Nucl ear Physi cs B340 ( 1990) 617- 632 Nort h-Hol l and CROSSOVER EXPONENT FORA WEAK MAGNETI C FI ELD AT THE LOCALI ZATI ON ORTHOGONAL FI XED POI NT M . BI AFORE Depar t ment of Physi cs, Massachuset t s I nst i t ut e of Technol ogy, Cambr i dge, MA 02139, USA C. CASTELLANI * and G. KOTLI AR Depar t ment of Physi cs, Rot ger s Uni ver si t y, Pi scat away, NJ 08854, USA Recei ved 4 December 1989 ( Revi sed 22 Januar y 1990) We r e- exam i ne t he non- l i near cr - model l agr angi an f or t he sof t modes of non- i nt er act i ng el ect r ons ina di sor der ed syst em ina weak magnet i c f i el d, cl ose to t wo di mensi ons . The magnet i c f i el d is shown to i nduce t wo scal i ng t erms in t he l ow- ener gy l agr angi an . The cr ossover exponent s ar e comput ed by usi ng a W i l son- Pol yakov non- l i near r enor mal i zat i on gr oup . The l ar gest one agr ees t o l owest or der with t he exponent expect ed f r om nai ve scal i ng argument s . 1 . I nt r oduct i on The non- l i near t r - model has r ecent l y been devel oped as a power f ul t ool f or anal yzi ng t he Ander son model of di sor der ed el ect r oni c syst ems near t wo di - mensi ons [ 1- 4] . In t hi s paper we appl y a non- l i near o- - model f ormal i sm to c om- put e t he cr ossover exponent in t he pr esence of a weak ext er nal magnet i c f i el d, a pr obl em first addr essed by Khmel ' ni t sky and Lar ki n [ 5] . In t he absence of a magnet i c f i el d t he conduct i vi t y cr scal es as tr _ 1/ 6` t- ` with ~ - (E - E, )- `' wher e E, denot es t he mobi l i t y edge . They ar gued t hat a smal l magnet i c f i el d i nt r oduces a new l engt h, t he "magnet i c l engt h" , L a = hcl eB , and t he conduct i vi t y is t hen a scal i ng f unct i on of L a / ~, i .e . 0- = ~- t`t - Z1f ( Lß/ 6 ) _ ( E - E c )l d- 2 mg(B/ (E - E, ) Z° ) , gi vi ng a cr ossover exponent cp = 2v . A r el at ed pr obl em is t he cr ossover exponent f or ot her t i me- r ever sal br eaki ng per t ur bat i ons . Opper - man [ 6] cal cul at ed t he cr ossover exponent f or a t i me- r ever sal br eaki ng- pot ent i al scat t er i ng per t ur bat i on to be cp = 2/ s + O( E) , r =d- 2 . The s ame cr ossover exponent was f ound f or a r andom, spi n- dependent but r eal , pot ent i al . Wegner [ 7] r econsi der ed t hi s pr obl em and f ound a cr ossover exponent cp = 2P + 0( , - ' ) f or * Per manent addr ess : Di part i ment o di Fi si ca Uni ver si t à di Roma " La Sapi enza" , I - 110185 Rome, I t al y . 0550- 3213/ 90/ $03 . 50 ©El sevi er Sci ence Publ i sher s B .V. ( Nor t h- Hol l and)