an extremely rich phase diagram structure is found and is explained by competing and frustrated

Transcription

1 Frustrate Further-Neighbor Antiferromagnetic an Electron-Hopping Interactions in the = 3 tj Moel: Finite-Temperature Global Phase iagrams from Renormalization-Group Theory C. Nair Kaplan, 1,,3 A. Nihat Berker, 4,5,6 an Michael Hinczewski 6,7 1 epartment of Physics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey, epartment of Physics, Koç University, Sarıyer 3445, Istanbul, Turkey, 3 Martin Fisher School of Physics, Braneis University, Waltham, Massachusetts 454, U.S.A., 4 Faculty of Engineering an Natural Sciences, Sabancı University, Orhanlı, Tuzla 34956, Istanbul, Turkey, 5 epartment of Physics, Massachusetts Institute of Technology, Cambrige, Massachusetts 139, U.S.A., 6 Feza Gürsey Research Institute, TÜBITAK - Bosphorus University, Çengelköy 34684, Istanbul, Turkey an 7 epartment of Physics, Technical University of Münich, Garching, Germany The renormalization-group theory of the = 3 tj moel is extene to further-neighbor antiferromagnetic or electron-hopping interactions, incluing the ranges of frustration. The global phase iagram of each moel is calculate for the entire ranges of temperatures, electron ensities, further/first-neighbor interaction-strength ratios. With the inclusion of further-neighbor interactions, an extremely rich phase iagram structure is foun an is explaine by competing an frustrate interactions. In aition to the τ tj phase seen in earlier stuies of the nearest-neighbor = 3 tj moel, the τ Hb phase seen before in the = 3 Hubbar moel appears both near an away from half-filling. PACS numbers: 71.1.F, 5.3.Fk, 64.6.e, 74.5.w I. INTROUCTION The simplest moel electron conuction moel, incluing nearest-neighbor hopping on a lattice an on-site Coulomb repulsion, is the Hubbar moel [1]. In the limit of very strong on-site Coulomb repulsion, seconorer perturbation theory on the Hubbar moel yiels the tj moel [, 3], in which sites oubly occupie by electrons o not exist. Stuies of the Hubbar moel [4] an of the tj moel [5], incluing spatial anisotropy [6] an quenche non-magnetic impurities [7] in goo agreement with experiments, have shown the effectiveness of renormalization-group theory, especially in calculating phase iagrams at finite temperatures for the entire range of electron ensities in = 3. These calculations have reveale new phases, ubbe the τ phases, which occur only in these electronic conuction moels uner oping conitions. The telltale characteristics of the τ phases are, in contrast to all other phases of the systems, a non-zero electron-hopping probability at the largest length scales (at the renormalization-group thermoynamic-sink fixe points an the ivergence of the electron-hopping constant t uner repeate rescalings. Furthermore, the phase iagram topologies, the oping ranges, an the contrasting quantitative τ an antiferromagnetic behaviors uner quenche impurities [7] have been in agreement with experimental finings [8, 9]. A benchmark for this renormalization-group approach has also been establishe by a etaile an successful comparison, with the exact numerical results of the quantum transfer matrix metho [1, 11], of the specific heat, charge susceptibility, an magnetic susceptibility in = 1 calculate with our metho.[1] Furthermore, results with this metho have inicate that no finite-temperature phase transition occurs in the tj moel in = 1. A phase separation at zero temperature has been foun in = 1 in Ref. [13]. Thus, the = 1 tj appears to have a first-orer phase transition at zero temperature that isappears as soon as temperature is raise from zero, as in other = 1 moels such as the Ising an Blume-Capel moels [14, 15]. A phase separation [16 18] occurs in = for low values of t/j, but not for t/j >.4.[5] In = 3, a narrow phase separation occurs, as seen in the ensity - temperature phase iagrams below. Two istinct τ phases have been foun in the Hubbar moel [4], τ Hb an τ tj, respectively occurring at weak an strong coupling. The calculate low-temperature behavior an critical exponent of the specific heat [4] have pointe to BCS-like an BEC-like behaviors, respectively. Only the τ tj phase was foun in the tj moel. The current work aresses the issue of whether both τ phases can be foun in the tj moel, via the inclusion of further-neighbor antiferromagnetic (J or furtherneighbor electron hopping (t interactions. We fin that, epening on the temperature an oping level, the further-neighbor interactions may compete with the further-neighbor effects of the nearest-neighbor interactions, namely that frustration occurs as a function of temperature an oping level. This competition (or reinforcement between the interactions of successive length scales unerpins the calculate evolution of the phase iagrams. Global phase iagrams are obtaine for the entire ranges of each type of further-neighbor interaction. With the inclusion of further-neighbor interactions, an extremely rich phase iagram structure is foun an is explaine by competing an frustrate interactions. Both τ Hb an τ tj phases are inee foun to occur in the tj moel with the inclusion of these further-neighbor interactions. Furthermore, istinctive lamellar phase iagram structures of antiferromagnetism interestingly sur-

2 roun the τ phases in the ope regions. II. THE tj HAMILTONIAN On a -imensional hypercubic lattice, the tj moel is efine by the Hamiltonian βh = P t J ij ij,σ ( c iσ c jσ + c jσ c iσ S i S j + V ij n i n j + µ i n i P, (1 rescaling factor b =. By neglecting the noncommutativity of the operators beyon three consecutive lattice sites, a trace over all states of even-numbere sites can be performe [19, ], Tr even e βh = Tr even e P i { βh(i,i+1} even i P even = Tr even e i { βh(i 1,i βh(i,i+1} Tr i e { βh(i 1,i βh(i,i+1} = even e P even i { β H (i 1,i+1} = e β H, i e β H (i 1,i+1 (4 where β = 1/k B T an, with no loss of generality [5], t is use. Here c iσ an c jσ are the creation an annihilation operators for an electron with spin σ = or at lattice site i, obeying anticommutation rules, n i = n i + n i are the number operators where n iσ = c iσ c iσ, an S i = σσ c iσ s σσ c iσ is the single-site spin operator, with s the vector of Pauli spin matrices. The projection operator P = i (1 n i n i projects out all states with oubly-occupie sites. The interaction constants t, J, V an µ correspon to electron hopping, nearestneighbor antiferromagnetic coupling (J >, nearestneighbor electron-electron interaction, an chemical potential, respectively. From rewriting the tj Hamiltonian as a sum of pair Hamiltonians βh(i, j, Eq. (1 becomes βh = ij P [ t σ ( c iσ c jσ + c jσ c iσ ] JS i S j + V n i n j + µ(n i + n j P ij { βh(i, j}, ( where µ = µ/. The stanar tj Hamiltonian is a special case of Eq. ( with V/J = 1/4, which stems from secon-orer perturbation theory on the Hubbar moel [, 3]. III. RENORMALIZATION-GROUP TRANSFORMATION A. = 1 Recursion Relations In = 1, the Hamiltonian of Eq. ( is βh = i { βh(i, i + 1}. (3 A ecimation eliminates every other one of the successive egrees of freeom arraye in a linear chain, with the partition function being conserve, leaing to a length where β H is the renormalize Hamiltonian. This approach, where the two approximate steps labele with are in opposite irections, has been successful in the etaile solutions of quantum spin [19 5] an electronic [4 7] systems. The anticommutation rules are correctly accounte within the three-site segments, at all successive length scales, in the iterations of the renormalizationgroup transformation. The algebraic content of the ecimation in Eq. (4 is e β H (i,k = Tr j e βh(i,j βh(j,k, (5 where i, j, k are three consecutive sites of the unrenormalize linear chain. The renormalize Hamiltonian is given by [ β H (i, k = P t σ ( c iσ c kσ + c kσ c iσ J S i S k + V n i n k + µ (n i + n k + G ]P, (6 where G is the aitive constant per bon, which is always generate in renormalization-group transformations, oes not affect the flow of the other interaction constants, an is necessary in the calculation of expectation values. The values of the renormalize (prime interaction constants appearing in β H are given by the recursion relations extracte from Eq. (5, which will be given here in close form, while Appenix A etails the erivation of Eq. (7 from Eq. (5: t = 1 ln γ 4 γ, J = ln γ 6 γ 7, V = 1 4 ln γ4 1γ 6 γ 3 7 µ = µ + 1 ln ( γ γ 4 γ 1 γ 4 γ4 4, G = b G + ln γ 1, (7,

3 3 where γ 1 = 1 + u 3 f( µ, ( γ = uf µ + 1 u x + 3 ( u vf J 8 + V + µ, γ 4 = u v + 1 ( 3J u xf, γ 6 = v 3 x + xf γ 7 = 3 vx v4 + vf 8 + V + µ, ( 3J 8 V µ ( J 8 V µ, (8 an v = exp ( J/8 + V/ + µ/, x = exp (3J/8 + V/ + µ/, u = exp (µ/, f(a = cosh t + A + A t + A sinh t + A. B. > 1 Recursion Relations (9 The Migal-Kaanoff renormalization-group proceure generalizes our transformation to > 1 through a bon-moving step [6, 7]. Eq. (7 can be expresse as a mapping of interaction constants K = {G, t, J, V, µ} onto renormalize interaction constants, K = R(K. The Migal-Kaanoff proceure strengthens by a factor of b 1 the bons of linear ecimation, to account for a bon-moving effect [6, 7]. The resulting recursion relations for > 1 are, which explicitly are K = b 1 R(K, (1 t = b 1 ln γ 4, J = b 1 ln γ 6, V = b 1 γ γ 7 4 ln γ4 1γ 6 γ7 3 µ = b 1 µ + b 1 ln ( γ γ 4 γ 1 γ 4 γ4 4, G = b G + b 1 ln γ 1., (11 This approach has been successfully employe in stuies of a large variety of quantum mechanical an classical (e.g., references in [4] systems. C. Calculation of Phase iagrams an Expectation Values The global flows of Eq. (1, controlle by stable an unstable fixe points, yiel the phase iagrams in temperature versus chemical potential [8]: The basin of attraction of each fixe point correspons to a single thermoynamic phase or to a single type of phase transition, Phase Interaction constants at sink t µ J V (ilute isorere (ense isorere AF V (antiferromagnetic 1 J 4 τ tj t (BEC-like superconuctor 1 J V 3 µ µ J 4 τ Hb (BCS-like superconuctor t µ 1 J µ V J 1 4 TABLE I: Interaction constants at the phase sinks. accoring to whether the fixe point is completely stable (a phase sink or unstable. Eigenvalue analysis of the recursion matrix at an unstable fixe point etermines the orer an critical exponents of the phase transitions at the corresponing basin. Table I gives the interaction constants t, J, V, µ at the tj moel phase sinks. The τ tj an τ Hb phases are the only regions where the electron-hopping term t oes not renormalize to zero at the phase sinks. On the contrary, in these phases, t an t, respectively. To compute temperature versus electron-ensity (oping phase iagrams, thermoynamic ensities are calculate by summing along entire renormalization-group flow trajectories.[9] A ensity, namely the expectation value of an operator in the Hamiltonian, is given by M α = 1 ln Z, (1 N K α where K α is an element of K = {K α }, Z is the partition function, an N is the number of lattice sites. The recursion relations for ensities are M α = b β M βt βα, where T βα K β K α. (13 In terms of the ensity vector M = {M α } an the recursion matrix T = {T βα }, T = Eq. (13 simply is b G t t t J t V t µ t G J t J J J V J µ J G V t V J V V V µ V G µ t µ J µ V µ µ µ, (14 M = b M T. (15 At a fixe point, the ensity vector M α = M α M α is the left eigenvector, with eigenvalue b, of the fixe-point

4 4 recursion matrix T (Table II. For non-fixe-points, iterating Eq. (15 n times, K M = b n M (n T (n T (n 1 T (1, (16 where, for n large enough, the trajectory arrives as close as esire to a completely stable (phase-sink fixe point an M (n M. The latter ensity vector M is the left eigenvector of the recursion matrix with eigenvalue b. When two such ensity vectors exist, the two branches of the phase separation of a first-orer phase transition are obtaine [9, 3], as illustrate with the phase separations foun below. Phase sinks Expectation values at sink P σ c iσ cjσ + c jσciσ ni Si Sj ninj 1 1 AF τ tj 3 τ Hb TABLE II: Expectation values at the phase sinks. The expectation values at a sink epitomize the expectation values throughout its corresponing phases, because, as explaine in Sec. IIIC, the expectation values at the phase sink unerpin the calculation of the expectation values throughout the corresponing phase which is constitute from the basin of attraction of the sink. IV. FURTHER-NEIGHBOR INTERACTIONS, TEMPERATURE- AN OPING-EPENENT FRUSTRATION AN GLOBAL PHASE IAGRAMS For the results presente below, we use the theoretically an experimentally ictate initial conitions of V/J = 1/4 an t/j =.5. The etails of the thermoynamic phases foun in this work, liste in Tables I an II, have been iscusse previously within context of the nearest-neighbor tj [5 7] an, for the τ Hb phase, Hubbar [4] moels. The τ Hb phase is seen here in the tj moel with the inclusion of the further-neighbor antiferromagnetic or electron-hopping interaction. Suffice it to recall here that the τ phases are the only phases in which: (1 the electron-hopping strength oes not renormalize to zero, but to infinity; ( the electron ensity oes not renormalize to complete emptiness or complete filling, but to partial emptiness/filling, leaving room for electron/hole conuctivity; (3 the nearest-neighbor electron occupation probability oes not renormalize to zero or unity, again leaving room for conuctivity at the largest length scales; (4 the electron-hopping expectation value is non-zero at the largest length scales; (5 the experimentally observe chemical potential shift as a function of oping occurs 1 3 FIG. 1: Construction of the further-neighbor moels. Part of a single plane of the three-imensional moel stuie here is shown. [6]; an (6 a low level ( 6% of quenche non-magnetic impurities causes total isappearance, in contrast to the antiferromagnetic phase ( 4% for total isappearance [7], again as seen experimentally. The low-temperature behavior an critical exponent of the specific heat [4] have pointe to BCS-like an BEC-like behaviors for the τ Hb an τ tj phases, respectively. The only approximations in obtaining the results below are the Suzuki-Takano an Migal-Kaanoff proceures, explaine above in Secs. IIIA an IIIB respectively. There are no further assumptions in Secs.IVA an IVB below. A. The t Moel The t moel inclues further-neighbor electronhopping interaction, as shown in Fig. 1. The threesite Hamiltonian, between the lattice noes at the lowest length scale, has the form: βh(i, j, k = βh(i, j βh(j, k ( t c iσ c kσ + c kσ c iσ, σ K (17 where βh(i, j is given in Eq. (, so that the first equation of Eq. (7 gets moifie as t = 1 ln γ 4 γ + t, (18 only for the first renormalization. Thus, for = 3, the first equation of Eq. (11 gets moifie as t = ln γ 4 γ + 4 t, (19 only for the first renormalization. Thus, the hoppingstrength t contributes to the first renormalization, but

5 τ τ Temperature 1/t /J <n i > FIG. : Global phase iagram of the further-neighbor t moel for t/j =.5 in temperature vs. chemical potential (first column an, corresponingly, temperature versus electron ensity (secon column. The relation t/j =.5 is use for all renormalization-group trajectory initial conitions. The t /t values are given in boxes. The ilute isorere (, ense isorere (, antiferromagnetic AF (lightly colore, τ tj (meium colore, an τ Hb (arkly colore phases are seen. Seconorer phase transitions are rawn with full curves, first-orer transitions with otte curves. Phase separation occurs between the ense ( an ilute ( isorere phases, in the unmarke areas within the otte curves in the electron ensity vs. temperature iagrams. These areas are boune, on the right an an the left, by the two branches of phase separation ensities, evaluate by renormalization-group theory as explaine in Sec.IIIC. Note that these coexistence regions between ense ( an ilute ( isorere phases are very narrow. ashe curves are not phase transitions, but isorer lines between the ense an ilute isorere phases. As explaine in the text, on each sie of the thick full curves (not a phase bounary, the further-neighbor electron hopping affects the τ phases oppositely. On the ash-otte curve (also not a phase bounary; overlaps, for t /t =, with the thick full curve electron hopping in the system is frustrate. is not regenerate by this first renormalization. Note that the quantitative memory of the further-neighbor interaction is kept in all subsequent renormalization-group steps, as the flows are moifie by the ifferent values of the first-renormalize interactions ue to the effect of the further-neighbor interaction. The subsequent global renormalization-group flows are in the space of t, J, V, µ, as is the case in position-space renormalizationgroup treatments [31 33] using a prefacing transformation. Which surfaces in this large (4-imensional flow space of t, J, V, µ are accesse is controlle by the original further-neighbor interaction. Thus, the further-neighbor

6 6.5 τ τ Temperature 1/t /J <n i > FIG. 3: The continuation of the global phase iagram in Fig.. interaction t shifts the value of t obtaine after the first renormalization-group transformation, as ictate by the physical moel (Fig.1. Since the value of the first-renormalize t in the absence of t alreay has a complicate epenence on the unrenormalize temperature an electron ensity, the variety of phase iagrams is obtaine. Inee, the effect of the further-neighbor interaction is epenent on the electron ensity, temperature, an other interactions in the system, ue to the presence of the first-term in Eq. (19, which is the key to the resulting spectacularly ifferent phase iagrams as the further-neighbor interaction is varie. (1 If the two terms in Eq. (18 are of the same sign, the nearestneighbor an further-neighbor electron hopping terms of the original system reinforce each other an the τ phases are enhance. ( If the two terms are of opposite signs, the nearest-neighbor an further-neighbor electron hopping terms of the original system compete with each other an, with the introuction of further-neighbor electron hopping, the τ phases are initially suppresse, but enhance as further-neighbor hopping becomes ominant. The two regimes (1 an ( are separate by the thick full lines in the phase iagrams in Figs. an 3. In the case ( of opposite signs, when the two terms cancel out each other, the system is frustrate, in which case, after the first renormalization, there is no electron hopping in the system. Since this conition is close uner renormalization, both on physical grouns an of course

7 7.5 τ τ Temperature 1/t /J <n i > FIG. 4: Global phase iagrams of the further-neighbor J moel for t/j =.5 in temperature vs. chemical potential (first column an, corresponingly, temperature versus electron ensity (secon column. The relation t/j =.5 is use for all renormalization-group trajectory initial conitions. The J /J values are given in boxes. The ilute isorere (, ense isorere (, antiferromagnetic AF (lightly colore, τ tj (meium colore, an τ Hb (arkly colore phases are seen. Seconorer phase transitions are rawn with full curves, first-orer transitions with otte curves. Phase separation occurs between the ense ( an ilute ( isorere phases, in the unmarke areas within the otte curves in the electron ensity vs. temperature iagrams. These areas are boune, on the right an an the left, by the two branches of phase separation ensities, evaluate by renormalization-group theory as explaine in Sec.IIIC. Note that these coexistence regions between ense ( an ilute ( isorere phases are very narrow. ashe curves are not phase transitions, but isorer lines between the ense an ilute isorere phases. As explaine in the text, on each sie of the thick full curves (not a phase bounary, the further-neighbor interaction affects the antiferromagnetic phase oppositely. On the ash-otte curve (also not a phase bounary; overlaps, for J /J =, with the thick full curve, antiferromagnetism in the system is frustrate. in our recursion relations (Eq. (7, no τ phase can occur in such a system. The ash-otte curves in Figs. an 3 inee show such systems. These competition, reinforcement, an frustration effects are temperature an oping epenent. These, an all other physical effects, o not epen on the sign of nearest-neighbor t of the original unrenormalize system, ue to the symmetry of hypercubic lattices [5] an as seen in Eq. (9.

8 8.5 τ τ Temperature 1/t AF /J <n i > FIG. 5: The continuation of the global phase iagrams in Fig. 4. Figs. an 3 give the global phase iagram of the t moel, as a function of temperature, electron ensity, chemical potential, an t /t. The values of the hopping-strength ratios t /t for the consecutive panels in these figures are chosen so that they sequentially prouce the qualitatively ifferent phase-iagram cross-sections, thereby revealing the evolution in the global phase iagram. Secon-orer phase transitions are rawn with full curves, first-orer transitions with otte curves. Phase separation occurs between the ense ( an ilute ( isorere phases, in the unmarke areas within the otte curves in the electron ensity vs. temperature iagrams. These areas are boune, on the right an an the left, by the two branches of phase separation ensities, evaluate by renormalization-group theory as explaine in Sec.IIIC. Note that these coexistence regions between ense ( an ilute ( isorere phases are very narrow. The cross-section t = is the phase iagram obtaine in previous work [5]. This iagram contains the τ tj phase between 33 37% hole oping away from halffilling an at temperatures 1/t <.1. The thick full curve here gives the systems evoi of electron hopping ue to the combine effects of temperature an oping on a nearest-neighbor-only interaction system. The first term of Eq. (18 is positive on the high ensity/chemical

9 9 potential, low temperature sie of the thick full curve an negative on the low chemical potential/ensity, high temperature sie of the thick full curve. Thus, the inclusion of t > will create competition an frustration (respectively reucing an eliminating the τ phases on the low chemical potential/ensity, high temperature sie of the curve iscusse here, reinforcement (enhancing the τ phases on the high chemical potential/ensity, low temperature sie of the same curve. The opposite occurs at t <. The thick full (no-hopping curve of t = is inclue, again as thick an full, in the t phase iagrams an the effects iscusse here are seen in the evolution, in both irections, of the global phase iagram. Pursuing the negative values of t, we see at t /t =.65 that the τ tj phase, being below the thick full curve, is inee reuce an bisecte into two isconnecte regions by the frustration (ash-otte curve. At the more negative value of t /t = -.15, only the higher oping region of the τ tj phase remains an is enhance as explaine after Eq. (18, extening through a wier range to 45 55% hole oping. The antiferromagnetic an isorere phases take part in a complex lamellar structure, in a narrow ban between 35 45% hole oping at low temperatures. At the even more negative values of t /t =.5 an.5, the τ tj phase appears in a wie range of hole oping, between 35 55%. Besies the complex lamellar structure of antiferromagnetic an isorere phases, we also see that the τ Hb phase participates in the lamellar phase structure an, separately, appears ajacently to the antiferromagnetic phase near half-filling. Particularly near half-filling, the τ Hb phase which evolves ajacently to the antiferromagnetic phase reaches to the higher temperatures of 1/t.5. This topology is ientical to that obtaine for the Hubbar moel [4]. For the positive values of t /t, the τ phases are enhance as explaine after Eq. (18 an the topology quickly evolves to that encountere in the Hubbar moel. The τ tj is not bisecte by the frustration (ashotte curve an appears between 33 37% hole oping as a continuation of the structure at t =. The τ Hb phase occurs again in two istinct regions an the one which lies nearer to half-filling again extens to high temperatures. B. The J Moel The J moel inclues further-neighbor antiferromagnetic interaction, as shown in Fig. 1. The three-site Hamiltonian, between the lattice noes at the lowest length scale, has the form: βh(i, j, k = βh(i, j βh(j, k J S i S k, ( ik where βh(i, j is given in Eq. (, so that the secon equation of Eq. (7 gets moifie as J = ln γ 6 γ 7 + J, (1 only for the first renormalization. Thus, for = 3, the secon equation of Eq. (11 gets moifie as J = 4 ln γ 6 γ J, ( only for the first renormalization. Again, the interaction J contributes to the first renormalization, but is not regenerate by this first renormalization. Reinforcement or competition occurs when J is, respectively, of same or opposite sign as the first term in Eq. (. These two regimes are again separate by the thick full lines in the phase iagrams of Figs. 3 an 4, while again frustration occurs on the ash-otte lines. In the reinforcement regime, we expect a large extent of the antiferromagnetic phase. The τ Hb phase is also expecte to grow in the reinforce region, for it is foun along the temperature extent of the antiferromagnetic phase. Figs. 4 an 5 show the global phase iagram of the J moel, as a function of temperature, electron ensity, chemical potential, an J /J. Again, the values of the coupling-strength ratios J /J for the consecutive panels in these figures are chosen so that they sequentially prouce the qualitatively ifferent phase-iagram cross-sections, thereby revealing the evolution in the global phase iagram. Again, the phase separation regions of the first-orer phase transitions are very narrow. For negative values of J /J, the antiferromagnetic phase is enhance, both near half-filling by the mechanism explaine after Eq. ( an, separately an to a lesser extent, isplacing the τ tj phase. The latter behavior is similar to that seen uner the introuction of quenche impurities, both experimentally [34 36] an from renormalization-group theory [7]. The τ Hb phase improves near the large antiferromagnetic region near half-filling. At J /J =, the τ Hb phase is foun in a wie range of hole oping, namely between 15 3%. Another interesting result is that the τ tj phase is epresse in temperature but remains stable in the interval of 33 37% hole oping. For positive values of J /J, the antiferromagnetic phase is reuce in the region near half-filling an enhance in the region near the τ tj phase, for reasons explaine after Eq. (. The τ Hb phase grows ajacently to the enhance antiferromagnetic region, being locate above the τ tj phase, causing a complex structure at higher hole opings an low temperatures. C. Conclusion We have shown that the tj moel with furtherneighbor antiferromagnetic (J or further-neighbor electron hopping (t interactions exhibits extremely rich

10 1 global phase iagrams. The phase separation regions of the first-orer phase transions are very narrow. Furthermore, these calculate phase iagrams are unerstoo in terms of the competition an frustration of nearest- an further-neighbor interactions. We fin that the two types of τ phases, previously seen in the Hubbar moel, occur in the tj moel with the inclusion of further-neighbor interactions. Acknowlegments This research was supporte by the Scientific an Technological Research Council of Turkey (TÜBİTAK an by the Acaemy of Sciences of Turkey. n p s m s Two-site eigenstates + φ 1 = 1 + 1/ 1/ φ = 1 { + } 1 1/ 1/ φ 4 = 1 { } φ 6 = 1 { } φ 7 = + 1 φ 9 = 1 { + } TABLE III: The two-site basis states, with the corresponing particle number (n, parity (p, total spin (s, an total spin z-component (m s quantum numbers. The states φ 3, φ 5, an φ 8 are obtaine by spin reversal from φ, φ 4, an φ 7, respectively. APPENIX A: ERIVATION OF THE ECIMATION RELATIONS The erivation of Eq. (7, first one in Ref.[5], is given in this Appenix. In Eq. (5 the operators β H (i, k an βh(i, j βh(j, k act on two-site an three-site states, respectively, where at each site an electron may be either with spin σ = or, or may not exist ( state. In terms of matrix elements, n p s m s Three-site eigenstates + ψ 1 = 1 + 1/ 1/ ψ =, ψ 3 = 1 { + } 1 1/ 1/ ψ 6 = 1 { } + ψ 8 = 1 { + } ψ 9 = 1 { + }, ψ 1 = 1 { } ψ 11 =, ψ 1 = 1 { + } + 1 ψ 13 = 1 { }, ψ 14 = 1 { + } 1 1 ψ 17 = 1 { } 1 ψ 18 = 1 { + } 3 + 1/ 1/ ψ = 1 6 { } 3 1/ 1/ ψ = 1 { } 3 + 3/ 3/ ψ 4 = 3 + 3/ 1/ ψ 5 = 1 3 { + + } TABLE IV: The three-site basis states, with the corresponing particle number (n, parity (p, total spin (s, an total spin z-component (m s quantum numbers. The states ψ 4 5, ψ 7, ψ 15 16, ψ 19, ψ 1, ψ 3, ψ 6 7 are obtaine by spin reversal from ψ 3, ψ 6, ψ 11 1, ψ 17, ψ, ψ, ψ 4 5, respectively. φ 1 G φ 1 φ φ 4 φ 6 φ 7 φ 9 φ t + µ + G φ 4 t +µ +G φ J + V + µ + G φ J + V + µ + G φ J + V + µ + G TABLE V: Block-iagonal matrix of the renormalize two-site Hamiltonian β H (i, k. The Hamiltonian being invariant uner spin-reversal, the spin-flippe matrix elements are not shown. u i v k e β H (i,k ū i v k = w j u i w j v k e βh(i,j βh(j,k ū i w j v k, (A1 where u i, w j, v k, ū i, v k are single-site state variables, so that the left-han sie reflects a 9 9 an the right-han sie a 7 7 matrix. Basis states that are simultaneous eigenstates of total particle number (n, parity (p, total spin magnitue (s, an total spin z-component (m s block-iagonalize Eq. (A1 an thereby make it manageable. These sets of 9 two-site an 7 three-site eigenstates, enote by { φ p } an { ψ q } respectively, are given in Tables III an IV. Eq. (A1 is thus rewritten as φ p e β H (i,k φ p = φ p u i v k u i w j v k ψ q ψ q e βh(i,j βh(j,k ψ q u,v,ū, v,w q, q ψ q ū i w j v k ū i v k φ p. (A There are five inepenent elements for φ p e β H (i,k φ p in Eq.(A (thereby leaing to five renormalize interaction constants {t, J, V, µ, G },

11 11 ψ 1 ψ 1 which we label γ p, γ p φ p e β H (i,k φ p for p = 1,, 4, 6, 7. (A3 ψ ψ 3 ψ µ t ψ 3 t µ ψ 9 ψ 1 ψ J + V + 3µ t ψ 1 t µ ψ 6 ψ 8 ψ 6 µ ψ J + V + 3µ ψ 11 ψ 1 ψ 11 µ t ψ 1 t 1 4 J + V + 3µ The iagonal matrix φ p β H (i, k φ p is given in Table V. The exponential of this matrix yiels the five renormalize interaction constants in terms of γ p, as given in Eq. (7. Furthermore, accoring to Eq. (A, each γ p is a linear combination of some ψ q e βh(i,j βh(j,k ψ q, ψ 13 ψ 14 ψ J + V + 3µ t ψ 14 t µ ψ 17 ψ 18 ψ J + V + 3µ ψ J + V + 3µ γ 1 = ψ 1 ψ 1 + ψ ψ + ψ 4 ψ 4, γ = ψ 3 ψ ψ 8 ψ 8 + ψ 1 ψ ψ 13 ψ 13, γ 4 = ψ 6 ψ ψ 9 ψ 9 + ψ 17 ψ ψ 18 ψ 18, ψ ψ J + V + 4µ ψ ψ V + 4µ γ 6 = ψ 1 ψ 1 + ψ ψ, ψ 4 ψ 5 ψ 4 1 J + V + 4µ ψ 5 1 J + V + 4µ γ 7 = ψ 11 ψ ψ ψ ψ 4 ψ 4, TABLE VI: iagonal matrix blocks of the unrenormalize three-site Hamiltonian βh(i, j βh(j, k. The Hamiltonian being invariant uner spin-reversal, the spin-flippe matrix elements are not shown. where ψ q ψ q ψ q e βh(i,j βh(j,k ψ q. In orer to calculate ψ q e βh(i,j βh(j,k ψ q the matrix blocks in Table VI are numerically exponentiate. [1] J. Hubbar, Proc. R. Soc. A 76, 38 (1963; 77, 37 (1964; 81, 41 (1964. [] P.W. Anerson, Science 35, 1196 (1987. [3] G. Baskaran, Z. Zhou, an P.W. Anerson, Soli State Commun. 63, 973 (1987. [4] M. Hinczewski an A.N. Berker, Eur. Phys. J. B 48, 1 (5. [5] A. Falicov an A.N. Berker, Phys. Rev. B 51, 1458 (1995. [6] M. Hinczewski an A.N. Berker, Eur. Phys. J. B 51, 461 (6. [7] M. Hinczewski an A.N. Berker, Phys. Rev. B 78, 6457 (8. [8] A.V. Puchkov, P. Fournier, T. Timusk, an N.N. Kolesnikov, Phys. Rev. Lett. 77, 1853 (1996. [9] C. Bernhar, J.L. Tallon, T. Blasius, A. Golnik, an C. Nieermayer, Phys. Rev. Lett. 86, 1614 (1. [1] G. Jüttner, A. Klümper, an J. Suzuki, Nucl. Phys. B 5, 471 (1998. [11] G. Jüttner, A. Klümper, an J. Suzuki, Physica B 59-61, 119 (1999. [1] M. Hinczewski, Ph.. Thesis, Massachusetts Institute of Technology, Chap. (5, [13] M. Ogata, M.U. Luchini, S. Sorella, an F.F. Assaa, Phys. Rev. Lett. 66, 388 (1991. [14] S. Krinsky an. Furman, Phys. Rev. Lett. 3, 731 (1974. [15] S. Krinsky an. Furman, Phys. Rev. B 11, 6 (1975. [16] S.A. Kivelson, V.J. Emery, an H.Q. Lin, Phys. Rev. B 4, 653 (199. [17] W.O. Putikka, M.U. Luchini, an T.M. Rice, Phys. Rev. Lett. 68, 538 (199. [18] E. agotto, Rev. Mo. Phys. 66, 763 (1994 [19] M. Suzuki an H. Takano, Phys. Lett. A 69, 46 (1979. [] H. Takano an M. Suzuki, J. Stat. Phys. 6, 635 (1981. [1] P. Tomczak, Phys. Rev. B 53, R5 (1996. [] P. Tomczak an J. Richter, Phys. Rev. B 54, 94 (1996. [3] P. Tomczak an J. Richter, J. Phys. A 36, 5399 ( 3. [4] C.N. Kaplan an A.N. Berker, Phys. Rev. Lett. 1, 74 (8. [5] O.S. Sarıyer, A.N. Berker, an M. Hinczewski, Phys. Rev. B 77, (8. [6] A.A. Migal, Zh. Eksp. Teor. Fiz. 69, 1457 (1975 [Sov. Phys. JETP 4, 743 (1976]. [7] L.P. Kaanoff, Ann. Phys. (N.Y. 1, 359 (1976. [8] A.N. Berker an M. Wortis, Phys. Rev. B 14, 4946 (1976. [9] S.R. McKay an A.N. Berker, Phys. Rev. B 9, 1315 (1984. [3] M.E. Fisher an A.N. Berker, Phys. Rev. B 6, 57 (198. [31] A.N. Berker, S. Ostlun, an F.A. Putnam, Phys. Rev. B 17, 365 (1978. [3] R.G. Caflisch an A.N. Berker, Phys. Rev. B 9, 179 (1984. [33] R.G. Caflisch, A.N. Berker, an M. Karar, Phys. Rev. B 31, 457 (1985. [34] M.-H. Julien, T. Fehér, M. Horvatić, C. Berthier, O.N. Bakharev, P. Ségransan, G. Collin, an J.-F. Marucco,

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