Energy of strongly correlated electron systems

Transcription

1 Avalable onlne at WSN 49(2) (2016) EISSN Energy of strongly correlated electron systems M. A. Reza a, K. A. I. L. Wjewardena Gamalath b Department of Physcs, Unversty of Colombo, Colombo 3, Sr Lana a,b E-mal address: 11sc12708@stu.cmb.ac.l, male@phys.cmb.ac.l ABSRAC he constraned path Monte Carlo method was used to solve the Hubbard model for strongly correlated electrons systems analytcally n arbtrary dmensons for one, two and three dmensonal lattces. he energy varatons wth electron fllng, electron-electron correlaton strength and tme as well as the netc and potental energes of these system were studed. A competton between potental and netc energes as well as a reducton of the rate of ncrease of the potental energy wth ncreasng correlaton were observed. he degenerate states of the lattce systems at zero correlaton and the ncrease n the energy separaton of the states at hgher correlaton strengths were evdent. he varaton of the energy per ste wth correlaton strength of dfferent lattce szes and dmensons were obtaned at half fllng. From these t was apparent that the most stable lattces were the smallest for all the dfferent dmensons. For one dmenson, the convergence of the results of the constraned path method wth the exact non-lnear feld theory results was observed. Keywords: Strongly correlated electron systems; constraned-path Monte Carlo method; Hubbard model; dynamcal mean feld theory 1. INRODUCION ranston metal oxdes such as ncel oxde and cobalt oxde even though havng partally flled d orbtals were reported by De Boer and Verwey n 1937 to have transparent nsulatng propertes, contradctng the mplcatons of conventonal band theory [1]. Mott suggested that the Coulombc repulson between the electrons prevented metallcty n these

2 materals and these materals were called Mott nsulators [2]. In strongly correlated materals, the potental energy arsng from ths Coulombc repulson, competes wth the electron netc energy to gve nterestng propertes, one beng the Mott metal to nsulator transton. In resstvty measurements conducted on transton metal compounds contanng magnetc mpurtes, a mnmum n resstvty was observed at a specfc temperature. Untl Jun Kondo n 1964 showed that ths mnmum arose from the competton between electron-phonon scatterng and spn-spn scatterng, t remaned a paradox [3]. hs type of behavour occurs manly due to strong electron-electron nteractons and correlatons. hese materals called strongly correlated electron systems, led to the emergence of a separate paradgm n condensed matter physcs. Apart from transton metal oxdes and perovstes, ths phenomenon has been found to be prevalent n graphenes, fullerenes and as well as n soft-matter le polymers [4]. he narrower the band, the longer wll an electron stay on the atom and thereby feels the presence of other electrons more. herefore a narrow bandwdth mples a stronger correlaton. In many materals wth partally flled d or f orbtals such as the transton metals, vanadum, ron, ncel and ther oxdes or rare earth materals such as cerum experence strong electron correlatons as the electrons occupy narrow orbtals [5]. he conventonal models faled to explan these unusual propertes due to the fact that the electrons were treated as separate non-nteractng enttes, whch wored for most materals used wdely such as slcon and alumnum. For strongly correlated materals, when the electron-electron nteractons are taen nto consderaton, the problem evolves nto a many-body problem whch s dffcult to solve even for smple lattces. he Hubbard Hamltonan ncludes the electron-electron repulsons and was ntally used to study the transton metal oxdes. Over the years the Hubbard model has been appled to more complex systems usng mean feld approaches and was used to study hgh temperature superconductors n the 1990s. Despte beng smple n form, t was successful n explanng the behavor of these materals to a larger extent. he fact that the Hubbard model cannot be solved analytcally n arbtrary dmensons led to varous numercal approaches such as the Lanczos algorthm at absolute zero and fnte-temperature auxlary feld Monte-Carlo for low temperatures. he Hubbard model n ts pure form s able to explan basc features of correlated electrons but t cannot account for the detaled physcs of real materals. he ndependent electron approxmaton used n densty functonal theory approaches s not used n ths approach, and the only approxmaton s that the lattce self-energy s momentum ndependent [6]. In ths paper, the strongly correlated electron systems were nvestgated by the Hubbard model approach [7] wth constraned path Monte Carlo method [8] n Slater determnant space. A lnear chan of electrons stes n one dmenson, a planar square lattce of 221 n two dmenson and a cubc lattce of 222 electron stes n three dmenson were studed through total energy, netc energy specfed by hoppng ntegrals and potental energy dctatng the form of the Hubbard-Stratonovch transformaton calculated from Hellman-Feynman theorem. her behavour wth tme and electron-electron correlatons were studed by settng the hoppng parameter n all drectons to unty. he number of Slater determnants was set to 100 and the tme step 0.01s. For dfferent lattce szes 2 1 1, and D lattces, 22 1, 32 1, D lattces and 22 2, 32 2, D lattces, the varatons of the energy per ste wth correlaton was obtaned. For -60-

3 the lattce n one dmenson, the results of the constraned path method wth the exact non-lnear feld theory was compared. 2. HUBBARD MODEL FOR SRONGLY CORRELAED SYSEMS he most complete model yet to descrbe strongly correlated systems s the Hubbard model ntroduced by John Hubbard n 1963 [7]. A regular array of fxed nuclear postons n a sold lattce s consdered neglectng the lattce vbratons and the electrons move around n ths lattce. Intally the atom s consdered as havng one energy level open for electron occupaton thereby gvng two electron stes of spn up and spn down possble. he electrons nteract va the screened Coulomb nteracton, and the largest nteracton s between two electrons at the same ste. he nteracton s modeled by consderng an addtonal term, the correlaton strength U when two electrons occupy the same ste and ths s set to zero f a ste s unoccuped or occuped by only one electron. Snce the nteractons between electrons on the same ste s much larger than the nteractons between electrons on neghbourng stes, the nteractons between electrons from neghbourng stes are not consdered. hs electronelectron nteractons gves rse to the potental energy term n the Hubbard Hamltonan. he netc energy term arses from the electron hoppng governed by the hoppng ntegral t, whch s the energy scale n most Hubbard calculatons. he hoppng s determned by the overlap of the two wave functons on a par of atoms. Snce electron wave functons de off exponentally wth dstance, only hoppng to nearest neghbours are consdered. c j s defned as the electron creaton operator, whch creates an electron of spn σ at lattce ste j and c s defned as the electron destructon operator, whch destroys an electron of spn σ at lattce ste. Hubbard hamltonan s defned as: H t c c U n n ( n n ) (1) j, j j j j j j j he frst term of the Hamltonan s the netc energy term whch denotes the destructon of an electron of spn at ste and a creaton of an electron of spn at lattce ste j. he term j, n the summaton denotes the fact that hoppng s allowed only between two adjacent stes. he second term s the potental energy term and t checs all stes and adds energy of U when t fnds a ste s doubly occuped. he thrd term s the chemcal potental controllng the fllng of electrons. Most nterestng phenomena are observed n strongly correlated materals when there s only one electron per ste ntally, whch represents the half fllng state. 3. CONSRAINED-PAH MONE CARLO MEHOD he constraned path Monte Carlo method combnes the concept of the Hubbard Stratonovch transformaton and Slater determnants wth branchng random wals [8]. In ths study t s assumed that the Hamltonan conserves the total z-component of the electron spn -61-

4 s z and that there s no transfer of spns. herefore the number of electrons wth each spn up and down component s fxed. he number of lattce stes n the Hubbard model was consdered as M and the th sngle-partcle bass state as. If N s the number of electrons n the system, then N s the number of electrons wth spn σσ ( = or ) and N M. For a sngle partcles wave functon, the coeffcents of the expanson n the sngle partcle bass s gven as the M-dmensonal vector : c 0 (2) he many-body wave functon whch s wrtten as a Slater determnant s formed from the N-dfferent sngle partcle orbtals by ther symmetrzed product: ˆ ˆ ˆ N (3) where the operator ˆ m c, m creates and electron n the m th sngle-partcle orbtal descrbed by equaton 1. s an M representng the coeffcents of the orbtals. he many-body ground state 0 and the N matrx referred to as a Slater determnant many-body wave functon are not necessarly a sngle Slater determnant. he constraned path Monte Carlo method wors wth a one-partcle bass n Slater determnant space. When the energy s taen n terms of the hoppng parameter t, t s useful to wrte the Hubbard Hamltonan n the form: ( H / t) c jc U / t n n ( / t) ( n n ) j j j j j, j j (4) he coulomb repulson from electrons on the same ste s U. It s useful to gve all energy related measurements n terms of the hoppng parameter t snce we are focusng on the correlatons whch depend on the strength of the nteracton between the electrons and not on the hoppng of the electrons. herefore the hoppng parameter t whch s a unt of energy s set to unty and the parameter U / t s the correlaton strength. he dfference n the Hubbard Hamltonan and the general electronc Hamltonan s the structure of the matrx elements of netc energy ˆK and potental energy V ˆ and ths dfference captures the propertes of strong correlaton. ˆK s specfed by hoppng ntegrals of the form K j whle the elements of potental energy V ˆ dctates the form of the Hubbard- Stratonovch transformaton. he ground-state wave functon 0 can be obtaned from a tral wave functon not orthogonal to 0 by repeated applcatons of the ground-state projecton operator -62-

5 p gs e ( Hˆ E ) (5) where s E s the best guess of the ground-state energy. If the wave functon at the n th tme step n, the wave functon at the next tme step s gven by n ( Hˆ E ) n e (6) 1 wth a small tme step. he second order rotter approxmaton: Hˆ ( Kˆ Vˆ ) Kˆ /2 Vˆ Kˆ /2 e e e e e (7) he netc energy, the one-body propagator e Vˆ Bˆ K /2 Kˆ e /2 and the potental energy propagator does not have the same form. A Hubbard Stratonovch (HS) transformaton transform the potental energy propagator to the desred form. In the Hubbard model, we can use the followng: e Un ( )/2 x ( n n ) n U n n e p( x ) e x 1 (8) where s gven by cosh exp( U / 2). px ( ) 1/ 2 s nterpreted as a dscrete probablty densty functon wth x 1. he exponent on the left, comes from the nteracton term V ˆ on the th ste s quadratc n n, ndcatng the nteracton of two electrons. he exponents on the rght, on the other-hand, are lnear n n, ndcatng two non-nteractng electrons n a common external feld characterzed by x. hus an nteractng system has been converted nto a non-nteractng system n fluctuatng external auxlary felds x, and the summaton over all such auxlary-feld confguratons recovers the many-body nteractons. he lnearzed operator on the rght hand sde n equaton s the spn ( n n ) on each ste. he frst component of the constraned path Monte Carlo method s the reformulaton of the projecton process as branchng, open-ended random wals n Slater determnant space nstead of updatng a fxed-length path n auxlary-feld space. We defne ˆ ( ) ˆ B x B ( x). Applyng the HS-transformed propagator: V V n1 E ( )[ ˆ ˆ ( ) ˆ n e P x B B x B ] (9) x K /2 V K /2 In the Monte Carlo realzaton of ths teraton, the wave functon at each stage s represented by a fnte ensemble of Slater determnants: -63-

6 n (10) n where labels the Slater determnants and an overall normalzaton factor of the wave functon has been omtted. hese Slater determnants wll be referred to as random walers. he teraton n equaton 9 s acheved stochastcally by Monte Carlo samplng of x. hat s, for each random waler n, an auxlary-feld confguraton x s chosen accordng to the probablty densty functon Px ( ) and propagate the determnant to a new determnant n1 ˆ ˆ ( ) ˆ n B B x B. hs procedure s repeated for all walers n the populaton. K /2 V K /2 hese operatons accomplsh one step of the random wal. he new populaton represents n1 n1. hese steps are terated untl suffcent data has been collected. After an equlbraton phase, all walers thereon are Monte Carlo (MC) samples of the ground-state wave functon and ground-state propertes can be computed. We wll refer to ths type of 0 approach as free projecton. In practce, branchng occurs because of the reorthonormalzaton of the Slater determnants. Computng the mxed estmator of the groundstate energy: E mxed Hˆ 0 (11) 0 requres estmatng the denomnator by where are random walers after equlbraton. Snce these walers are sampled wth no nowledge of, terms n the summaton over can have large fluctuatons that lead to large statstcal errors n the MC estmate of the denomnator, thereby n that of E mxed. o elmnate the decay of the sgnal-tonose rato, we mpose the constraned path approxmaton. It requres that each random waler at each step to have a postve overlap wth the tral wave functon : 0 (12) ( n) he constraned path approxmatons easly mplemented by redefnng the mportance functon: O ( ) max,0 (13) he mxed estmator for the ground-state energy for an ensemble s gven by -64-

7 Emxed wel[, ] w (14) where the local energy E L s: E L[, ] Hˆ (15) hs quantty can be easly evaluated for any waler (Slater determnant) as follows. For any par of Slater determnants Green's functon as: c c j 1 j ( ) c c and, we can calculate the one-body equal-tme (16) hs mmedately enables the computaton of the netc energy term t c c. j j he potental energy term U c c c c 16, but can be reduced to that form by an applcaton of Wc's theorem: j does not have the form of equaton c c c c c c c c c c c c c c c c (17) and he reducton occurs because the up and down spn sectors are decoupled n both. hs s not the case n a parng or generalzed Hartree-Foc wave functons. he former s the desred form for U < 0. he latter can be used to mprove the qualty of the tral wave functon for U > 0, and s necessary f the Hamltonan contans spn-orbt couplng. he potental energy value U c c c c when combned wth the netc energy value j j t c c provdes an unbased estmate for the total energy, but the potental and netc energy terms alone are based. From the netc energy term t mples that the netc energy has no varaton wth correlaton strength, but t s nown to vary wth the correlaton. herefore the potental energy s calculated usng an alternatve method, the Hellman- Feynman theorem: dh de PE ( U) U ( U) U (18) du du -65-

8 Here de / du s calculated from the fnte-dfference formula. he netc energy value s then obtaned by deductng the potental energy from the total energy. Snce both the total energy and potental energy are unbased estmators, the netc energy s then unbased too. he dervatve was obtaned usng the fve-pont dfference formula: de E( U 2 U ) 8 E( U U ) 8 E( U U ) E( U 2 U ) du 12U (19) 4. OAL ENERGY AND ELECRON FILLING he total energy of the electron system wth tme for full fllng and half fllng were calculated for one (1D), two (2D) and three (3D) dmensonal lattces. A lnear chan of stes ( L 2, L 1, L 1) for x y z the 1D case, a planar square lattce of 22 1 electron stes ( L 2, L 2, L 1) x y z for the 2D case and a cubc lattce of 22 2 stes ( L 2, L 2, L 2) for x y z 1(a): Half-fllng the 3D case were consdered. L denotes the number of stes n the th drecton. he hoppng parameter was set to unty n all drectons. he number of Slater determnants was set to 100 and the tme step to 0.01s. he nterval between energy 1(b): Full fllng measurements was set to 20 Fgure 1. otal Energy E/ t aganst tme for the 1D 21 1 blocs. An energy measurement s made every 0.2 seconds. he varaton of total energy as a functon of tme for the half fllng and full fllng cases obtaned for the 1D, 2D and 3D lattces. hese are shown n fgures 1, 2 and 3. From these fgures we can see that for 1D, 2D and 3D lattces, the total -66-

9 energy vares wth tme n the half-fllng cases only, whle t s a constant n tme for the full fllng case. he energy oscllates n a specfc regon for the half-fllng case and at correlaton strength of U / t 4, the total energy of the system at full fllng s postve, whch mples that the system s not bounded and s unstable for full fllng under ths formulaton. Around half-fllng s where the strange propertes of the strongly correlated materals maybe found, whch s confrmed from most expermental results. herefore from here on n the study, only half fllng cases were consdered for all the lattce systems. 2(a): Half-fllng 2(b): Full fllng Fgure 2. otal Energy E/t aganst tme for the 2D 221lattce -67-

10 3(a): Half-fllng 3(b): Full fllng Fgure 3. otal Energy E/t aganst tme for the 3D 222lattce -68-

11 5. OAL ENERGY, CORRELAION SRENGH AND IME he total energy of the electron system wth tme and the electron-electron correlaton strength for the half fllng case was obtaned for the same one, two and three dmensonal lattces. he hoppng parameters n x, y and z- drectons were set to unty ( t 1, t 1, t 1). he x y z number of Slater determnants was set to 100 and the tme step 0.01s. he on-ste repulson strength U/t was set to ncrease from 0 to 8.0 n steps of 0.5. he magnary tme was set to run from 0 to 6 s n the 1D case and 0 to 12 s n the 2D case and 0 to 18 s n the 3D case n steps of 0.01 s. he total energy n terms of hoppng parameter and correlaton strength as well as tme for one, two and three dmensonal lattces are shown n fgures 4(a), 4(b) and 4(c) respectvely. hese fgures show that for 1D, 2D and 3D systems, the total energy of the system ncreases wth the correlaton strength whle oscllatng wth tme. he varaton wth correlaton strength s more sgnfcant than the varaton wth tme. he oscllaton of the total energy of the system wth tme s more promnent at hgher correlatons strengths. hs explans the fact that when there s zero correlaton strength wth tme, f an electron from a sngly occuped ste jumps to another sngly occuped ste, gvng a double occuped ste and an empty ste snce the correlaton strength s zero, there s no change n energy. herefore although the confguraton of Fgure 4(a): For 1D 2x1x1 lattce Fgure 4(b): For 2D 2x2x1 lattce Fgure 4(c): For 3D 2x2x2 lattce Fgure 4. otal Energy E/t, correlaton strength U/t and tme for 1D (2x1x1), 2D (2x2x1) and 3D (2x2x2) lattces -69-

12 the system has changed, the energy remans the same and the two states are degenerate n energy. But when the correlaton strengths are non-zero and ncreasng, the doubly occuped state has more energy than the state wth two sngly occuped stes, therefore when the electron jumps to a sngly occuped ste gvng that ste double occupancy, the energy of the system ncreases and t goes to a hgher energy state, and when the electron jumps bac, the system goes to a lower energy state, thus gvng oscllatons n the total energy of the system wth tme as correctly predcted for the three cases n the above plots. 6. POENIAL ENERGY AND KINEIC ENERGY AGAINS CORRELAION he varaton of the potental energy (PE) and netc energy (KE) of the electron systems wth correlaton strength were calculated for the same 1D, 2D and 3D lattce systems at half fllng. he hoppng parameters were set to unty ( t 1, t 1, t 1). he x y z number of Slater determnants was set to 100 and the tme step 0.01s. he on-ste repulson strength U/t was set to ncrease from 0 to 8.0 n steps of he energy measurements were taen 10 tmes and the standard error was taen as the error n the energy values. Second order polynomals were ftted to the data. he netc energes and potental energes as a functon of correlaton strength U/t for 1D, 2D and 3D are shown n fgures 5, 6 and 7 respectvely. From the above fgures t can be seen that the potental energy ntally ncreases wth the correlaton strength, but as the system enters the strong couplng regme, the potental energy saturates and decreases gradually. It s seen that netc energy contnues to ncrease n the wea, Fgure 5(a): Potental enery aganst correlaton n terms of hoppng parameter t for 1D 2x1x1 lattce Fgure 5(b): Knetc enery aganst correlaton n terms of hoppng parameter t for 1D 2x1x1 lattce -70-

13 ntermedate and strong couplng regmes. It can also be seen that the errors n the potental energy s slghtly hgher than that of the netc energy. Also t shows that the potental energy and netc energy values agree sgnfcantly wth the second order polynomal, therefore we can say that the energes are of the second order wth correlaton. he netc energy contnues to ncrease n the wea, ntermedate and strong couplng regmes. Fgure 6(a): Potental enery aganst correlaton n terms of hoppng parameter t for 2D 2x2x1 lattce Fgure 6(b): Knetc enery aganst correlaton n terms of hoppng parameter t for 2D 2x2x1 lattce -71-

14 Fgure 7(a): Potental enery aganst correlaton n terms of hoppng parameter t for 3D 2x2x2 lattce Fgure 7(b): Knetc enery aganst correlaton n terms of hoppng parameter t for 3D 2x2x2 lattce 7. ENERGY PER LAICE SIE VS. CORRELAION SRENGH he energy of the 1D, 2D and 3D lattce systems are governed by the number of electrons, the fllng, and the correlaton strength, when the temperature and hoppng parameter are fxed. Snce the energy depends on the number of electrons or the number of lattce stes, the energy per lattce ste for a system s a better measure for the comparson of the energy n lattce systems. he varaton of the energy per lattce ste wth correlaton strength both n terms of the hoppng parameter were compared for the 1D 2 1 1, 2D 22 1 and 3D 22 2 lattce systems. he hoppng parameter was set to unty n all drectons. he number of Slater determnants were set to 100 and the tme step 0.01s. -72-

15 he on-ste repulson strength U/t was set to ncrease from 0 to 8.0 n steps of 0.5. he energy values were calculated for 10 seconds and averaged for each correlaton value. he standard error was taen as the error n energy. A second order polynomal was ftted to the data. hese are shown n the fgure 8 wth 1D n blue, 2D n green and 3D n red. Fgure 8. Energy per ste aganstcorrelaton per hoppng parameter for 1D, 2D,3D lattces he above plot shows that the energy per ste of the 3D system s more negatve than the 2D system whch s more negatve than the 1D lattce system. hs shows that the 3D system s the most stable when compared to the 2D and 1D systems. Also t shows that the 1D, 2D and 3D energy values strongly agreed wth the ftted second order polynomals. Fgure 9. Energy per ste aganstcorrelaton n terms of hoppng parameter for 21 1(blue), 411 (green) and 81 1 (red) 1D lattces -73-

16 Fgure 10. Energy per ste aganst correlaton n terms of hoppng parameter for 22 1(blue), 32 1(green), and 42 1(red)2D lattces Fgure 11. Energy per ste aganst correlaton n terms of hoppng parameter for 22 2 (blue), 32 2 (green) and 32 2 (red) 3D lattces he energy varatons per ste wth correlaton was obtaned for dfferent szes of the latces namely, 2 1 1, 4 1 1, D lattces, 22 1, 32 1, D lattces and 22 2, 32 2, D lattces. A second order polynomal was ftted to the data and these are presented n fgure 9, 10 and 11 respectvely. he comparson of 1D lattces showed that the lattce (blue) was more stable than the (red) and the (green) lattces. here s no fxed relatonshp between the magntude of the energy and the length of the 1D chan. For the 2D lattces the 22 1 lattce was the most stable, followed by the 32 1 lattce. he smallest lattce was more stable and the lattces had smlar -74-

17 energes ntally, but as the correlaton strength ncreases, the energes of the three lattces too dfferent values. In the 3D case the smallest lattce s agan the most stable, and the larger lattces are of hgher energes and less stable. Also the ntal value of energy s not the same at zero correlaton, unle the 2D case. Further as n the prevous cases, the data values agreed wth the second order polynomal ft. herefore the energy per ste for all the lattces consdered are strongly second order. 8. CONVERGENCE OF HE CONSRAINED-PAH MONE CARLO MEHOD It s mportant to chec whether the constraned-path Monte Carlo method, the method used n ths study, converges to the actual values, so that ts fndngs may be consdered as accurate. here s no exact soluton to the Hubbard model usng dynamcal mean feld theory except n one dmenson. he exact soluton for the energy of a D lattce system wth one spn up and one spn down electron obtaned from nonlnear feld theory [10] s: E U U 2 ( 64) / 2 (20) herefore the total energy results obtaned for the D lattce system can be compared wth the exact soluton above to test the convergence of the constraned-path Monte Carlo method. he number of Slater determnants were set to 100 and the tme step Δτ = 0.01s. he nterval between energy measurements was set to 20 blocs. An energy measurement s made every 0.2 seconds. Repeated measurements were taen and the average value and Monte Carlo standard error was taen as the energy and error n energy. he on-ste repulson strength U/t was set to ncrease from 0 to 8.0 n steps of he exact calculatons obtaned from equaton 21 and the values calculated from constraned-path Monte Carlo method are presented n fgure 12 by blue and red respectvely. Even wth 100 Slater determnants, the total energy values obtaned from the constraned path method agree very closely wth the energy values obtaned from the exact non-lnear feld theory method. Fgure 12. otal energy E/taganst correlaton strength U/t from exact soluton (blue) and the constraned-path Monte Carlo method (red)for the 1D 21 1 lattce. -75-

18 9. CONCLUSIONS For the 1D, 2D and 3D lattces at fxed temperature and at correlaton strength of 4 tmes the hoppng parameter, the total energy was negatve and the systems were bound at half fllng. But at full fllng, the total energy of the system was constant wth tme and postve ndcatng an unbounded nature of the system n ths formulaton. he 1D lattce at fxed temperature approached ferromagnetc orderng at hgh correlaton strengths. It was also conclusve that for the same 1D, 2D and 3D lattces at half fllng and fxed temperature the varaton of the system wth correlaton was more sgnfcant than the varaton wth tme and the oscllaton of the total energy of the systems wth tme was more promnent at hgher correlatons strengths. At correlaton strengths of four to eght tmes the hoppng parameter, the oscllatons of the total energy wth tme was more sgnfcant. he degeneracy wth respect to the sngly and doubly occuped states were present at zero correlaton and were partally removed at hgher correlaton strengths. A competton between the potental energy and netc energy s promnent n the 1D, 2D and 3D lattces at fxed temperature n the range of correlaton strengths of zero to eght tmes the hoppng parameter, although the potental energy contrbuton was less n magntude to the netc energy. he varaton n netc and potental energes wth correlaton strength can be ftted to a second order polynomal. herefore the netc and potental energes of the 1D, 2D and 3D lattces were of second order n correlaton strength, but wth a certan degree of dsagreement. Further t was observed that the energy per ste was the most negatve that s most stable for the smaller lattces and as the lattce szes ncreased, the negatvty of the energes s reduced. he varaton of the total energy of the 1D system obtaned from the constraned path method and from the exact non-lnear feld theory method agreed well even wth only 100 Slater determnants, ndcatng that the constraned path Monte Carlo method can be used for correlated systems. References [1] J. Boer, E. J. W. Verwey, Sem-conductors wth partally and wth completely flled 3d lattce bands, Proc. Phys. Soc., vol. 49, no. 4S, pp , [2] N. F. Mott, Metal-Insulator ranston, Rev. Mod. Phys., vol. 40, no. 4, pp , [3] J. Kondo, Resstance Mnmum n Dlute Magnetc Alloys, Prog. heo. Phys. vol. 32, no. 1, pp , [4] H. Barman, Dagrammatc perturbaton theory based nvestgatons of the Mott transton physcs, (Ph.D. hess, heoretcal scence unt, Jawaharlal Nehru centre for advanced scentfc research, Bangalore, Inda, 2013) [5] V. N. Antonov, L. V. Beenov, and A. N. Yareso, Electronc Structure of Strongly Correlated Systems, Advances Cond. Matt. Phys , pp.1-107, [6] D. Vollhardt, Dynamcal mean-feld theory for correlated electrons. Annalen Der Phys, vol. 524, no. 1, pp. 1-19,

19 [7] G. Kotlar, A. Georges, Hubbard model n nfnte dmensons, Phys. Rev. B, vol. 45, no. 12, pp , [8] A. omas, QUES: QUantum Electron Smulaton oolbox. Avalable at: [9] H. Nguyen, H. Sh, J. Xu, and S. Zhang, CPMC-Lab: A Matlab pacage for Constraned Path Monte Carlo calculatons, Comp. Phys. Comm., vol. 185, no. 12, pp , [10] F. D. M. Haldane, Nonlnear Feld heory of Large-Spn Hesenberg Antferromagnets: Semclasscally Quantzed Soltons of the One-Dmensonal Easy-Axs Néel State. Phys. Rev. Lett. 50(15) (1983) ( Receved 10 May 2016; accepted 26 May 2016 ) -77-