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This is an expansion in 1/9, starting with the order (1/S)~?. This suggests that in the classical limit, S — oo, we can get away with keeping only the first two terms (keeping only the first is too crude since it is a constant energy). The roots of operators are rather inconvenient and the practical usefulness of the scheme lies in the expansion of these roots in orders of ala; /2S. For example, the nearest-neighbor Hamiltonian and is thus the fully polarized ferromagnetic ground state. Holstein-Primakoff bosons are therefore better suited for expansions around this polarized state, while Schwinger bosons are better suited in the paramag- netic phase. Both schemes are exact, though.

Figure 38 This is an expansion in 1/9, starting with the order (1/S)~?. This suggests that in the classical limit, S — oo, we can get away with keeping only the first two terms (keeping only the first is too crude since it is a constant energy). The roots of operators are rather inconvenient and the practical usefulness of the scheme lies in the expansion of these roots in orders of ala; /2S. For example, the nearest-neighbor Hamiltonian and is thus the fully polarized ferromagnetic ground state. Holstein-Primakoff bosons are therefore better suited for expansions around this polarized state, while Schwinger bosons are better suited in the paramag- netic phase. Both schemes are exact, though.

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This is, not surprisingly, the field of a magnetic dipole. We can thus identify m; with the magnetic dipole moment of the current loop. Tha. snanaetin (AGHA lAY) mranewant 4a fave] Writing unit vectors with a hat, A = a/a, we obtain Compare the orbital angular momentum of the electron: We had found 1 = m,R?w. We obtain the relation
This is, not surprisingly, the field of a magnetic dipole. We can thus identify m; with the magnetic dipole moment of the current loop. Tha. snanaetin (AGHA lAY) mranewant 4a fave] Writing unit vectors with a hat, A = a/a, we obtain Compare the orbital angular momentum of the electron: We had found 1 = m,R?w. We obtain the relation
In quantum physics, it is common to write angular momenta as dimensionless quantities by drawing out a factor of h. Thus we replace s > sh, s/h > s,m, > m, = —gyps etc. We use this convention from now on. The interaction of the electronic charge with the electromagnetic field it generates leads to small corrections (“anomalus magnetic moment”), which can be evaluated within QED to very high accurancy. One finds The derivation of the Pauli equation gives s-s = 3/4. We know from introductory quantum mechanics that any angular momentum operator L has to satisfy the commutation relations [L,, Li] = 1 >0,,, €kimLm, [Lx, L?] = 0, which define the spin algebra su(2). (These commutation relations are to a large extend predetermined by the commutation relations of rotations in three-dimensional space—a purely classical concept. This does not course not fix the value of h, though.) It is also shown there that this implies that L? can have the eigenvalues /(1 +1) with J = 0, 3 1, 3, 2,... and Ly, k = x,y, z, can than have the eigenvalues m=-—l,-l+1,...,0-1,1.
In quantum physics, it is common to write angular momenta as dimensionless quantities by drawing out a factor of h. Thus we replace s > sh, s/h > s,m, > m, = —gyps etc. We use this convention from now on. The interaction of the electronic charge with the electromagnetic field it generates leads to small corrections (“anomalus magnetic moment”), which can be evaluated within QED to very high accurancy. One finds The derivation of the Pauli equation gives s-s = 3/4. We know from introductory quantum mechanics that any angular momentum operator L has to satisfy the commutation relations [L,, Li] = 1 >0,,, €kimLm, [Lx, L?] = 0, which define the spin algebra su(2). (These commutation relations are to a large extend predetermined by the commutation relations of rotations in three-dimensional space—a purely classical concept. This does not course not fix the value of h, though.) It is also shown there that this implies that L? can have the eigenvalues /(1 +1) with J = 0, 3 1, 3, 2,... and Ly, k = x,y, z, can than have the eigenvalues m=-—l,-l+1,...,0-1,1.
We now call the electrons with spin parallel to S “spin up” (t) and the others “spin down” (|). Furthermore s; and 1; commute. We can thus replace, in the expectation value, s; by S/2S' for spin up and by —S/2S for spin down, respectively. (Note that s; has magnitude 1/2.) Thus
We now call the electrons with spin parallel to S “spin up” (t) and the others “spin down” (|). Furthermore s; and 1; commute. We can thus replace, in the expectation value, s; by S/2S' for spin up and by —S/2S for spin down, respectively. (Note that s; has magnitude 1/2.) Thus
The typical timescale of this rotation should be h/|A|. For “slow” experiments like magnetization mea- surements, only the time-averaged moment mops will be observable. To find it, we project my onto the direction of the constant J: 3ut my does not commute with the Hamiltonian because of the spin-orbit coupling term AL-S. (J does commute but S does not.) Thus m, is not a constant of motion, whereas J is. We can think of S and L ind thus m, as rotating around the fixed vector J:
The typical timescale of this rotation should be h/|A|. For “slow” experiments like magnetization mea- surements, only the time-averaged moment mops will be observable. To find it, we project my onto the direction of the constant J: 3ut my does not commute with the Hamiltonian because of the spin-orbit coupling term AL-S. (J does commute but S does not.) Thus m, is not a constant of motion, whereas J is. We can think of S and L ind thus m, as rotating around the fixed vector J:
We have thus found that Q@ = 0. We now make the radius of S arbitrarily small and find that there is no singularity at r = 0. The first factor Q is a matrix acting on my. But Q does not distinguish any direction in space and thus has to be of the form @ = q1 with q € R. On the other hand, the trace of Q is What we have achieved so far is to make the singularity explicit as the second term. The resulting interaction energy between the nuclear moment and the spin moment of an electron is
We have thus found that Q@ = 0. We now make the radius of S arbitrarily small and find that there is no singularity at r = 0. The first factor Q is a matrix acting on my. But Q does not distinguish any direction in space and thus has to be of the form @ = q1 with q € R. On the other hand, the trace of Q is What we have achieved so far is to make the singularity explicit as the second term. The resulting interaction energy between the nuclear moment and the spin moment of an electron is
partially filled d-shell on the outside of ion partially filled f-shell on the inside of ion Crystal field effects are, as the name implies, the effects of the crystal on an ion. We consider the most important cases of 3d (4d, 5d) and 4f (5f) ions. In stable states, these ions typically lack the s-electrons from the outermost shell and sometimes some of the d- and/or f-electrons. d- and f- ions behave quite differently in crystals. We consider the examples for Fe’* and Gd?"
partially filled d-shell on the outside of ion partially filled f-shell on the inside of ion Crystal field effects are, as the name implies, the effects of the crystal on an ion. We consider the most important cases of 3d (4d, 5d) and 4f (5f) ions. In stable states, these ions typically lack the s-electrons from the outermost shell and sometimes some of the d- and/or f-electrons. d- and f- ions behave quite differently in crystals. We consider the examples for Fe’* and Gd?"
Here, 6 is a number. How does this single-ion-anisotropy term split the 2J + 1 states of a multiplet? the last two terms are proportional to the idendity operator and thus irrelevant for the splitting. The first term we rewrite as
Here, 6 is a number. How does this single-ion-anisotropy term split the 2J + 1 states of a multiplet? the last two terms are proportional to the idendity operator and thus irrelevant for the splitting. The first term we rewrite as
Working in the eigenbasis {|m.y)} of J,, we see that J{ only connect states that differ in mz by 4. Further- more, the states |m.) are eigenstates of the mixed terms J}. J*, J, J_J,J_ etc. and of J+. This means that we can express all the mixed terms as polynomials of J,. Since spin inversion commutes with the anisotropy term, all terms in these polynomials have to be even. The resulting matrix (mJ|J? + Jj + J2|m’;) has the general structure
Working in the eigenbasis {|m.y)} of J,, we see that J{ only connect states that differ in mz by 4. Further- more, the states |m.) are eigenstates of the mixed terms J}. J*, J, J_J,J_ etc. and of J+. This means that we can express all the mixed terms as polynomials of J,. Since spin inversion commutes with the anisotropy term, all terms in these polynomials have to be even. The resulting matrix (mJ|J? + Jj + J2|m’;) has the general structure
Over a larger range of J we find the following spectra, where each dot now represents a multiplet without indication of its multiplicity:
Over a larger range of J we find the following spectra, where each dot now represents a multiplet without indication of its multiplicity:
If the symmetry is not cubic, anisotropy terms tend to appear already at second order in r and, conse- quently, in J. For an orthorhombic lattice the second-order terms are
If the symmetry is not cubic, anisotropy terms tend to appear already at second order in r and, conse- quently, in J. For an orthorhombic lattice the second-order terms are
If Hund’s 1st and 2nd rules win over the crystal field, the only possible values for LD are 0, 2, 3. Together with the schemes on p. 18 we obtain the following splittings: If the symmetry is lower than cubic, the levels will be split further. For sufficiently low symmetry, all degeneracies (except for spin degeneracy) are lifted. In cases where the ground state would still be degenerate for a putative crystal structure, the crystal will often distort to break the degeneracy and lower the symmetry. This is the Jahn-Teller effect. For example, in the 3d° configuration of Cu?* or in the 3d* configuration of Mn?*, the ground state would be twofold degenerate in a cubic environement. A distortion of the crystal causes two energy contributions:
If Hund’s 1st and 2nd rules win over the crystal field, the only possible values for LD are 0, 2, 3. Together with the schemes on p. 18 we obtain the following splittings: If the symmetry is lower than cubic, the levels will be split further. For sufficiently low symmetry, all degeneracies (except for spin degeneracy) are lifted. In cases where the ground state would still be degenerate for a putative crystal structure, the crystal will often distort to break the degeneracy and lower the symmetry. This is the Jahn-Teller effect. For example, in the 3d° configuration of Cu?* or in the 3d* configuration of Mn?*, the ground state would be twofold degenerate in a cubic environement. A distortion of the crystal causes two energy contributions:
Thus the total ground-state energy generically has a minimum for non-zero distortion: 3.3.2 Quenching of the orbital angular momentum
Thus the total ground-state energy generically has a minimum for non-zero distortion: 3.3.2 Quenching of the orbital angular momentum
(This argument fails if the ground state is degenerate since then the ion can be in a superposition of the degenerate states with complex coefficients.) We can draw the important conclusion that in transition-metal salts the magnetization is mainly carried by the electron spins, not the orbital angular momentum. The remaining term —A? > a0 Si ApvSp in Hs describes the anisotropy energy of the spin as opposed to the orbital angular momentum discussed earlier. Higher-order anisotropy terms are obtained in higher orders of pertubation theory. Not surprisingly, the allowed anisotropy terms are dictated by crystal symmetry so that our earlier discussion carries over. is imaginary. Since we have already seen that it must be real, we conclude
(This argument fails if the ground state is degenerate since then the ion can be in a superposition of the degenerate states with complex coefficients.) We can draw the important conclusion that in transition-metal salts the magnetization is mainly carried by the electron spins, not the orbital angular momentum. The remaining term —A? > a0 Si ApvSp in Hs describes the anisotropy energy of the spin as opposed to the orbital angular momentum discussed earlier. Higher-order anisotropy terms are obtained in higher orders of pertubation theory. Not surprisingly, the allowed anisotropy terms are dictated by crystal symmetry so that our earlier discussion carries over. is imaginary. Since we have already seen that it must be real, we conclude
We thus find a crossover from LS behavior for kgT < AF to predominantly HS behavior for kpT >> AE, driven by the higher degeneracy of the HS state, i.e., by entropy. where Gig and Gyg are the degeneracies of the LS and HS state, respectively. If these are only due to the spin we have G = 25+ 1. In our example with Sis = 0 and Syg = 2 we obtain
We thus find a crossover from LS behavior for kgT < AF to predominantly HS behavior for kpT >> AE, driven by the higher degeneracy of the HS state, i.e., by entropy. where Gig and Gyg are the degeneracies of the LS and HS state, respectively. If these are only due to the spin we have G = 25+ 1. In our example with Sis = 0 and Syg = 2 we obtain
This shows that Kinym. — 1/2 Jm,m. > 0 so that the corrected Coulomb term is reduced but still repulsive. Even more importantly, we obtain a spin-spin interaction of the form with Jm,m,. > 0. This interaction prefers parallel alignment of spins, i.e., it is ferromagnetic. This is the derivation of Hund’s first rule: The total spin of electrons in a partially-filled shell of one ion tends to be maximal. Note that all terms containig Jm,m,. are quantum-mechanical in origin: They appear because we have written the density p = —eW'W as a bilinear form in the field operator, which made unconventional pairings of orbital indices m even possible. There is no analogy in classical physics.
This shows that Kinym. — 1/2 Jm,m. > 0 so that the corrected Coulomb term is reduced but still repulsive. Even more importantly, we obtain a spin-spin interaction of the form with Jm,m,. > 0. This interaction prefers parallel alignment of spins, i.e., it is ferromagnetic. This is the derivation of Hund’s first rule: The total spin of electrons in a partially-filled shell of one ion tends to be maximal. Note that all terms containig Jm,m,. are quantum-mechanical in origin: They appear because we have written the density p = —eW'W as a bilinear form in the field operator, which made unconventional pairings of orbital indices m even possible. There is no analogy in classical physics.
where U > 0 This is the famous “Hubbard U-term”, which will become important later. For a single relevant orbital ¢(r), we of course just get 4.1.2 Inter-ion exchange interaction Essentially everything from the previous discussion goes through if we allow the ionic sites Ri,...,Ra to be different. We just have to treat R; as another quantum number besides m;. We here restict ourselves to a model with a single, non-degenerate (except for spin) orbital per site. We can then drop the orbital quantum numbers m;. As noted above, we assume the orbitals at different sites to have negligible overlap, i.e., they are orthogonal. We will have a first-order perturbation if R; = R4 and Rz2 = R3 or if R; = Rg and Rz = Ry. In complete analogy to the previous subsection we obtain
where U > 0 This is the famous “Hubbard U-term”, which will become important later. For a single relevant orbital ¢(r), we of course just get 4.1.2 Inter-ion exchange interaction Essentially everything from the previous discussion goes through if we allow the ionic sites Ri,...,Ra to be different. We just have to treat R; as another quantum number besides m;. We here restict ourselves to a model with a single, non-degenerate (except for spin) orbital per site. We can then drop the orbital quantum numbers m;. As noted above, we assume the orbitals at different sites to have negligible overlap, i.e., they are orthogonal. We will have a first-order perturbation if R; = R4 and Rz2 = R3 or if R; = Rg and Rz = Ry. In complete analogy to the previous subsection we obtain
We find that the singlet (5 = 0) is lower in energy than the triplet (S = 1), i.e., there is an antiferro magnetic interaction. This results from the lowering of the kinetic energy for antiparallel spins. For paralle spins the hopping is blocked by the Pauli principle, which is why t does not even appear in the eigenenergies of the triplet. Therefore, this mechanism is called kinetic exchange. An example is the Hz molecule, whict has a singlet ground state. For U/|t| > 00, the corresponding eigenstate approaches (| t, |) —| |,¢))/W2, i-e., the spin singlet. For finite U, it has some admixture of doubly occupied states. Thus the spectrum looks like this:
We find that the singlet (5 = 0) is lower in energy than the triplet (S = 1), i.e., there is an antiferro magnetic interaction. This results from the lowering of the kinetic energy for antiparallel spins. For paralle spins the hopping is blocked by the Pauli principle, which is why t does not even appear in the eigenenergies of the triplet. Therefore, this mechanism is called kinetic exchange. An example is the Hz molecule, whict has a singlet ground state. For U/|t| > 00, the corresponding eigenstate approaches (| t, |) —| |,¢))/W2, i-e., the spin singlet. For finite U, it has some admixture of doubly occupied states. Thus the spectrum looks like this:
Hopping of a hole from Cu?* via O?~ to the next Cu?? (and back) leads to a strong antiferromagnetic exchange interaction proportional to t+. The kinetic exchange from direct Cu-Cu hopping is proportional to (t’)? but is nevertheless tiny in comparison because of |t’| < |. An important example is provided by the cuprates: Cu?*t-ions form a square lattice (maybe slightly eformed) with O?~-ions centered on the bonds. In ionic crystals, the magnetic ions are always separated by non-magnetic anions. Thus both the direct exchange interaction (note that the exchange integral contains @R, (t1)@R,(r1)) and the kinetic exchange interaction (note that J contains t?) become very small. A larger exchange interaction results from electrons hopping from a magnetic cation to an anion and then to the next cation. \l ) VU Ib HO VEeLulelesno lity iit CULIIVALISUIL UCUaAUDSY UL 14% | wy 1¢|> This type of magnetic interaction is called superexchange. Note that this nomenclature is not consistently used. Goodenough calls this “semicovalent exchange”. Since it is the most important exchange mechanism for ionic crystals, it has been studied extensively. Detailed calculations have led to a number of rules of thumb to estimate the strength and sign of the superexchange, which need not be antiferromagnetic in all cases. These are the Goodenough-Kanamori rules. We adapt them from the formulation given by Anderson (1963):
Hopping of a hole from Cu?* via O?~ to the next Cu?? (and back) leads to a strong antiferromagnetic exchange interaction proportional to t+. The kinetic exchange from direct Cu-Cu hopping is proportional to (t’)? but is nevertheless tiny in comparison because of |t’| < |. An important example is provided by the cuprates: Cu?*t-ions form a square lattice (maybe slightly eformed) with O?~-ions centered on the bonds. In ionic crystals, the magnetic ions are always separated by non-magnetic anions. Thus both the direct exchange interaction (note that the exchange integral contains @R, (t1)@R,(r1)) and the kinetic exchange interaction (note that J contains t?) become very small. A larger exchange interaction results from electrons hopping from a magnetic cation to an anion and then to the next cation. \l ) VU Ib HO VEeLulelesno lity iit CULIIVALISUIL UCUaAUDSY UL 14% | wy 1¢|> This type of magnetic interaction is called superexchange. Note that this nomenclature is not consistently used. Goodenough calls this “semicovalent exchange”. Since it is the most important exchange mechanism for ionic crystals, it has been studied extensively. Detailed calculations have led to a number of rules of thumb to estimate the strength and sign of the superexchange, which need not be antiferromagnetic in all cases. These are the Goodenough-Kanamori rules. We adapt them from the formulation given by Anderson (1963):
with the time-independent but possibly spatially inhomogeneous vector Dj2 associated with the bond be- tween S; and Sj. When is this term allowed by symmetry? We have to consider all symmetry operations of the crystal that leave the center point C on the bond between the two spins fixed. The whole Hamiltonian and in particular Hp)y must remain the same if we apply any of these symmetry operations.
with the time-independent but possibly spatially inhomogeneous vector Dj2 associated with the bond be- tween S; and Sj. When is this term allowed by symmetry? We have to consider all symmetry operations of the crystal that leave the center point C on the bond between the two spins fixed. The whole Hamiltonian and in particular Hp)y must remain the same if we apply any of these symmetry operations.
Thus # is only invariant if Di2= 0. In this case there cannot be a Dzyaloshinsky-Moriya term. If the symmetry is lower, the term is possible. Moriya has given rules for the allowed directions of Dj. We consider one example, a two-fold rotation axis C2 through C perpendicular to the bond. Herein, the underlined term changes sign under C2, the others do not. Thus H is invariant only if Dj2, = 0. Therefore, Dj2 must be perpendicular to the C2 symmetry axis, i.e., the z-axis. To find its orientation in the zy-plane we have to actually calculate it.
Thus # is only invariant if Di2= 0. In this case there cannot be a Dzyaloshinsky-Moriya term. If the symmetry is lower, the term is possible. Moriya has given rules for the allowed directions of Dj. We consider one example, a two-fold rotation axis C2 through C perpendicular to the bond. Herein, the underlined term changes sign under C2, the others do not. Thus H is invariant only if Dj2, = 0. Therefore, Dj2 must be perpendicular to the C2 symmetry axis, i.e., the z-axis. To find its orientation in the zy-plane we have to actually calculate it.
A technical problem encountert for the Heisenberg model is that in the terms in parentheses do not commute. The last term is obviously diagonal in the usual basis spanned by the product states |$1,™m1), |S2,m2) ... |Sn,my) but the other two terms change the quantum numbers ™4, M5. .1 Ground state in the ferromagnetic case
A technical problem encountert for the Heisenberg model is that in the terms in parentheses do not commute. The last term is obviously diagonal in the usual basis spanned by the product states |$1,™m1), |S2,m2) ... |Sn,my) but the other two terms change the quantum numbers ™4, M5. .1 Ground state in the ferromagnetic case
EE NES SRA SR RSS ee Se RE ene AEE MEN nncl ees NEST nener Stew nit NeneEntiann Meer Qtr: ewer RE Sa Ys In the context of magnetism, a system is bipartite if all sites can be divided into two disjoint subsystems A and B in such a way that Ji; = 0 ift,7 € A or ifi,7 € B. This means that the only interactions are between sites from different subsystems. The most important case of course involves spins on a crystal lattice, in which case a bipartite lattice has the obvious definition. Example: the square lattice with nearest-neighbor interactions is bipartite: An obvious guess would be that, for Jj; < 0 on a bipartite lattice, all spins are fully polarized but in opposite directions for the two sublattices. This is called a Néel state. We consider the state If some J;; are negative, the fully polarized states are still eigenstates of H but generally no longer ground states. Practically nothing is known rigorously about the ground state of the general Heisenberg model. An important subclass for which rigorous results exist are antiferromagnets on bipartite systems.
EE NES SRA SR RSS ee Se RE ene AEE MEN nncl ees NEST nener Stew nit NeneEntiann Meer Qtr: ewer RE Sa Ys In the context of magnetism, a system is bipartite if all sites can be divided into two disjoint subsystems A and B in such a way that Ji; = 0 ift,7 € A or ifi,7 € B. This means that the only interactions are between sites from different subsystems. The most important case of course involves spins on a crystal lattice, in which case a bipartite lattice has the obvious definition. Example: the square lattice with nearest-neighbor interactions is bipartite: An obvious guess would be that, for Jj; < 0 on a bipartite lattice, all spins are fully polarized but in opposite directions for the two sublattices. This is called a Néel state. We consider the state If some J;; are negative, the fully polarized states are still eigenstates of H but generally no longer ground states. Practically nothing is known rigorously about the ground state of the general Heisenberg model. An important subclass for which rigorous results exist are antiferromagnets on bipartite systems.
It is often useful to map a certain model onto a different but equivalent one. In many cases in many-body theory this procedure has led to new insights. Several such mappings take a spin (e.g., Heisenberg) model into a bosonic one. Bosonic models are often easier to study since boson creation and annihilation operators have simpler commutation relations than spin operators. Also, a bosonic model has a straightforward physical interpretation as a system of, usually coupled, harmonic oscillators. Tec Beet Oe cee bec ope eee Adee eee ees oD eee The Schwinger bosons are introduced as follows:
It is often useful to map a certain model onto a different but equivalent one. In many cases in many-body theory this procedure has led to new insights. Several such mappings take a spin (e.g., Heisenberg) model into a bosonic one. Bosonic models are often easier to study since boson creation and annihilation operators have simpler commutation relations than spin operators. Also, a bosonic model has a straightforward physical interpretation as a system of, usually coupled, harmonic oscillators. Tec Beet Oe cee bec ope eee Adee eee ees oD eee The Schwinger bosons are introduced as follows:
Thus P,P; = P; so that P; is a projection operator. It clearly projects onto the subspace of triad spin 3/2. On the other hand, In the dimer states |+), none of the triad spins can be 3/2 since two of the three spins form a singlet. Thus P;|+) = 0 for all 7. Therefore, The usefulness of parent Hamiltonians such as the Majumdar-Ghosh model is that they allow us to derive exact results. While these do not directly apply to other models such as the nearest-neighbor Heisenberg model, they do give insight into the generic behavior of families of spin models.
Thus P,P; = P; so that P; is a projection operator. It clearly projects onto the subspace of triad spin 3/2. On the other hand, In the dimer states |+), none of the triad spins can be 3/2 since two of the three spins form a singlet. Thus P;|+) = 0 for all 7. Therefore, The usefulness of parent Hamiltonians such as the Majumdar-Ghosh model is that they allow us to derive exact results. While these do not directly apply to other models such as the nearest-neighbor Heisenberg model, they do give insight into the generic behavior of families of spin models.
We now assume the fluctuations S; — (S;) to be small in the sense that the contribution of the last term in AT is negligible. Neglecting this term we obtain We have thus replaced the exchange interaction by a Zeeman term describing spins in an effective B-field (or “molecular field” )
We now assume the fluctuations S; — (S;) to be small in the sense that the contribution of the last term in AT is negligible. Neglecting this term we obtain We have thus replaced the exchange interaction by a Zeeman term describing spins in an effective B-field (or “molecular field” )
The function Bgs(a) is plotted for several values of S in the following graph: We find that Beg is everywhere parallel to (S;). Since the (S;) are the spin averages in the effective field Beg and H does not contain any anisotropy terms, this is required of a consistent approximation. Furthermore, Beg(R,) and (S;) have to point in the same (and not opposite) direction. This is the case if J(Q) > 0. This inequality is satisfied, as we will see.
The function Bgs(a) is plotted for several values of S in the following graph: We find that Beg is everywhere parallel to (S;). Since the (S;) are the spin averages in the effective field Beg and H does not contain any anisotropy terms, this is required of a consistent approximation. Furthermore, Beg(R,) and (S;) have to point in the same (and not opposite) direction. This is the case if J(Q) > 0. This inequality is satisfied, as we will see.
Solutions correspond to intersections between the straight line M — M (the identity function) and the curv M > SBs(J(Q)M/kegT). A non-trivial solution exists if SBgs has a larger slope at M = 0 than the identit finction. j.e.. if for the unknown M € [0,S]. Since Bg(0) = 0, there is always a solution M = 0. Non-trivial solutions have to be found numerically or graphically. The mean-field approximation now amounts to solving the equation
Solutions correspond to intersections between the straight line M — M (the identity function) and the curv M > SBs(J(Q)M/kegT). A non-trivial solution exists if SBgs has a larger slope at M = 0 than the identit finction. j.e.. if for the unknown M € [0,S]. Since Bg(0) = 0, there is always a solution M = 0. Non-trivial solutions have to be found numerically or graphically. The mean-field approximation now amounts to solving the equation
YUCHULY, LAA YZ \UY)] -— U- To see that the non-trivial solution is the stable one, if it exists, one has to consider the free energy. The result within mean-field theory is that for T < T, the free energy has a minimum at M > 0 but a maximum at M = 0. Thus the average of J(q) is zero. If there is any exchange interaction, J(q) is not identically zero. Conse- quently, max J(q) > 0.
YUCHULY, LAA YZ \UY)] -— U- To see that the non-trivial solution is the stable one, if it exists, one has to consider the free energy. The result within mean-field theory is that for T < T, the free energy has a minimum at M > 0 but a maximum at M = 0. Thus the average of J(q) is zero. If there is any exchange interaction, J(q) is not identically zero. Conse- quently, max J(q) > 0.
Analytical results exist in limiting cases: To actually find the magnetization M, we have to solve the mean-field equation numerically. The solution is of the form sketched here: As noted above, any rotation gives another equilibrium solution. In addition, the phase a is also not constrained by the mean-field theory. Thus we obtain a hugely degenerate space of equilibrium solutions— another example of spontaneous symmetry breaking.
Analytical results exist in limiting cases: To actually find the magnetization M, we have to solve the mean-field equation numerically. The solution is of the form sketched here: As noted above, any rotation gives another equilibrium solution. In addition, the phase a is also not constrained by the mean-field theory. Thus we obtain a hugely degenerate space of equilibrium solutions— another example of spontaneous symmetry breaking.
Since the magnetization is uniform, it depends on the interaction J(q = 0) at q = 0 and not on max J(q). We define the paramagnetic Curie temperature We are here interested in the case of a weak field B, for which linear-response theory is valid. In the paramagnetic phase for T > T,, the applied field induces a magnetization
Since the magnetization is uniform, it depends on the interaction J(q = 0) at q = 0 and not on max J(q). We define the paramagnetic Curie temperature We are here interested in the case of a weak field B, for which linear-response theory is valid. In the paramagnetic phase for T > T,, the applied field induces a magnetization
In general, one expect a divergence of the form as T approaches the critical point at T.. Thus in mean-field theory, we find y = 1. This exponent is alsc changed by fluctuations ignored i in mean- field theory. See also Sec. 7. 2. — — — a oe pene For general helical order, Ty < T. and x grows for T — T* but does not diverge. (The linear response to a field modulated with wavevector Q does diverge at T,..) The following sketch shows 1/y for three different materials:
In general, one expect a divergence of the form as T approaches the critical point at T.. Thus in mean-field theory, we find y = 1. This exponent is alsc changed by fluctuations ignored i in mean- field theory. See also Sec. 7. 2. — — — a oe pene For general helical order, Ty < T. and x grows for T — T* but does not diverge. (The linear response to a field modulated with wavevector Q does diverge at T,..) The following sketch shows 1/y for three different materials:
Note that To < 0 is possible since J(q = 0) can be negative. This is often the case for antiferromagnets.
Note that To < 0 is possible since J(q = 0) can be negative. This is often the case for antiferromagnets.
The first example concerns systems with weak and strong interactions. Let us consider a lattice of dimers of spin S with strong ferromagnetic exchange interaction J, which are coupled to each other by a much weaker ferromagnetic exchange interaction J’ < J. The mean-field Hamiltonian reads
The first example concerns systems with weak and strong interactions. Let us consider a lattice of dimers of spin S with strong ferromagnetic exchange interaction J, which are coupled to each other by a much weaker ferromagnetic exchange interaction J’ < J. The mean-field Hamiltonian reads
with Z:= D>, e~ 8m, Note that (e~8! — e~¥»)/(E, — Em) is positive for all Em # En. Thus (A, B) i a scalar product on the space of linear operators. Furthermore, where C is another linear operator. Then
with Z:= D>, e~ 8m, Note that (e~8! — e~¥»)/(E, — Em) is positive for all Em # En. Thus (A, B) i a scalar product on the space of linear operators. Furthermore, where C is another linear operator. Then
Thus the magnetization in 3D need not vanish for B + 0*. The Mermin-Wagner theorem does not make any statement on the existence of long-range order in 3D. Note that it is really the dimension that enters, not the coordination number z. Thus the theorem forbids long-range order for the triangular lattice with z = 6 but not for the simple cubic lattice also with z = 6.
Thus the magnetization in 3D need not vanish for B + 0*. The Mermin-Wagner theorem does not make any statement on the existence of long-range order in 3D. Note that it is really the dimension that enters, not the coordination number z. Thus the theorem forbids long-range order for the triangular lattice with z = 6 but not for the simple cubic lattice also with z = 6.
For the example of the nearest-neighbor ferromagnetic Heisenberg model one obtains where z is the coordination number of the lattice. (Note that higher-order terms depend on details of the lattice structure that cannot be expressed in terms of z alone.) There is a slight difference in the definition compared to the one we used for the Curie-Wei® Law, essentially a factor of volume. TR meen xan aww ley Diaad Dk memmnetiaow; dice T fxs
For the example of the nearest-neighbor ferromagnetic Heisenberg model one obtains where z is the coordination number of the lattice. (Note that higher-order terms depend on details of the lattice structure that cannot be expressed in terms of z alone.) There is a slight difference in the definition compared to the one we used for the Curie-Wei® Law, essentially a factor of volume. TR meen xan aww ley Diaad Dk memmnetiaow; dice T fxs
S*" have the form of plane waves. Note that the z- and y-components are out of phase by 7/2 or 90°. Classically, S;(¢) can be said to precess on a cone. and absorbing f Into Jj; and {4B we Obtain equations without the quantum parameter /2). Since we are interested in low-energy excitations, we expect the state of the system to be close to the fully polarized ground state. What this actually means is defined by the approximation we are making: we replace S? in products of two spin operators by S.. This gives for the first two equations Note that these equations also make sense for a purely classical model (by restoring the units of the spin and absorbing f into Jj; and 4g we obtain equations without the quantum parameter fi).
S*" have the form of plane waves. Note that the z- and y-components are out of phase by 7/2 or 90°. Classically, S;(¢) can be said to precess on a cone. and absorbing f Into Jj; and {4B we Obtain equations without the quantum parameter /2). Since we are interested in low-energy excitations, we expect the state of the system to be close to the fully polarized ground state. What this actually means is defined by the approximation we are making: we replace S? in products of two spin operators by S.. This gives for the first two equations Note that these equations also make sense for a purely classical model (by restoring the units of the spin and absorbing f into Jj; and 4g we obtain equations without the quantum parameter fi).
In the classical limit, this kind of excitations is called a spin wave. Its frequency is evidently For example, in the absense of an applied field and for nearest-neighbor exchange on the simple cubic lattice, we have
In the classical limit, this kind of excitations is called a spin wave. Its frequency is evidently For example, in the absense of an applied field and for nearest-neighbor exchange on the simple cubic lattice, we have
However, the difference between ng (hw) and np(fw,) is of higher order in 1/5 since the difference between hufi¥ and hwg is of higher order. Such terms are not correctly described by the Holstein-Primakoff Hamiltonian truncated after the S° term, anyway. We can thus set
However, the difference between ng (hw) and np(fw,) is of higher order in 1/5 since the difference between hufi¥ and hwg is of higher order. Such terms are not correctly described by the Holstein-Primakoff Hamiltonian truncated after the S° term, anyway. We can thus set
We can now calculate thermodynamic quantities in analogy to the ferromagnetic case. We only consider the staggered magnetization in 3D: Unlike for the ferromagnet, the dispersion is linear. This is a general result for antiferromagnets, not restricted to our particular models. The spin-wave velocity is thus a meaningful quantity for small q. For the models considered here it is csw = —2VdJSa/h. We also find hwg—o = 0, again satisfying the Goldstone theorem. for the chain/square/simple cubic nearest-neighbor models. Thus for small q we finc In the previous subsection, we found that the non-interacting-magnon approximation predicts two degenerat: magnon species with dispersion
We can now calculate thermodynamic quantities in analogy to the ferromagnetic case. We only consider the staggered magnetization in 3D: Unlike for the ferromagnet, the dispersion is linear. This is a general result for antiferromagnets, not restricted to our particular models. The spin-wave velocity is thus a meaningful quantity for small q. For the models considered here it is csw = —2VdJSa/h. We also find hwg—o = 0, again satisfying the Goldstone theorem. for the chain/square/simple cubic nearest-neighbor models. Thus for small q we finc In the previous subsection, we found that the non-interacting-magnon approximation predicts two degenerat: magnon species with dispersion
U rotates spin j by an angle 277/N about the z-axis. The action of U is most easily pictured for a ferromagnetic state (which is otherwise irrelevant for the theorem):
U rotates spin j by an angle 277/N about the z-axis. The action of U is most easily pictured for a ferromagnetic state (which is otherwise irrelevant for the theorem):
Now the operator S?7S/,,— S%S*., changes sign under rotation by 7 around (1,1,0), whereas the singlet j |0) is invariant under this (or any) rotation. Thus the expectation value of S7S%., — S¥S*., vanishes. We j obtain We see that the energy expectation value in the twisted state |1) is bounded from above by Ey)—27?.J$?/N = Eo for N > o so that it approaches the ground-state energy for N — oo. Since (1|0) = 0, |1) can be writter as a superposition of eigenstates not including |0) so that at least one of these must have an eigenenergy approaching Ey for N — oo. This completes the proof. “ ~_ ~ We have seen above that spontaneously broken symmetries lead to gapless excitations; this is Goldstone’s theorem. The converse is evidently not true: the 1D antiferromagnetic Heisenberg chain has a gapless excitation spectrum but no long-range order. 8.4.2 The Jordan-Wigner transformation This is a good point to introduce another mapping of a spin model onto a simpler model, in this case a fermionic one. The scheme discussed here is most suited to the antiferromagnetic spin chain. We start from a slightly more general spin Hamiltonian,
Now the operator S?7S/,,— S%S*., changes sign under rotation by 7 around (1,1,0), whereas the singlet j |0) is invariant under this (or any) rotation. Thus the expectation value of S7S%., — S¥S*., vanishes. We j obtain We see that the energy expectation value in the twisted state |1) is bounded from above by Ey)—27?.J$?/N = Eo for N > o so that it approaches the ground-state energy for N — oo. Since (1|0) = 0, |1) can be writter as a superposition of eigenstates not including |0) so that at least one of these must have an eigenenergy approaching Ey for N — oo. This completes the proof. “ ~_ ~ We have seen above that spontaneously broken symmetries lead to gapless excitations; this is Goldstone’s theorem. The converse is evidently not true: the 1D antiferromagnetic Heisenberg chain has a gapless excitation spectrum but no long-range order. 8.4.2 The Jordan-Wigner transformation This is a good point to introduce another mapping of a spin model onto a simpler model, in this case a fermionic one. The scheme discussed here is most suited to the antiferromagnetic spin chain. We start from a slightly more general spin Hamiltonian,
For the XY model (A = 0), we clearly obtain a system of free fermions, which has a simple exact solutior in terms of Slater determinants. With
For the XY model (A = 0), we clearly obtain a system of free fermions, which has a simple exact solutior in terms of Slater determinants. With
where & is from the first Brillouin zone, k €]—7, 7]. For A ¥ 0 we find a nearest-neighbor density-density interaction. The model is still exactly solvable using the Bethe ansatz, but this solution is much more complicated than for A = 0. We know, however, that the ground state |0) is a spin singlet. Thus
where & is from the first Brillouin zone, k €]—7, 7]. For A ¥ 0 we find a nearest-neighbor density-density interaction. The model is still exactly solvable using the Bethe ansatz, but this solution is much more complicated than for A = 0. We know, however, that the ground state |0) is a spin singlet. Thus
without an additional change of the energy or the total spin S*. Note that the kinks correspond to 7 phase jumps of the Néel order. A single kink costs an energy of the order of |.J|/2. Now one considers superpositions of such kinks over the whole chain and finds again that this leads to excitations with energy going to zero for gq + 0. These excitations are called spinons. We see that a spin-flip, which is the physical excitation for example in neutron scattering, generates a pair of spinons, which are then free to propagate independently from one another. One says that they are deconfined. A spinon is evidently a spin-1/2 excitation since it is “half a magnon”. Also, in the above sketch the spinons do indeed carry an excess spin of 1/2. can be split into two kinks (domain walls) in the Néel order,
without an additional change of the energy or the total spin S*. Note that the kinks correspond to 7 phase jumps of the Néel order. A single kink costs an energy of the order of |.J|/2. Now one considers superpositions of such kinks over the whole chain and finds again that this leads to excitations with energy going to zero for gq + 0. These excitations are called spinons. We see that a spin-flip, which is the physical excitation for example in neutron scattering, generates a pair of spinons, which are then free to propagate independently from one another. One says that they are deconfined. A spinon is evidently a spin-1/2 excitation since it is “half a magnon”. Also, in the above sketch the spinons do indeed carry an excess spin of 1/2. can be split into two kinks (domain walls) in the Néel order,
The above discussion does not carry over to higher dimensions: the separation of a spin-flip into a pair of kinks, or of a magnon into a pair of spinons, is associated with an energy increase in higher dimensions. Thus spinons are confined for d > 2. The reason is that the two spinons are connected by a string of flipped spins with frustated bonds to their unflipped neighbors:
The above discussion does not carry over to higher dimensions: the separation of a spin-flip into a pair of kinks, or of a magnon into a pair of spinons, is associated with an energy increase in higher dimensions. Thus spinons are confined for d > 2. The reason is that the two spinons are connected by a string of flipped spins with frustated bonds to their unflipped neighbors:
This dispersion has been observed with neutron scattering for KCuF3 by Tennant et al. in 1993
This dispersion has been observed with neutron scattering for KCuF3 by Tennant et al. in 1993
The unperturbed states of free electrons can be written as products of plane waves y,(r) = V2 gk and spinors | t), | |). The first-order correction to an eigenstate |kc) is with €, := h?k?/2m. Note that the magnetization vanishes in the unperturbed state and that the first-order correction to the energy also vanishes if q 4 0. Inserting |ko)“), we obtain
The unperturbed states of free electrons can be written as products of plane waves y,(r) = V2 gk and spinors | t), | |). The first-order correction to an eigenstate |kc) is with €, := h?k?/2m. Note that the magnetization vanishes in the unperturbed state and that the first-order correction to the energy also vanishes if q 4 0. Inserting |ko)“), we obtain
Note that this integral is singular if g < 2k. In this case we should treat it as a principal-value integral After some calculation we find
Note that this integral is singular if g < 2k. In this case we should treat it as a principal-value integral After some calculation we find
For a parabolic dispersion, the density of states is
For a parabolic dispersion, the density of states is
The singularity in xq is actually a “nesting” feature due to the Fermi surface portions at k and k + q being locally parallel. For q — 0 we have f — 2 so that we recover the Pauli susceptibility. Note that the q-dependent susceptibility xq has a singularity at q = 2k; the derivate dyq/dq diverges there. Since 2kp is the diameter of the Fermi sea, a magnetic field modulated with wave vector q of magnitude gq = 2kr mixes electrons at opposite points on the Fermi surface.
The singularity in xq is actually a “nesting” feature due to the Fermi surface portions at k and k + q being locally parallel. For q — 0 we have f — 2 so that we recover the Pauli susceptibility. Note that the q-dependent susceptibility xq has a singularity at q = 2k; the derivate dyq/dq diverges there. Since 2kp is the diameter of the Fermi sea, a magnetic field modulated with wave vector q of magnitude gq = 2kr mixes electrons at opposite points on the Fermi surface.
The density of states is thus constant. For B > 0, it is replaces by equidistant d-function peaks: The magnetic field transforms the continous spectrum of the two-dimensional electron gas, Ex = h?(k?2 + k2) /2m, into a discrete spectrum of Landau levels enumerated by n. For B = 0, the density of states is The energies do not depend on k, and are thus hugely degenerate. Note that the apparent asymmetry between the x- and y-direction in H is gauge-dependent and therefore without physical consequence. We could have chosen the vector potential A to point in any direction within the zy-plane. The isotropy of two-dimensional space is not broken by the choice of a special gauge.
The density of states is thus constant. For B > 0, it is replaces by equidistant d-function peaks: The magnetic field transforms the continous spectrum of the two-dimensional electron gas, Ex = h?(k?2 + k2) /2m, into a discrete spectrum of Landau levels enumerated by n. For B = 0, the density of states is The energies do not depend on k, and are thus hugely degenerate. Note that the apparent asymmetry between the x- and y-direction in H is gauge-dependent and therefore without physical consequence. We could have chosen the vector potential A to point in any direction within the zy-plane. The isotropy of two-dimensional space is not broken by the choice of a special gauge.
We can obtain the degeneracy of the Landau levels as follows: since the total number of states does no change, the Landau levels must accomodate these states, thus the degeneracy of the first one (and all th others) is _ At low temperatures, the low-energy states are filled successively until all N electrons are accomodated. If
We can obtain the degeneracy of the Landau levels as follows: since the total number of states does no change, the Landau levels must accomodate these states, thus the degeneracy of the first one (and all th others) is _ At low temperatures, the low-energy states are filled successively until all N electrons are accomodated. If
The limit limg_,o M does not exsist and neither does the limit limg_,o y for the susceptibility , = 0M/0B This unphysical result is due to our assumption of T = 0. At any T > 0, the thermal energy kgT will b: large compared to the energy spacing iw, between the Landau levels for sufficiently small B. In this regime we can neglect the discreteness of the Landau levels and write shows oscillations that are periodic in 1/B. These are the de Haas-van Alpen oscillations. Here, B; is the field for which v = 2, i.e., for which the lowest Landau level is completely filled,
The limit limg_,o M does not exsist and neither does the limit limg_,o y for the susceptibility , = 0M/0B This unphysical result is due to our assumption of T = 0. At any T > 0, the thermal energy kgT will b: large compared to the energy spacing iw, between the Landau levels for sufficiently small B. In this regime we can neglect the discreteness of the Landau levels and write shows oscillations that are periodic in 1/B. These are the de Haas-van Alpen oscillations. Here, B; is the field for which v = 2, i.e., for which the lowest Landau level is completely filled,
We are interested in the susceptibility for small fields. Thus we have hu. < kgT and we of course still assume kpT < yp for a typical metal. The rigorous calculation is rather involved. It is presented for example in “Solid State Physics” by Grosso and Pastori Parravicini. We here give a simplified and less rigorous Vveraion.
We are interested in the susceptibility for small fields. Thus we have hu. < kgT and we of course still assume kpT < yp for a typical metal. The rigorous calculation is rather involved. It is presented for example in “Solid State Physics” by Grosso and Pastori Parravicini. We here give a simplified and less rigorous Vveraion.
For kpT < hw,, the low-energy states are again filled up until all electrons are accomodated. We again expect to find special features whenever the chemical potential 1 reaches hw.(n + 1/2) with n = 0,1,2... Since yz is roughly constant, while hw, « B, we find in the total energy E(B) and thus in the magnetization M(B) = —(1/V)0E/OB features periodic in 1/B. These are again de Haas-van Alphen oscillations. They are visible if B is large.
For kpT < hw,, the low-energy states are again filled up until all electrons are accomodated. We again expect to find special features whenever the chemical potential 1 reaches hw.(n + 1/2) with n = 0,1,2... Since yz is roughly constant, while hw, « B, we find in the total energy E(B) and thus in the magnetization M(B) = —(1/V)0E/OB features periodic in 1/B. These are again de Haas-van Alphen oscillations. They are visible if B is large.
for the free electron gas.
for the free electron gas.
CACHAN MIVEraCllOll DELWCCH LIC ClCCULOMs Il a Weta. Following Bloch, we now consider special variational states for which the spin-up and spin-down electrons each fill a Fermi sea but with generally different volume. We assume a free-electron dispersion for simplicitiy. The Fermi seas are then characterized by the two Fermi wave numbers kr+ and kry. The total density of spin-o electrons is no = k}., /61? so that keg = (61?n,)/°. The unperturbed energy is We see that the energy is reduced by each pair of electrons with parallel spins. There is thus a ferromagnetic exchange interaction between the electrons in a metal.
CACHAN MIVEraCllOll DELWCCH LIC ClCCULOMs Il a Weta. Following Bloch, we now consider special variational states for which the spin-up and spin-down electrons each fill a Fermi sea but with generally different volume. We assume a free-electron dispersion for simplicitiy. The Fermi seas are then characterized by the two Fermi wave numbers kr+ and kry. The total density of spin-o electrons is no = k}., /61? so that keg = (61?n,)/°. The unperturbed energy is We see that the energy is reduced by each pair of electrons with parallel spins. There is thus a ferromagnetic exchange interaction between the electrons in a metal.
which separates two phases with different properties:
which separates two phases with different properties:
this Hamiltonain defines the Stoner model. Hstoner describes electrons in an effective field Bog = 2/(guB)kpO(N; — N,)/Ne. This field or the polarization (N+ — N,)/Ne has to be determined selfconsis- tently. In the ground state this is done by minimizing the energy
this Hamiltonain defines the Stoner model. Hstoner describes electrons in an effective field Bog = 2/(guB)kpO(N; — N,)/Ne. This field or the polarization (N+ — N,)/Ne has to be determined selfconsis- tently. In the ground state this is done by minimizing the energy
Note that the right-hand expression has slope 2/3 at ¢ = 0. We plot both sides of this equation:
Note that the right-hand expression has slope 2/3 at ¢ = 0. We plot both sides of this equation:
There are three cases:
There are three cases:
Zero-energy excitations are dangerous, since they suggest an instability of the mean-field ground state. One has to go beyond Stoner theory to see that they do not destroy the ferromagnetic ground state in this case. A situation with crossing Fermi spheres is sketched here: Note that in the crescend-shaped region to the left we find excitations that increase the total spin. We can now sketch the dispersion of the particle-hole continuum for partial polarization:
Zero-energy excitations are dangerous, since they suggest an instability of the mean-field ground state. One has to go beyond Stoner theory to see that they do not destroy the ferromagnetic ground state in this case. A situation with crossing Fermi spheres is sketched here: Note that in the crescend-shaped region to the left we find excitations that increase the total spin. We can now sketch the dispersion of the particle-hole continuum for partial polarization:
Note that the particle-hole continuum is also visible in y(q,w), specifically as a non-zero imaginary part Im y(q,w) 4 0. Moreover, the spin waves do not survive as propagating modes when they are degenerate in energy with the particle-hole continuum. This is plausible: a magnon can here decay into many particle-hole excitations. This mechanism is called Landau damping. Hence, the full dispersion lools like this: 1 Sec. 4.2 we have studied the half-filled Hubbard model for the case of weak hybridization, i.e., large U/t. ‘he result was an antiferromagentic exchange interaction
Note that the particle-hole continuum is also visible in y(q,w), specifically as a non-zero imaginary part Im y(q,w) 4 0. Moreover, the spin waves do not survive as propagating modes when they are degenerate in energy with the particle-hole continuum. This is plausible: a magnon can here decay into many particle-hole excitations. This mechanism is called Landau damping. Hence, the full dispersion lools like this: 1 Sec. 4.2 we have studied the half-filled Hubbard model for the case of weak hybridization, i.e., large U/t. ‘he result was an antiferromagentic exchange interaction
We note that Aycisenberg describes a Heisenberg model for the singly occupied sites, except for the rather trivial term —n,;n; /4.
We note that Aycisenberg describes a Heisenberg model for the singly occupied sites, except for the rather trivial term —n,;n; /4.
On the other hand, if ¢;; only contains nearest-neighbor hopping, AH does not frustrate the order, since AH contains the product t;j;tj;~ describing next-nearest-neighbor hopping. AH is often neglected in practice. Note that it has been suggested and tested by numerical calculations but not rigorously shown that the ground state of the t-J model in a certain range of hole concentrations is a superconductor.
On the other hand, if ¢;; only contains nearest-neighbor hopping, AH does not frustrate the order, since AH contains the product t;j;tj;~ describing next-nearest-neighbor hopping. AH is often neglected in practice. Note that it has been suggested and tested by numerical calculations but not rigorously shown that the ground state of the t-J model in a certain range of hole concentrations is a superconductor.
The spin excitation is clearly just the kink we have discussed in Sec. 8.4, the superositions of kinks make up the spinons. The charge excitation indicated by © does not carry a spin; the total spin in a symmetric interval centered at the charge is zero. This excitation is called a holon. 10.5 Nagaoka ferromagnetism We have seen that mobile holes doped into a half-filled ¢- model in two or three dimensions frustrate the antiferromagnetic order. We might ask how effectively the holes suppress antiferromagnetism. Experimen- tally, the CuQ2 planes in the cuprates, which are reasonably well described by a two-dimensional t-J model, loose antiferromagnetic order for only 2% of hole doping, i.e. 0.02 holes per lattice unit.
The spin excitation is clearly just the kink we have discussed in Sec. 8.4, the superositions of kinks make up the spinons. The charge excitation indicated by © does not carry a spin; the total spin in a symmetric interval centered at the charge is zero. This excitation is called a holon. 10.5 Nagaoka ferromagnetism We have seen that mobile holes doped into a half-filled ¢- model in two or three dimensions frustrate the antiferromagnetic order. We might ask how effectively the holes suppress antiferromagnetism. Experimen- tally, the CuQ2 planes in the cuprates, which are reasonably well described by a two-dimensional t-J model, loose antiferromagnetic order for only 2% of hole doping, i.e. 0.02 holes per lattice unit.
See Nielsen and Bhatt, arXiv:0907.3671. Although this phase diagram follows from mean-field theory, it appears to be qualitatively correct in that for finite U/t, the antiferromagnetic ground state survives over an extended hole-doping range. However, both experiments and more advanced numerical calculations indicate that it is then replaced by a state with short-range antiferromagnetic and possibly charge-density order.
See Nielsen and Bhatt, arXiv:0907.3671. Although this phase diagram follows from mean-field theory, it appears to be qualitatively correct in that for finite U/t, the antiferromagnetic ground state survives over an extended hole-doping range. However, both experiments and more advanced numerical calculations indicate that it is then replaced by a state with short-range antiferromagnetic and possibly charge-density order.
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