A String Theory Approach to the Three-dimensional Ising Model
| dc.contributor.advisor | Mojurnder, M.A . (PhD) | |
| dc.contributor.author | Tadesse, Mesfin | |
| dc.date.accessioned | 2022-03-24T06:01:20Z | |
| dc.date.accessioned | 2023-11-09T11:27:13Z | |
| dc.date.available | 2022-03-24T06:01:20Z | |
| dc.date.available | 2023-11-09T11:27:13Z | |
| dc.date.issued | 1989-06 | |
| dc.description.abstract | Th<? t hree-d i mens io nal Ising mod p.l (3D1M) ha s been one or the un solved problems i n physic s . Polyak ov has s uggested lhaL it may be pos s ible to solve this model via the st ri ng th'!Ol'Y meth ods . The 3D 1M in the critical regime can be ass u~d t ~ correspond to the ferm i oni c s t ring which >.pans a 20 world-sheet. The 3D 1M a lso co rr esponds to the conventional , .2 a~d . 4 La gr an gia field t heori es . It can be shown that t he scaling di~ens i o ns of these two and three-d i mensi onal fie l ds are proportional. The co ns t an t of proportionalttz is he uristically ide nt ified wi th the Ha usdorff dimens 'ion o f the world -sheet . Computer cal cu la tio" of th e Hausdorff di mension and t he scal i ng dimensions of the 3D conven tio nal field , the resu l ts of which are available tn the literature , enable numer i cal evaluation of the critical expo nent~ " and B 0'" t he · Ising mode l. These agree well with ex peri me ntal results . .The l si ng mod e l f irst appaa r ~d i n the t hesi s of Ernst I si ng i n 1925. Thi s mode l was. of cour se . one di mens i ona l and. to 1 5 in 9 '~ di sap poi ntment, di d no t e xh ib i t a phas e t r ansiti on. The t wo and t hr ee-d i me nsional Is ing mode l s are i ndeed only t wo of s t at i stica l mo de l s wh ich undergo phase trans it i ons •. The two-dimens i onal Is i ng model was first consi der ed by ,"" Pei erl s i n 1936. In 1941. Kramers and . Wanni-er de t ermined the critical t emperature (the t r ans i tion point) by assum ing i ts existe nce . The year 1944 cou ld be thought of as the beg i rt ~ n g of a new era in the s tu dy of the phys i cs of phase tran s i t i ons. -- .... 1n t hi s year, L.Ons ager pre s en t ed a paper i n which i t has been sh own r ig oro us ly thdt the two - dimens i onal ISing medel 8)!hibit ed a phase t ra ns ition. The pa r t i tion funct i on f or the two-dimens i onal Is ing model was oota i ned by co nve rt ing t he model into a system of fr ee ~', a jorana f ermi ons . This shows tha t t he two-dimen s ional 1's'; ng mode l is equivalent t o a free quantum field theory of fe rm i /) i'iS, 1 , 2 ,.,e t hods . 3 The model is also treated by usi ng combinatorial The three-dmensional Ising madel, ' (301M) still rema i ns one of the problems in phySics for which no exact solutio ns have been pos sib le yet. Even if the partition func t ion ha s ' not been found the critica l exponen t s have been determined by t he ser i es and r eno r mal is a t ion group methods . The values of J the c r itica l ~xpon e nts obtained i n th i s manner have beenc om pared witt, those obtained experimentally . Recent develop- , mellts :n the string theory i,a ve l ed t o the expectation tha t the 3DIM might be ~oJ va b le in the not too distant f ut ure ty t ~chntij~es t hat are bei ng developed cu;rently. It, hDwe~er . a pp ear~ th at the current leuel of development has not yet reached the stage where one could bPply t he st ri ng tl,eory to calculate t he partition function or even the critica l expone nts . Ho wever, as we shall see . by a combination of the ideas of the , string t heory with the results of renormalisation group stud i es one could predict some of the critical exponents. This paper is organized as follows: Chapter 1 i s a r eview of the 301M and its possible representation by the string theory. Of course. here. we made use of analogy to the two-dimensional Ising model in deriving the partition function. Polyakov' s interpretation is menti oned. Chapter Ii deal s with thf bQsonic string, Also, in this chapter. the s upersymme trised form of the string~that is to say the fermloni c str i ng- is considered. Chapter III is a discussion of the attempts at the calculati on of the' values of the critical exponent. In ChapterIV we have a short discussion of the a,pplication tJ:ki ~ the r enormalization group t~ the Ising model. A heuristic application of the ideas of the string model is made to some result of the re norma lization group for caltulattng the numerical values of the critical exponents. This is done in Chapter V. Finally, we discuss ' briefly the results and the fut ~re ' possib'litte~. | en_US |
| dc.identifier.uri | http://10.90.10.223:4000/handle/123456789/30787 | |
| dc.language.iso | en | en_US |
| dc.publisher | Addis Ababa University | en_US |
| dc.subject | String Theory Approach | en_US |
| dc.title | A String Theory Approach to the Three-dimensional Ising Model | en_US |
| dc.type | Thesis | en_US |