fig.. The same as i Fig. for the -spi system with the atiferromagetic exchage iteractio.

7 . p a.. = E (t) b Fig.. The same as i Fig. - for the three spi system with the ferromagetic exchage iteractio.. p a = - - E (t) b Fig.. The same as i Figs. - for the frustrated -spi system.

8 E (t=) Fig.. The eergy levels of the -spi system at the begiig of the quatum aealig. - - E (t =τ) Fig.. The eergy of the statioary states of the - spi system with the ferromagetic exchage iteractio E (t =τ) Fig.. The -spi chai with the alterate ferro- atiferromagetic couplig (solid lie - J =, dоuble lie - J = ) Fig.. The eergy of the statioary states of the ofrustrated -spi system with the alterate exchage iteractio.

9 . p... Fig.. Probabilities of the basis states, which cotribute to the groud state of the o-frustrated -spi system with the alterate exchage iteractio. Fig.. The structure of the basis state =, which makes the mai cotributio to the groud state. Circles idicate spis poitig up. E (t =τ). C (t =τ) Fig.. The eergy of the statioary states of the frustrated -spi system with the alterate exchage iteractio Fig.. Amplitudes of the basis states, which cotribute to the groud state of the frustrated -spi system with the alterate exchage iteractio.

10 = = = = Fig.. The structure of the basis states, which make the mai cotributio to the groud state. Circles idicate spis poitig up.