Doped antiferromagnets in high dimension

Transcription

1 PHYSICAL REVIEW B VOLUME 57, NUMBER JUNE 1998-I Dope antiferromagnets in high imension E. W. Carlson, S. A. Kivelson, an Z. Nussinov Department of Physics, University of California, Los Angeles, California V. J. Emery Department of Physics, Brookhaven National Laboratory, Upton, New York Receive 9 September 1997; revise manuscript receive 18 November 1997 The groun-state properties of the t-j moel on a -imensional hypercubic lattice are examine in the limit of large. It is foun that the unope system is an orere antiferromagnet, an that the ope system phase separates into a hole-free antiferromagnetic phase an a hole-rich phase. The latter is electron free if J4t an is weakly metallic an typically superconucting if J4t. The resulting phase iagram is qualitatively similar to the one previously erive for 2 by a combination of analytic an numerical methos. Domainwall or stripe phases form in the presence of weak Coulomb interactions, with perioicity etermine by the hole concentration an the relative strength of the exchange an Coulomb interactions. These phases reflect the properties of the hole-rich phase in the absence of Coulomb interactions, an, epening on the value of J/t, may be either insulating or metallic i.e., an electron smectic. S In this paper, the zero-temperature properties of the t-j moel of a ope antiferromagnet on a -imensional hypercubic lattice are evaluate using a systematic expansion in powers of 1/. For each property of interest the leaing behavior in the large- limit is compute, an in some cases, corrections up to orer 1/ 5 are obtaine. These results are obtaine by breaking the full Hamiltonian into an unperturbe piece H 0 an a perturbation H 1 an then reorganizing conventional perturbation theory in powers of H 1 into a 1/ expansion. Of course the partition of the Hamiltonian may be chosen for calculational convenience, since it oes not affect the results. The convergence of this expansion will not be aresse. Our proceure iffers from the extensive recent work on the relate problem of the Hubbar an Falicov-Kimball moels in large imension 1 in the way the large imension limit is taken. First of all, we o not assume that the ratio J/t of the exchange integral J an the hopping amplitue t is parametrically small as. The previous stuies assume that t is proportional to 1/ so that, when J is expresse in terms of the on-site interaction U, it follows that J/t 4t/U1/. The phase iagram will be stuie for parametrically small values of J/t in Sec. VIII, but our results are less complete in this case, because of the ifficulty of controlling perturbation theory in this limit. Seconly, the hypercubic lattice is bipartite, i.e., it can be broken into two sublattices, which we label black an re, such that the Hamiltonian has interactions only between sites on ifferent sublattices. This favors the classical Néel state, which has a uniaxial magnetization with opposite sign on the two sublattices. By contrast, earlier stuies, which were primarily concerne with the Mott transition an possible non-fermiliqui states of the Hubbar moel, assume a nonbipartite lattice which frustrates the Néel state. For both reasons, this previous work oes not she much light on the behavior of ope antiferromagnets. A notable exception is the work of van Dongen 2 on the small-u limit of the Hubbar moel on a hypercubic lattice which foun, as we o, that the weakly ope antiferromagnetic phase is unstable to phase separation, even if the parameters are scale as J eff /t4t/u 1/. Throughout this paper, units are chosen such that the lattice constant an Boltzmann s constant are all equal to one. I. SUMMARY OF RESULTS A. Results in large imension Our principal result is the global zero-temperature phase iagram as a function of J/t an hole concentration x, inthe limit of large, as shown in Fig. 1. It is immeiately clear that in most of the phase iagram, the unope orere antiferromagnetic phase coexists with a hole-rich phase. For J/t4, the hole-rich phase is electron free; otherwise it contains an exponentially small but nonvanishing concentration of electrons. In the intermeiate-coupling regime, 2J/t 4, the resiual attraction (J) between electrons is great enough to overcome the har-core repulsion, an leas to a BCS instability of the ilute metal, proucing an s-wave superconucting state at exponentially low-energy scales. At smaller values of J/t, the net interaction between electrons is repulsive. This implies that the system either remains metallic own to zero temperature or exhibits higher-angularmomentum pairing 3 via the Kohn-Luttinger mechanism. 4 A peculiarity of the phase iagram in Fig. 1 is that the bounary of the two-phase region intersects the J/t0 axis at a nonzero value of x. This is not likely to be correct in any finite imension. For small x an large but finite imension, we expect that in the limit J/t 0, the groun state is a ferromagnetic Fermi liqui, an hence the moel oes not phase separate. In Sec. VIII, we iscuss the behavior of the moel for J/t parametrically small, J/t1/. Here the 1/ /98/5723/ /$ The American Physical Society

2 57 DOPED ANTIFERROMAGNETS IN HIGH DIMENSION FIG. 1. Phase iagram of the t-j moel in the limit : Here x is the hole concentration (1x is the electron concentration. The phase bounary is given by Eq. 53, artificially setting 2. Two-phase labels the two-phase region, where a uniform ensity phase is thermoynamically unstable, SC labels a region of s-wave superconuctivity, an M labels a region of metallic behavior with repulsive interactions, which presumably has an ultralow temperature superconucting instability ue to the Kohn- Luttinger effect Ref. 4. expansion is slightly more ifficult to control, so our results, summarize in Fig. 2, are incomplete. The resulting conjectural phase iagram for large but finite emboies all the insights gaine from stuying the limit, but corrects the unphysical features of the phase iagram in Fig. 1. We have also stuie the behavior of one or two ope holes an the character of charge omain walls in the antiferromagnet. It will be seen that the latter are stabilize by a long-range Coulomb interaction. These stuies bring out an important characteristic of our large- expansion. Whenever FIG. 2. Conjecture phase iagram of the t-j moel for large but finite : This figure shoul be viewe as a blowup of the small J/t portion of Fig. 1. The horizontal line represents the small J/t extension of the phase bounary in Fig. 1; in fact, in large, this line woul be exponentially close to the top of the figure, but we have rawn it, as in Fig. 1, at a position obtaine by setting 2 in the large expression. The bounary of the fully polarize ferromagnetic metallic phase labele F is rawn in accor with the large expression in Eq. 75. There might be other, lower-energy phases, e.g., high-ensity stripe phases that coul occur below these two phase bounaries, in the region marke two-phase, especially close to the point of intersection. a hole moves in the antiferromagnetic backgroun, it may break a number of bons of orer at each hop. Consequently, for such processes, the physics is exchange ominate for large an it amounts to an expansion in powers of t/j. This is true of the motion of one or two holes an of omain-wall fluctuations in which holes hop into the environment. However the questions of phase separation, omain-wall phase equilibrium, an superconuctivity at low electron concentration are not subject to this limitation. The ensuing iscussion will be organize by orer of increasing hole concentration. To leaing orer in 1/ the states of minimum energy of a single hole lie precisely on the magnetic Brillouin zone which also is the Fermi surface of the noninteracting system with a half-fille ban. The massive egeneracy of these low-energy hole states is lifte by terms of O(1/ 4 ), an it is foun that the absolute minimum occurs at k(/2)1,1,1,...together with points relate by the point-group symmetry. Moreover, as euce previously by Trugman 5 in stuies of two holes in a twoimensional antiferromagnet, we fin that propagation of pairs of holes is no less frustrate than is the propagation of a single hole, because of a subtle effect of Fermi statistics. There is, however, an effective attraction 1/ between two holes ue to the fact that two nearest-neighbor holes break one less antiferromagnetic bon than two far-separate holes; this attraction always leas to a two-hole boun state. An interesting metastable state is a charge magnetic omain wall i.e., a 1 imensional hypersurface with finite hole concentration an suppresse magnetic orer. We have foun that the most stable omain wall has an electron ensity which is, to leaing orer in 1/, equal to that of the hole-rich phase which can exist in equilibrium with the antiferromagnet. Thus omain walls can be viewe as a form of local phase separation. Also the omain-wall configuration with the lowest surface tension i.e., energy per unit hyperarea of wall is the vertical site-centere antiphase iscommensuration in the antiferromagnetic orer; i.e., it is parallel to a single nearest-neighbor vector an o uner reflection through a site-centere vertical hyperplane. We have consiere the effect of weak, long-range Coulomb interactions as a perturbation. While this stuy is not exhaustive, we conclue that, for a substantial range of parameters, the groun state consists of a perioically orere array of optimal omain walls of the sort escribe above, especially when x is small but not too small. In this range of x, the groun state is insulating for J/t4, an metallic for J/t4. The latter phase is an electron-smectic 6 which exhibits crystalline orer in one irection an liquilike behavior in the transverse (1) irections. The liqui features are associate primarily with the motion of electrons along the omain wall, an they may be metallic or conense into a superconucting state. We have argue previously that the competition between a local tenency to phase separation in a ope antiferromagnet an the long-range Coulomb repulsion between holes prouces a large variety of intermeiate scale structures, incluing arrays of omain walls, which are significant features of ope antiferromagnets that we have calle frustrate phase separation. 7,8 However, these phenomena have not previously been erive from a microscopic magnetic moel. 9 It is particularly striking that, in the appropriate

3 CARLSON, KIVELSON, NUSSINOV, AND EMERY 57 TABLE I. Comparison of the results of exact numerical stuies Ref. 11 the row labele exact on the two-imensional spin- 1/2 Heisenberg antiferromagnet with the perturbative results in powers of 1/ erive in the present paper. We have been unable to fin corresponing exact three-imensional results. The imension is inicate by the arguments of the compute quantities. The rows labele upper an lower give the rigorous upper an lower bouns on the energies obtaine in the text. The approximate results are obtaine by setting yj /J z 1, V0, an 2 or 3 in the series expansion, evaluate to the state orer. All energies are measure in units of J/, an the magnetization m is quote in units in which g B 1, where B is the Bohr magneton. E AF (2) m(2) E AF (3) m(3) E 2leg (2) Exact Ref. 12 Upper Lower range of parameters, charge- an spin-ensity wave orer coexist with metallic, an even superconucting behavior. B. How large are 2 an 3? Large is, of course, only of acaemic interest; we are intereste in the physical imensions, 1, 2, an 3. The properties of the one-imensional electron gas 1DEG are well unerstoo 10 by now, an exhibit behavior that is quite imension specific. Moreover, for most of the conceivable orere states, the lower critical imension for long-range orer at zero temperature is one, so the 1DEG is not likely to be well unerstoo in terms of aiabatic continuity from large imension. However, long-range orer at zero temperature is quite robust in both two an three imensions, so there is every reason to expect that a 1/ expansion will capture the essential physics of many of the zero-temperature thermoynamic states. To test this conjecture, we woul like to make both qualitative an quantitative comparisons between the results of the large theory an any available exact, or well-controlle FIG. 3. Zero-temperature phase iagram of the two-imensional t-j moel, euce from numerical stuies of finite-size systems with up to 60 electrons, as well as from various analytic results. This figure is abstracte from Hellberg an Manousakis Ref. 14. numerical or analytic results in two an three imensions. Table I gives a quantitative comparison between the 1/ expansion an well-establishe numerical results for the unope system, i.e., for the spin-1/2 Heisenberg antiferromagnet. It can be seen that the groun-state energy can be obtaine from the low-orer expansion in powers of 1/ to 0.6% accuracy or better. By carrying the series to higher orer, an possibly oing a Paé analysis of the series, much improve accuracy for all physical quantities coul be expecte. In Sec. XI comparison will be mae between numerical results an the results of perturbation theory about the Ising limit Table II. Qualitative comparisons can be mae with the phase iagram of the two-imensional t-j moel which has been euce from combine analytic an numerical 13,14 stuies. Figure 3, abstracte from the work of Hellberg an Manousakis, 14 shows the phase iagram euce from numerical stuies of systems with up to 60 electrons. As in large, there is no thermoynamically stable zerotemperature phase with ilute holes for any J/t. Inee, asie from the behavior of the bounary of the two-phase region at very small J/t, the phase iagrams in Figs. 1 an 3 are similar. As suggeste above, when the pathologies of the formal limit are remove by taking into account the new processes that become important at parametetrically small values of J/t 1/2, one obtains for large but finite the phase TABLE II. Comparison of the results of exact numerical stuies the row labele exact on the two-imensional spin-1/2 Heisenberg antiferromagnet, with the perturbative results in powers of yj /J z erive in the present paper. The imension is inicate by the arguments of the compute quantities. The approximate results are obtaine by setting y1, V0, an 2 or 3 in the series expansion, evaluate to the state orer. All energies are measure in units of J/, an the magnetization m is quote in units in which g B 1, where B is the Bohr magneton. E AF (2) m(2) E AF (3) m(3) E 2leg (2) y y y Exact Ref Ref Ref. 12

4 57 DOPED ANTIFERROMAGNETS IN HIGH DIMENSION iagram shown in Fig. 2, which is topologically equivalent to Fig. 3. Of course, in 2, parametrically small values of J/t are not all that small, so there is no reason to expect quantitative agreement with the large- results. The critical value of J/tY c at which the phase-coexistence line eviates from x1 is 14 Y c in 2, an is rather well approximate by the value Y c 4 as. However, the slope of the phase coexistence line in 2 is much steeper than woul be euce from the large- theory. Similar etaile information on the three-imensional t-j moel is not available at this time, although arguments presente hitherto 7,13 suggest that the phase iagram is qualitatively similar to that in 2, consistent with the expectations from the 1/ expansion. Our calculation of the spectrum of one hole in an antiferromagnet may be compare to the numerical calculations of Dagotto et al. 15 on 2 systems with 1616 sites an J/t 0.4. They foun that the one-hole spectrum is well represente by the two-imensional version of the expression in Eq. 38, confirming the qualitative accuracy of the large expression. However the values of the parameters obtaine to leaing orer in 1/ are quantitatively quite far from the exact results, an this iscrepancy is mae worse by the inclusion of higher-orer terms. This is not unexpecte in view of the fact that large rives the motion of a single hole into the exchange-ominate limit. In particular, it is clear from Eq. 37 that the large- expansion gives a negative value for the banwith W in 2, unless J0.93t. Thus it is essential to compare the large- expansion to numerical results at large J/t. Specifically, from Eq. 36 with y1, the banwith for 2 is given by W/t2.125t/J1.83(t/J) 3.It woul be interesting to compare this result with numerical calculations for large J/t extrapolate to the thermoynamic limit. Martinez an Horsch 16 have foun that an approximate treatment of the motion of a single hole gives W/t2t/J for large J/t, which agrees very well with our large- result. Via a variational calculation, Boninsegni an Manousakis 17 fin W/t in the thermoynamic limit for 2 an J/t5, while Eq. 36 gives Finally, we can extrapolate to two imensions the character of the orere arrays of charge omain walls at low oping concentration an weak Coulomb interaction. Domain walls in two imensions are one-imensional lines an such orere arrays are known as stripe phases. Directly extrapolating the optimal large imensional omainwall structures to 2, we woul expect the stripes to be site-centere, vertical, antiphase omain walls in the antiferromagnetic orer, an to be metallic an possibly superconucting for J/tY c an insulating for J/tY c. In particular, if we extrapolate the leaing-orer expression for the electron ensity in the hole-rich phase, Eq. 54, to2, an then evaluate it for tj, we fin that such stripes shoul have approximately 0.31 ope holes per site along the stripe, an are thus metallic. Transverse to the stripe irection, such a phase is a generalize charge an spin-ensity wave state, in which the perio of the charge ensity wave is half that of the spin-ensity wave. 19 However, because of the electronic motion along the stripe, this phase is actually an electron smectic. 6 Unfortunately, there are no etaile microscopic two-imensional calculations to compare with these results, so we compare them with experiments on ope antiferromagnets. 20 C. Rigorous results In aition to our perturbative results in powers of 1/, we have obtaine rigorous upper an lower bouns on the groun-state energy of the unope system. These bouns, which are also quote in Table I, are shown to converge in the limit. D. Relation to experimental results on ope antiferromagnets in quasi-two an three imensions By now there are many examples of antiferromagnetic insulators that can be chemically ope. One prominent feature of these materials is the occurrence of high-temperature superconuctivity, a phenomenon for which the present results provie little irect insight. 21 However various spinan charge-orere states, as well as nearly orere fluctuating versions of such structures, have been observe in these systems 22,23 by irect structural probes, especially neutron scattering. Two concrete, an well stuie examples of this are the quasi-two-imensional perovskites La 2x Sr x NiO 4 an La 1.6x N x Sr x CuO 4, in both of which the ope hole concentration is equal to the Sr concentration x. The unope parent compouns with x0) are antiferromagnetic insulators with spin S1 for the nickelates an S 1/2 for the cuprates. In both cases, upon oping, the system forms 24,23 a stripe phase, in which the ope holes are concentrate in antiphase omain walls in the antiferromagnetic orer. At present it is not known whether the omain walls are site or bon centere in general. At higher oping concentration in the nickelates, there is strong evience that both types of omain wall coexist ue to interactions between the walls 25. However, there is a crucial ifference between the omain walls in the two materials: In the nickelates, there is one ope hole per site along the omain walls, an the ope system is, corresponingly, insulating. In the cuprate, the hole concentration along the omain wall is roughly one ope hole per two sites along the omain wall, an the system is corresponingly metallic, an even superconucting, espite the presence of almost static charge- an spin-ensity wave orer. This latter behavior is very suggestive evience of an electron-smectic phase. 6 In aition, the omain walls are iagonal in the nickelates 24 an vertical in the cuprates. 23,26 We feel that the occurrence of charge stripes in lightly ope antiferromagnets, the fact that these stripes are antiphase omain walls in the antiferromagnetic orer, an that they can be metallic or insulating, epening on the ratio of J/t, are physically robust features of the large theory which we expect to apply mutatis mutanis in 2. However, the preference for vertical versus iagonal stripes, an site-centere versus bon-centere stripes is likely to epen on microscopic etails, even in large imensions. Of more profoun importance is the fact that, while in large imensions the charge omain walls always crystallize at low temperature into an orere ensity wave, in low imensions, especially in two imensions, there is the very real possibility that the omain walls will be quantum

5 CARLSON, KIVELSON, NUSSINOV, AND EMERY 57 isorere ,6 In such a melte state, which might be either fully isorere isotropic or still retain orientational orer electron nematic, the sort of charge an spinorere states that are characteristic of the large theory occur as local correlations in the fluctuation spectrum; a microscopic electronic theory of such quantum isorere states is not available at present. II. THE MODEL The moel we consier is the straightforwar generalization of the usual t-j moel or t-j-v moel 30 : H 1 JS i S j Vn i n j t i, j c i, c j, H.c., i,j, 1 where S i, c i,, c i, is the spin of the electron on site i, n i c i, c i, is the number of an electron on site i, c i, creates an electron with a z component of spin equal to 1/2, are the Pauli matrices, there is a constraint of no ouble occupancy of any site, n i 0,1, an i, j signifies nearest-neighbor sites on the -imensional hypercubic lattice. In comparing results of ifferent calculations, it is important to note that there is more than one efinition of the t-j moel. Most commonly, 13,14 the t-j moel is efine as in Eq. 1 with VJ/4, but without the prefactor of 1/. Where it can be one reaily, we will quote results for arbitary V, but where this leas to complications, we will, for simplicity, analyze only the canonical case VJ/4. The aitional factor of 1/ is inclue so that the groun-state energy ensity remains finite in the limit; thus, in making a comparison with previous results on the 2 t-j moel, all energies compute here shoul be multiplie by 2. III. THE UNDOPED ANTIFERROMAGNET The unope system has one electron per site so that the electron hopping term (t) has no effect, an the system is manifestly insulating; the only remaining egrees of freeom are escribe by a spin-1/2 Heisenberg antiferromagnet with exchange coupling J. A. Rigorous bouns It is possible to obtain upper an lower bouns on the groun-state energy of the spin-1/2 Heisenberg moel which approach each other in the large -limit. An upper boun is obtaine by calculating the variational energy of the Néel state, which has alternating up an own spins on alternate sites, an gives a groun-state energy per site of E Néel J/4V. A lower boun for the groun-state energy can be obtaine 31 as follows: We express the full Hamiltonian as a sum of pieces, H jblack H j, 2 3 where the sum is over all sites on the black sublattice an H j is the exchange interactions between site j an its nearest neighbors which are necessarily on the re sublattice. The Hamiltonians H j are reaily iagonalize, but not simultaneously since they o not commute with each other. Nonetheless, the sum of the groun-state energies of H j gives the lower boun E lower (1 1 )J/4V for the groun-state energy per site. These results, combine, prove that the groun-state energy per site E AF of the Heisenberg moel approaches that of the classical Néel state in the limit of infinite imension, J/411/E AF VJ/4. B. Perturbative expression for the groun-state energy an sublattice magnetization We now embark on the erivation of results in a systematic expansion in powers of 1/. For this purpose, we will consier the Heisenberg moel as the isotropic limit of a Heisenberg-Ising moel. To begin with, we use Rayleigh- Schröinger perturbation theory to evaluate the properties of interest in powers of the XY coupling, an then reorganize this perturbation theory in powers of 1/. Thus, we take as our unperturbe Hamiltonian the Ising piece of the interaction, an treat the XY piece, H 0 1 J z S z i S z j Vn i n j, i, j H 1 J S x i S x j S y i S y j, 6 i, j as a perturbation. The groun state of H 0 is the twofol egenerate Néel state. H 1 has the effect of flipping pairs of spins, which because of the large coorination in high imensions means that the intermeiate states have energies that are proportional to. We have evaluate the perturbative expression for the groun-state energy per site E AF, an the grounstate sublattice magnetization m to fourth orer in y J /J z, but it woul be straightforwar using moern methos of high-temperature series expansion to exten these results to higher orer. The results are E AF V 4 J 1 y 2 21 y Oy 6 an m 2 1 1y y Oy It is clear that successive powers of y bring aitional powers of 1/ from the aitional energy enominators, as promise, so that the O(y 6 ) terms are actually O(y 6 / 5 ) an 4 5 7

6 57 DOPED ANTIFERROMAGNETS IN HIGH DIMENSION O(y 6 / 3 ) for the energy an magnetization, respectively. Reorganizing these expressions in powers of 1/ yiels an E AF V J 4 1 y 2 2 y y 2 8y y 2 163y O1/ 5 m y y 2 8y O1/ The appropriate expressions for the Heisenberg moel can now be obtaine by taking the limit y 1. C. Golstone moes an the long-wavelength physics Because the Néel state involves a broken continuous symmetry, we know that there must exist a gapless Golstone moe, the magnon. In the presence of Ising anisotropy, the magnon is massive, an is perturbatively relate to the single spin flip. Thus, we coul imagine using the same ecomposition of the Hamiltonian into an Ising an XY piece to compute the magnon spectrum perturbatively, an then reanalyze the expression in terms of the 1/ expansion. This is impractical, but it is instructive to see why. In 0th orer i.e., in the Ising moel, there is a set of N/2 egenerate excite states with excitation energy J z /2 an S z 1 obtaine by flipping a spin on the black sublattice, an there is a complementary set of excite states with S z 1 obtaine by flipping a spin on the re sublattice. These states resolve themselves into the two polarizations of the magnon ban upon performing egenerate perturbation theory in powers of yj /J z. The results of egenerate perturbation theory can be summarize in terms of an effective Hamiltonian, H eff J i, j b i b j, i& jblack 11 where b j creates a spin flip on site j an obeys boson commutation relations, b i,b j ij. To be concrete, we have consiere the magnon with S z 1, so we take the Hamiltonian to operate in the 1 spin-flip sector, jblack b j b j 1. The effective Hamiltonian can be solve by Fourier transform to give a magnon energy mag (k) jblack J 0,j expik R j. If we were actually intereste in the case in which there was substantial Ising anisotropy, we coul simply compute J i, j to the esire orer, since if i an j are n steps apart on the lattice, J i, j J z y/ n, an hence for small y, H eff is short range. It woul also seem that the same logic woul justify the self-same expansion for large, an inee as is implicit in the iscussion of the grounstate energy this is cruely true. However, even though J i, j falls rapily with n, the number of nth Manhattan neighbors grows just as rapily, i.e., as n. For nonzero wave vector, this oes not matter, as the far neighbors contribute to mag with rapily varying phases, an so the long-range tails of J i, j are unimportant. However, for k very near k0 or, equivalently, near k,,,...), all terms in the Fourier transform a in phase, so J i, j must be compute to infinite orer. Of course the point is that the low-energy Golstone moes have exceeingly small phase space in large imension, although they always ominate the temperature epenence of thermoynamic quantities at low enough temperature an the asymptotic ecay of correlation functions at large enough istances. Thus the Golstone moes are entirely unimportant in high imensions, except for physical quantities that strongly accentuate the lowest energy excitations. The way to stuy the Golstone behavior is in terms of a spin-wave expansion, again suitably reinterprete in terms of the 1/ expansion. We thus start by consiering the spin-s Heisenberg antiferromagnet in imensions using the stanar 32 Holstein-Primakoff bosons to obtain the spinwave spectrum in powers of 1/S. We will confine ourselves, here, to the lowest-orer theory, as it aequately illustrates the point. The sublattice magnetization is thus S in the classical Néel state, but receives a correction of orer S 0 from spin-wave fluctuations as ms 1 1 k 2S k O 1 S 2, 12 where the integral over k is over the first Brillouin zone, k 1 a1 cosk a 13 is the normalize structure factor, an the spin-wave energy is mag k JS 1 2 k 1O1/S. 14 Expaning the integran in powers of an employing k 2n k /2 2n!/4 n n!, 15 we obtain ms 1 1 2S 1 4 O 1 2O 1 2 S Clearly, in the limit S1/2, the spin-wave expansion can be re-expresse as an expansion in powers of 1/. Inee this expression agrees with the earlier result of perturbation theory in y, in the limit y1, as, of course, it must. For k near 0 or equivalently, near ), the spin-wave spectrum for S1/2) can be expane in powers of k to give the usual linear ispersion of the Golstone moe with spin-wave velocity cj//21o1/. 17

7 CARLSON, KIVELSON, NUSSINOV, AND EMERY 57 We can also compute the transverse spin-spin correlation function to leaing orer in 1/S, an examine the resulting expression at large. Using stanar results, it is easy to see 32 that S i x S j x S i y S j y Se i R ij k 2 e 1k i R ij 1k cosk R ij 1 2 k 1OS, 1 18 where R ij R i R j. In the large R limit, this integral can be evaluate by approximating the integran by its small k expression, 1 2 (k) 1 k 2, which yiels the Golstone behavior S i x S j x S i y S j y S R e 2S R 1 0 e R i R ij, e i R ij 19 where RR ij, R 0 /2e, is the gamma function, an in the secon line we have use Stirling s formula for large. It is easy to see that the integral evaluate above is ominate by values of k 2 2 /R 2, an since the small k approximation is vali only so long as k 2, the Golstone behavior is only vali for RR 0, as is suggeste by the form of the result. The most efficient way to evaluate properties of the system at low but nonzero temperature is to use the 1/ expansion to compute the fully renormalize zero-temperature parameters that enter the O3 nonlinear- moel which governs the Golstone moes, namely the spin-wave stiffness s an the transverse uniform susceptibility 0. The susceptibility can reaily be compute perturbatively in powers of 1/S, an the resulting expression reexpresse in powers of 1/ as 0 1 4J z SO 3 1 2O 1 S 2 S In terms of this, s can be compute from the relation 33 s c 2 0, where c is the spin-wave velocity given in Eq. 17. IV. ONE HOLE IN AN ANTIFERROMAGNET 21 The one-hole problem has a structure that is nominally like that of the one-magnon problem, but ae factors of 1/ make the perturbative approach tractable for all values of k. We efine as the unperturbe Hamiltonian H 0, the Ising limit of the t-j moel, with J t0. A. The minimal hole with S z 1/2 There are 2N egenerate one-hole groun states of H 0, where the factor of 2 is ue to the global egeneracy of the Néel state, an the factor of N which is the number of lattice sites comes from the locations of the empty site the hole. Henceforth we focus only on the states in which the magnetization is up on the black sublattice. These states are, in turn, separate into isjoint Hilbert spaces labele by the conserve quantum number, the total z component of spin, since a hole on a black sublattice site has S z 1/2 an one on the re sublattice has S z 1/2. For concreteness, we will focus on the N/2 egenerate states corresponing to a hole on the black sublattice. We now use egenerate perturbation theory to construct the effective Hamiltonian of one hole H eff 1 t ij c i c j t ij c i c j, i& jblack i&jre 22 where c j is the fermionic creation operator for a hole on site j, an for Hermiticity, t ij t ji. Once the effective Hamiltonian is compute, its eigenstates an eigenvalues can be foun by Fourier transform: hole k jblack t 0 j expik R j. 23 To begin with, we stuy the perturbative expressions for the iagonal term, 0 t ii. 0 2V J z y z2 21 Oy 4 Oz 4 Oy 2 z 2 2V J z 21 y2 16z 2 4 y2 32z O1/ 3, 24 where zt/j z an yj /J z. The corrections to the hole self-energy which are inepenent of y are the famous string corrections to the hole self-energy, of which the retraceable paths of Brinkman an Rice 34 are a subclass. Next we stuy the coupling between secon Manhattan neighbor sites nearest-neighbors on the black sublattice. There are 2(1) true secon neighbors, reache by taking a step to the nearest-neighbor site in one irection, an then a secon step in an orthogonal irection, an there are 2 straight line secon neighbors reache by taking two steps in the same irection. For i an j true secon neighbors, t ij 2 Ising, where the factor of 2 in the efinition takes account of the fact that there are two minimal paths to the true secon neighbor. Here is given by

8 57 DOPED ANTIFERROMAGNETS IN HIGH DIMENSION tyz y N y z D Oy 3 Oz 4 Oz y 2 tyz y 41 5 y 22z O1/ 2, 25 with D N an N , 32 an D ; 27 comes from processes in which, in lowest orer, the hole hops twice, followe by a spin exchange which repairs the resulting amage to the spin orer. In aition, there is a contribution 32tz 5 Ising Oz2 tz5 6 61O 1, 28 which comes from a process in which a hole circles a plaquette one an a half times, thus eating its own string. This process, which was iscovere by Trugman 5 for 2, survives even in the Ising limit y0. However, Ising is higher orer in powers of 1/ an so is negligible, even when tj. For straight-line secon neighbors, t ij Ising where where 4tyz 21431y 1 N 2 43 y2 D 2 z N 3 2 D 3 Oy3 Oz 4 Oz y 2 tyz y 41 5 y 22z O1/ 2, 29 N , 30 D , 33 an Ising is the corresponing Trugman term which is O(tz 9 / 9 ). Because they are higher orer in 1/, we will henceforth ignore Ising an Ising relative to. However, we shall see below that the ifference 4tzy OyOz 2 3tzy y O 2, 34 6 although smaller than by a factor of 1/, plays a critical role in etermining the ban structure of the minimum energy hole in an antiferromagnet. The sign of these terms eserves some comment. Since the lattice structure efine by t ij is not bipartite, the sign of the matrix elements is physically significant. In the present case, since the leaing-orer contributions to come from thir-orer perturbation theory, the resulting matrix element is positive. These expressions may be combine to obtain an expansion for the banwith W of a single hole by aing the contributions from the ifferent hopping processes. Each hop contributes a factor 2 or 2 to W, so W Then, using Eqs. 25 an 29 an expaning in powers of 1, W t 2yz1 4 y 5yz 2 1 y 22z O This result oes not make sense unless W0 or z y 1 2. y 37 Since zt/j z, this illustrates the fact that the motion of a single hole is exchange ominate in the large- expansion.

9 CARLSON, KIVELSON, NUSSINOV, AND EMERY 57 It is important to note that the leaing-orer expression for contains one more power of 1/ than the corresponing matrix element in the effective Hamiltonian for one magnon. Inee, the leaing-orer behavior of the matrix elements connecting sites separate by 2n nearest-neighbor steps (2nth nearest Manhattan neighbors is t 2n tz 2n1 y n / 3n, where the factor of tz 2n1 t 2n comes from the minimum number of hops for the hole to propagate this istance, the factor of y n J n reflects the minimum number of spin flips neee to restore the spins to a groun-state configuration following the passage of a hole, the factor of (3n1) comes from the accompanying energy enominators at this orer of perturbation theory, an one aitional factor of 1/ comes from the overall normalization of the Hamiltonian. Because of these extra factors in the expression for t ij, its contributions to the hole energy fall rapily with istance in high imensions, where the number of 2n-step paths is 2(2 1) 2n1, an so the number of 2nth Manhattan neighbors can grow no faster than (2) 2n. Thus, the longer range pieces of t ij can be neglecte for any value of k in large enough imension. The effective Hamiltonian obtaine by retaining only terms out to secon Manhattan neighbors is given by hole k k 4 cos 2 k a. a1 38 If we ignore the small ifference, (), then the minimum energy hole states are locate on the 1 imensional hypersurface, (k)0; since the ban structure for the noninteracting tight-bining moel is free 2t(k), an that moel is, in turn, particle-hole symmetric, this hypersurface is precisely the Fermi surface of the half-fille ban in the absence of interactions. With higher-orer terms in powers of 1/ which prouce a nonzero value of ()0) the minimum energy of a single hole occurs at k(1/2) an the 2 1 symmetry-relate points. We emphasize that, although we have treate the effects of t perturbatively, we have not mae a small-t approximation. The present results are vali for arbitrary t/j, so long as it is not parametrically large i.e., so long as tj). Nonetheless, because each hop of the hole may break O() bons, the large- limit is exchange ominate. B. Magnetic polarons with larger spin In low imensions, an for tj, it is believe that a single hole in an antiferromagnet prouces a ferromagnetic bubble in its vicinity, or, more precisely, that there is a series of level crossings as a function of t/j at which the total z component S z of the spin of the groun state for a single hole state steaily increases. However, in the large- limit, the antiferromagnetic energy always ominates unless t is parametrically larger than J, i.e., unless t x J, where x is a positive exponent which we will estimate below. Such parametrically large values are beyon the scope of the present analysis. To estimate the magnitue of t/j at which ferromagnetic bubbles first appear, we consier a straightforwar generalization to 2) of the calculation of Emery, Kivelson, an Lin 13 for a hyperspherical ferromagnetic polaron in the large size limit where iscrete lattice effects can be neglecte. We balance the magnetic energy lost in the volume of the polaron against the zero-point energy compute in the effective-mass approximation to localize the hole in the interior of the polaron. This results in a polaron with a raius, L2t/A (VE AF J/4) 1/(2), where A is the area of the unit -imensional hypersphere A 2 //2/2e/, 39 an in the final line, we have use Stirling s approximation for large ; the polaron has spin S z A L /. By efinition, the spin of the polaron must be substantially greater than 1, which means that t 2 J. We conclue that in the large- limit, the low-energy hole branch is always the naive, S z 1/2 vacancy state, an that local ferromagnetism is never a relevant piece of the one-hole physics. V. EFFECTIVE INTERACTIONS BETWEEN TWO HOLES Broaly speaking, the effective interactions between two holes are of two kins, potential, which are inuce by istortions in the antiferromagnetic orer, an ynamic, which minimize the zero-point kinetic energy of a hole. At long istances, the effective potential of interaction can be compute by consiering change in the magnetic Hamiltonian inuce by two static holes at lattice sites 0 an i, i H 2 J j 0 S i S j k S 0 S k, 40 where j (i) signifies the sum over the nearest-neighbor sites of i. Then, on integrating out the magnetic egrees of freeom, we obtain i V eff R i J 2 / t j 0 k S 0 S k S 0 S k, TS i t S j ts i S j 41 where are higher-orer terms in powers of J, which are also of shorter range as a function of R i, t is the imaginary time, an T is the imaginary-time orering operator. It is straightforwar 35 to etermine from linear spin-wave theory that V eff 1/R 21, 42 which is a short-range potential in the sense that the integral over space of V eff is noninfinite in all imensions greater than 1. The integral over space of V eff itself is easily seen to be 0. Moreover, the long-istance tails of V eff ecrease in importance as increases. For this reason, we will ignore the power-law tails of V eff an simply consier its ominant, short-istance pieces.

10 57 DOPED ANTIFERROMAGNETS IN HIGH DIMENSION The nearest-neighbor interaction between two holes which is certainly attractive in the canonical case, V J/4) can reaily be compute from perturbation theory in powers of yj /J z : V eff ê 1 V J z 4 1Oy 4 V J z 4 1O1/3. 43 There is a consierably weaker interaction which, in fact, is repulsive between secon nearest-neighbor holes: J V eff z y 2 ê 1 ê Oy 2 J zy O1/. 44 Inee, to this orer, the effective interaction is the same for all secon Manhattan neighbors, V eff (2ê 1 )V eff (ê 1 ê 2 )1 O(1/ 3 ). Clearly, for further Manhattan neighbors, the effective interactions are own by aitional powers of 1/. The kinetic terms, in general, generate fairly complicate interactions of the form T eff ijkl T ijkl c i c j c k c l, 45 where, as before, c i is a fermionic creation operator for a hole at site j. However, in large, it is strongly ominate by its short-range components, of which the ominant terms are a potential interaction between nearest-neighbor holes which renormalizes V eff (ê 1 ) an a pair-hopping term. Inee, combining the potential an kinetic terms to leaing orer in 1/, we fin a two hole contribution to the effective Hamiltonian i.e., the interaction part of the effective Hamiltonian, of which Eq. 22 is the noninteracting piece: H 2 eff U eff i, j c i c j c j c i T eff i, j,k c j c i c j c k O1/ 3, 46 where in the pair-hopping term i, j,k signifies a set of sites such that i an k are both nearest neighbors of j which we efine to inclue the case ik), an U eff V J z 8z Oy 4 Oz 4 an Oy 2 z 2 V J z 1 4 4z2 2 O1/ 3 T eff 11 tz y 41 Oy2 Oz 2 tz y 4O1/ The pair-hopping term T eff has an interesting history: In early work on high-temperature superconuctivity, it was often claime that, whereas the motion of a single hole is inhibite by antiferromagnetic orer, pair motion appears to be entirely unfrustrate. It was suggeste that this might inicate a novel nonpotential source of an attraction between holes which coul be the mechanism of high-temperature superconuctivity. At first sight, the fact that T eff 2, while the single-particle hopping term is 3, appears to support the valiity of this iea in large. However, the fallacy of this argument was reveale in the work of Trugman, 5 who showe that this moe of propagation of the hole pair was frustrate by a quantum effect which originates in the fermionic character of the hole. In large, this frustration effect is particularly graphic. Pair bining is enhance if we ignore single-particle hopping, an iagonalize H 2 eff. Of course, any state in which the two holes are farther than one lattice site apart are eigenstates of H 2 eff, so long as terms of this range are neglecte because they are of higher orer as in Eq. 46. For the states in which the two holes are nearest neighbors, H 2 eff can be block iagonalize by Fourier transform, with the result that there are bans of eigenstates labele by a ban inex an a Bloch wave vector k. It is straightforwar to see that none of these bans isperses their energies are inepenent of k) an that 1 of these bans have energy U eff, while the remaining ban has energy U eff 2T eff. This final ban, which feels the effect of pair propagation, has the highest energy. On the other han, if the holes were bosons, this latter ban woul have energy U eff 2T eff, which is much closer to what one might have expecte. It follows from this argument that coherent propagation of a pair is not an effective mechanism of pair bining an that the short-range attraction between two holes in an antiferromagnet arises from the fact that two nearest-neighbor holes break one less antiferromagnetic bon than two far-separate holes. This interaction is sufficient to prouce a two-hole boun state because the one-hole spectrum has an essentially egenerate ban minimum along the 1 imensional magnetic Brillouin zone. As in the Cooper problem, this gives a constant ensity of states at low energy an any attractive interaction is sufficient to prouce a boun state. VI. FINITE HOLE CONCENTRATION We have suggeste 13,8 that, in general, a ope antiferromagnet in 2 will phase separate into a hole-free antiferromagnetic region an a hole-rich region. There is now substantial evience, both numerical 13,14 an analytical, 13,36,37 that this is the case for the t-j moel in 2. Phase separation is, of course, a first-orer transition, so it must be stuie by comparing the total energy of various caniate homogeneous an inhomgeneous states to fin the true groun state at fixe hole ensity. In large imension, there are many metastable states which are, in a sense, local groun states of given character. While the large- limit allows us to compute the energy an character of a given caniate groun state exactly, it is almost never possible to prove that we have actually ientifie the global groun state. Specifically, we have compute the energy of various caniate states as iscusse below an foun that, of these, the lowest-energy state is phase separate into an unope antiferromagnet an a hole-

11 CARLSON, KIVELSON, NUSSINOV, AND EMERY 57 rich phase with a very low electron ensity, i.e., with hole concentration equal to or nearly equal to 1. Moreover, given that the interaction between two holes is strongly attractive of orer the hole banwith at short istances, an weakly attractive at long istances, we feel that it is extremely unlikely that any ilute hole-liqui or hole-crystal phase in the antiferromagnet is stable in large. Below, we show that the same instability shows up in a ilute omain-wall phase in which the holes are concentrate on an array of wiely separate omain walls. What this means is that, uner the assumption that we have not overlooke a lower-energy state which we consier unlikely, we can obtain a complete an exact unerstaning of the zero-temperature phase iagram in the limit by consiering the phase coexistence between the unope antiferromagnet which we have alreay characterize an a very hole-rich or ilute electron phase. We emphasize that the physics of phase separation an omain walls Sec. VII in large- is not exchange ominate because it oes not involve breaking a large number of bons. A. Properties of ilute electrons in large We now consier the groun-state properties of ilute electrons on a -imensional hypercubic lattice, for large. Near the bottom of the ban, the electron ispersion relation is approximately quaratic, i.e., (k)2ttk 2 / is a goo approximation so long as each component of k is small compare to 1, or typically, that k 2. Also in this limit, the interactions between electrons are weak since they rarely approach each other, an so can be ignore to first approximation. Thus, we will begin by consiering the properties of the noninteracting, quaratically ispersing electron gas in imensions. For this problem, the Fermi momentum as a function of the chemical potential,, isk F (2t)/t for 2t, an the corresponing ensity is 1O, 1 49 n 2A k F 2 2 e 2 F k where A is given in Eq. 39, a spin-egeneracy factor of 2 has been inclue, an the final equality uses the large expression for A. Note that whenever the conition k F 2 is satisfie, the electron ensity is exponentially small for large, so our approximations are exponentially accurate. The energy per site of this system can be compute reaily: E gas 2tn1k F 2 /24. B. Conitions of thermoynamic equilibrium 50 In general, for two phases to be in thermoynamic equilibrium, they must have equal chemical potentials,. However, here, the unope antiferromagnetic phase is incompressible, so that the zero-temperature chemical potential is unetermine. Then the conition for the electron gas to be in equilibrium with the antiferromagnet is of. Thus, if E AF 2t, the only thermoynamically stable zero-temperature phases of the t-j moel are the unope antiferromagnet an the vacuum no electrons. On the other han, if E AF 2t, phase coexistence is possible between the antiferromagnet an a metallic phase with allowe electron ensities, nn max, where n max is the electron ensity at the equilibrium value of. We shall see shortly that both E gas an n are exponentially small at large, so that to exponential accuracy, E AF. 52 If we use the large expression for E AF, we conclue that the metallic state is stable only if J4t1O(1/), an that if this conition is satisfie, the maximum stable ensity of the metallic phase is n max 2 e4tj 4t 1O Notice that this quantity is small an hence our approximations are justifie, even in the limit tj, where n max 2/ e /2. 54 C. Effective interactions in the metallic state, an the conitions for superconuctivity Since the electron ensity in the metallic state is small, interaction effects are ominate by pairwise collisions between electrons. In the triplet channel, there is a nearestneighbor electron-electron repulsion of strength (4V J)/4, while in the singlet channel there is an infinite, on-site repulsion, an a nearest-neighbor attraction of strength (4V3J)/4. For the canonical choice of V J/4, which we aopt in most of this section for simplicity of notation, the attractive interaction in the singlet channel is simply J/. We can look for evience of a simple s-wave instability of the metallic state in the low-ensity limit by stuying the conitions for the existence of a solution to the BCS gap equation. First, consier the unperturbe noninteracting thermal Green function at low, but finite temperature Gk tanh 2 k, 55 2k where (k)2t(k) an (k) is efine in Eq. 13. Now it is straightforwar to show that, in the singlet channel, the BCS equation for the transition temperature T c may be written in terms of the corresponing real-space Green functions G 0 1 N k Gk, 56 E AF E gas /1n, 51 where E gas is the groun-state energy per site of the electron gas an n is the electron ensity, both of which are functions G 1 2 N k Gk a1 cosk a tn k k Gk, 57