Magnetic Orders and Magnetic Phase Transitions

Abstract

In previous sections (Chaps. 4 and 6) the cases of isolated magnetic moments, with the related phenomena of atomic diamagnetism and paramagnetism, have been addressed.

Appendices

Appendix 17.1 Phase Diagram and Related Effects in 2D Quantum Heisenberg Antiferromagnets (2DQHAF)

Since the discovery that La$_2$CuO$_4$, the parent of high temperature superconductors (see Sect. 18.8), is the experimental realization of the model for two-dimensional (2D) quantum ($S = 1/2$) Heisenberg antiferromagnet (2DQHAF), a great deal of interest was triggered towards low-dimensional quantum magnetism. As sketched in Fig. 17.7, the system we are going to discuss is basically a planar array of $S = 1/2$ magnetic ions onto a square lattice, in antiferromagnetic interaction and described by the magnetic Hamiltonian

$$\begin{aligned} \mathcal {H}= J\sum _{i,j}\,'\, \mathbf {S_i}\cdot \mathbf {S_j}, \end{aligned}$$

(A.17.1.1)

with $J> 0$ and summation limited to the nearest neighbors spins.

Fig. 17.7

Sketch of a planar antiferromagnet with weak inter-planes interaction $J_{\perp }$

$S= 1/2$ characterizes the quantum condition and Heisenberg character means the absence of single ion anisotropy.

We shall devote our attention to a variety of aspects involving static and dynamical properties of that 2D array: (i) the temperature dependence of the in-plane magnetic correlation length $\xi _{2D}$ entering in the equal-time correlation function $<\mathbf {S_i}(0)\cdot \mathbf {S_j}(0)>$; (ii) the critical spin dynamics driving the system towards the long-range ordered state (at $T= 0$ in pure 2DQHAF in the absence of interplanar interaction $J_{\perp }$); (iii) the validity of the dynamical scaling, where $\xi _{2D}$ controls the relaxation rate $\varGamma$ of the order parameter according to a law of the form $\varGamma \propto \xi _{2D}^{-z}$, with critical exponent z (see Sect. 15.1).

We shall also comment on the modifications induced by spin dilution (or spin doping) namely when part of the $S= 1/2$ magnetic ions are substituted by non-magnetic $S= 0$ ions, as well as mentioning the effects related to charge doping, namely the injection (for instance by hetero-valent substitutions) of $S= 1/2$ holes, thus creating local singlets which can itinerate onto the plane, locally destroying the magnetic order and inducing novel spin excitations. These aspects are of particular interest in the vicinity of the percolation thresholds, where the AF order is about to be hampered at any finite temperature. This can be considered a situation similar to a quantum critical point (QCP), where no more the temperature but rather the Hamiltonian parameters can drive the transition.

In Fig. 17.8 the phase diagram for the system sketched in Fig. 17.7 is shown. This diagram results from a variety of experimental studies in synergistic interplay with theoretical descriptions, that will not be recalled in detail.⁷ We only present it and define some characteristic parameters.

Fig. 17.8

Phase diagram reporting the regimes theoretically proposed for 2DQHAF, as a function of temperature and of the parameter g related to the strength of quantum fluctuations

In the Figure g is a dimensionless parameter measuring the strength of quantum fluctuations and it can be related to spin wave velocity $c_{sw}$ and to the spin stiffness $\rho _s$:

$$\begin{aligned} g= \frac{\hbar c_{sw} \sqrt{2\pi }}{k_B \rho _s a} \end{aligned}$$

(A.17.1.2)

(a lattice parameter, see Fig. 17.7). The spin stiffness $\rho _s$ measures the increase in the ground state energy for rotation of the magnetization of the two sublattices by an angle $\theta$ ($\varDelta E = \rho _sk_B\theta ^2/2$). It can be written $\rho _s\propto (c_{sw}^2/\chi _{\perp })$ $\chi _{\perp }$ being the transverse spin susceptibility. The parameter g is expected to increase upon doping and disorder.

It has been customary to map the Hamiltonian (A.17.1.1) onto the so-called non-linear $\sigma$ model, that for $T\rightarrow 0$ is the simplest continuum model with the same symmetry and the same spectrum of excitations. In this way the diagram in Fig. 17.8 has emerged.

Below a given value $g_c$ the ground state, at $T= 0$, is the Néel AF state, which extends to finite temperature $T_N$ because of the interplane interaction $J_{\perp }\ll J$.

The percolation threshold for the AF state at $T= 0$ as a function of g is at $g_c$. For $g< g_c$, upon increasing temperature, above $T_N$, one enters in the renormalized classical (RC) regime. Here the effect of the quantum fluctuations is to renormalize $\rho _s$ and $c_{sw}$ with respect to the mean field values for the “classical” 2D Heisenberg paramagnet. Thus the in-plane correlation length goes as

$$\begin{aligned} \xi _{2D}\propto e^{2\pi \rho _s/T} . \end{aligned}$$

(A.17.1.3)

Weakly damped spin waves exist for wave vectors $q\ge \xi ^{-1}$ while for longer wave lengths only diffusive spin excitations of hydrodynamic character are present (see Sect. 15.4). For $T\ge J/2\simeq 2\rho _s$ instead of entering into the classical limit (that would be reached for $T\gg J$) the planar QHAF should cross to the quantum critical (QC) regime. In this phase, typical of 2D and 1D quantum magnetic systems, the only energy scale is set by temperature and $\xi \propto J/T$. On increasing g, according to proposals still under debate (in the scenario of quantum phase transitions theories) the increase in quantum fluctuations can inhibit an ordered state even at $T = 0$. The system is then in the quantum disordered (QD) regime, the correlation length being short and temperature independent. In the spectrum of spin excitations a gap of the order of $hc_{sw}/\xi _{2D}$ opens up.

The somewhat speculative phase diagram illustrated in Fig. 17.8 is still debated. In particular the validity of the non-linear $\sigma$ model is not entirely accepted for large T and/or for large g regions. Furthermore, there is not clear evidence of crossovers from RC to QC or to QD regimes. Also the real nature of the low-energy excitations remains an open question. The cluster-spin-glass (SG) phase is the one for $g> g_c$ in which the experiments indicate the presence of mesoscopic “islands” of AF character separated by domains walls, with effective magnetic moments undergoing collective spin freezing, without long range order even at temperature close to zero.

Above a certain amount of charge doping (e.g. hole injection as in La$_{2-y}$Sr$_y$CuO$_4$, see Sect. 18.8), as discovered by Müller and Bednorz, the systems become superconductors (SC phase), with the so-called underdoped and overdoped regimes characterized by a transition temperatures $T_c < T_{max}$ (the one pertaining to the optimal doping).

In SC underdoped phases a gap in the spin excitations at the AF wave vector $\mathbf {q_{AF}}= (\pi /a, \pi /a)$ has been experimentally observed to arise at a given temperature. The spin-gap (and charge pseudo-gap) region has possibly to be related to superconducting fluctuations (see Sect. 18.11) of “anomalous” character or to AF fluctuations locally creating a “tendency” towards a mesoscopic Mott insulator. Exotic excitations of various nature have been considered to occur in the regions of high g’s. We shall not go into detail involving these aspects, which are still under debate and less settled than the ones for low g, namely for the doped non-superconducting 2DQHAF.

Summarizing conclusions that can be drawn from the studies in pure 2DQHAF are the following: (i) the absolute value and the temperature dependence of the in-plane magnetic correlation length follows rather well the theoretical expression given by A.17.1.3; (ii) in La$_2$CuO$_4$ and in similar 2DQHAF the RC regime appears to hold, up to temperature of the order of $1.5\,J$; (iii) no evidence of crossover to QC or QD regimes has been clearly observed.

Some more quantitative comments can be given about the effect of spin dilution. As already mentioned, in La$_2$CuO$_4$ spin dilution is obtained by $S= 0$ Zn$^{2+}$ (or Mg$^{2+}$) for $S= 1/2$ Cu$^{2+}$ substitutions. While in La$_{2-y}$Sr$_y$CuO$_4$ the Néel temperature drops very fast with the Sr content, analogous effect but at much lower rate, is driven by the spin dilution (see Fig. 17.9).

Fig. 17.9

Doping dependence of the Néel temperature in diluted 2DQHAF (La$_2$Cu$_{1-x}$(Zn,Mg)$_x$O$_4$) and in hole-doped 2DQHAF (La$_{2-x}$Sr$_x$CuO$_4$), from a variety of measurements

In the limit of weak doping, the dilution model should hold. The dilution model modifies the Hamiltonian A.17.1.1 simply by considering the probability that a given site is spin-empty:

$$\begin{aligned} \mathcal {H}= J\sum _{i,j}\,'\, p_i\mathbf {S_i}\cdot p_j\mathbf {S_j}= J(0)(1-x)^2 \sum _{i,j}\,'\, \mathbf {S_i}\cdot \mathbf {S_j} . \end{aligned}$$

(A.17.1.4)

Then the spin stiffness should depend on doping according to $\rho _s(x)= \rho _s(0) (1-x)^2$, the correlation length becoming (see Eq. A.17.1.4)

$$ \xi _{2D}(x,T)\simeq \xi _{2D}(0,T) e^{-(2-x) x 1.15 J(0)/T} . $$

An indication for the value of the correlation length at $T_N$ can be obtained from the mean field argument:

$$ \xi _{2D}^2(x,T_N) J_{\perp }(x)= T_N(x) $$

Then for the correlation length one can write

$$ \xi _{2D}(x,T_N)\simeq \xi _{2D}(0,T_N) \frac{(1-4x)^{1/2}}{1-x} $$

In Fig. 17.10 the doping dependence of the spin stiffness⁸ is compared with the prediction of the dilution model. As it could be expected, the dilution model is reasonably well obeyed for light doping while for x amount of the non-magnetic ions larger than about 0.1 it evidently fails.

Fig. 17.10

Spin stiffness $\rho _s(x)$ in spin diluted La$_2$CuO$_4$ and comparison with the dependence expected within the dilution model

It should be remarked that although in the strong dilution regime the reduction of the spin stiffness dramatically departs from the one predicted by the dilution model, still the transition to the AF state occurs when the correlation length reaches an in-plane value around 150 lattice steps, as in pure or lightly doped systems.

Another quantity of interest for the quantum effects in disordered 2DHQAF is the zero-temperature staggered magnetic moment $<\mu (x, T\rightarrow 0)>$ along the local quantization axis, in other words the dependence of the sub-lattice magnetization on spin dilution. The staggered magnetic moment is different from the classical $S= 1/2$ value because of the quantum fluctuations, which in turn are expected to increase with spin dilution. The quantity

$$ R(x,T=0)= \frac{<\mu (x,0)>}{<\mu (0,0)>} $$

has been obtained to a good accuracy from the magnetic perturbation due to the local hyperfine field on $^{139}$La NQR spectra, from $\mu$SR precessional frequencies and from neutron scattering also close to the percolation threshold of Zn-Mg doped La$_2$CuO$_4$ (see Fig. 17.11).

Fig. 17.11

The zero temperature normalized sublattice magnetization in spin diluted La$_2$CuO$_4$. Comparison with spin-wave theory (SWT), Quantum Monte Carlo (QMC) and classical spin ($S\rightarrow \infty$) theoretical behaviour is presented

While the classical doping dependence (for $S\rightarrow \infty$) as well as the one predicted by the quantum non-linear $\sigma$ model are not supported by the experimental findings, the data indicate a doping dependence of the form $R= (x_c- x)^{\beta }$, with critical exponent $\beta = 0.45$, close to the behaviour deduced from spin wave theory.

As regards the temperature dependence of R, it appears that both in the light doping regime as well as for strong dilution a universal law of the form

$$ R(x= const, T)\propto [T_N(x)- T]^{\beta } $$

holds, with a small critical exponent $\beta$ that appears to be around 0.2 for light doping while on approaching the percolation it increases to 0.3 (Fig. 17.12).

Fig. 17.12

Temperature dependence of the sublattice magnetization in spin diluted La$_2$CuO$_4$

Details and further experimental data can be found in the article by Rigamonti, Carretta and Papinutto in Novel NMR and EPR Techniques, J. Dolinsek, M. Vilfan and S. Zumer (Eds.) Springer (2006).

Appendix 17.2 Remarks on Scaling and Universality

An example of scaling and universality is offered by the equation of state of real gases. As it is known from elementary physics, all the fluids obey to the same equation of state once that the thermodynamic variables, pressure (P), temperature (T) and volume (V), are scaled in terms of the correspondent critical variables $P_c$ and $T_c$ (see Sect. 15.1). The Van der Waals equation takes a universal form, independent on the microscopic parameters that control pressure and volume:

$$ \mathbf {P}= f(\mathbf {T}, \mathbf {V}), $$

with $\mathbf {P}= P/P_c$, $\mathbf {T}= T/T_c$ (and $\mathbf {V}= V/V_c$, the volume being the inverse density). The law of the correspondent states can be considered an example of universality in the thermodynamic relationships.

It is conceivable that also at the phase transitions of the second order, when on approaching the critical temperature the correlation length tends to infinity, the thermodynamic behaviour becomes independent on the detailed parameters of the short range interaction. Then some kind of universal laws can be written, disregarding irrelevant terms and/or variables. This is also the root of the so-called renormalization group theory (RG) which has been devised by Wilson in order to determine the partition function or the critical exponents very close to $T_c$ by means of recursion relations.

To give some remarks about the property of scale invariance and an example of class of universality of the thermodynamic potentials, let us return to the free energy for homogeneous system (Eq. 15.19):

$$\begin{aligned} f[m,T]= f_0(T)+ a_0(T-T_c) m^2 + \frac{1}{2} b_0 m^4 + \cdots \end{aligned}$$

(A.17.2.1)

From $h= (\partial f/\partial m)_T$ the equation of state involving the field h and the magnetization m around the transition is written

$$\begin{aligned} h= 2 a_0 T_c \varepsilon m + 2 b_0 m^3 , \end{aligned}$$

(A.17.2.2)

with $\varepsilon = (T-T_c)/T_c$, namely in the form

$$\begin{aligned} h= q\left[ 2\left( \frac{m}{p}\right) + 4\left( \frac{m}{p}\right) ^3\right] , \end{aligned}$$

(A.17.2.3)

having defined new factors $p= (2a_0 T_c \varepsilon / b_0)^{1/2}$ and $q= (a_0 T_c \varepsilon )^{3/2}/ (b_0/ 2)^{1/2}$.

From Eq. (A.17.2.3) it is noted that

$$\begin{aligned} \biggl (\frac{m}{p} \biggr )= g_{\pm }({h}/{q}) \end{aligned}$$

(A.17.2.4)

where $g_{\pm }$ is universal function ($g_-$ corresponding to $\varepsilon <0$, $g_+$ to $\varepsilon >0$), namely the same function for any system (provided with the same symmetry properties). In other words, when the magnetization is scaled by $\varepsilon ^{1/2}$ it is no more a function of temperature and field separately but it becomes a function only of the ratio $h/\varepsilon ^{3/2}$. Then the equation of state A.17.2.3 involves a single independent variable instead of two, $\varepsilon$ and h. In a similar way it is possible to rewrite the free energy A.17.2.1 in terms of (h / q) and (m / p) to get a universal function, with f[m, T] scaled by $p q\propto \varepsilon ^2$ in terms of the variable $(h/ \varepsilon ^{3/2})$.

The possibility to rescale the independent variables of a function in a way to decrease their number is a characteristic of the homogeneous functions. By following a conjecture by Widom we reformulate the scaling properties sketched above by writing the equation of state in the form

$$\begin{aligned} h= m \psi (\varepsilon , m^{1/\beta } ) \end{aligned}$$

(A.17.2.5)

where $\psi$ is homogeneous function of degree $\gamma$, meaning $\psi (\lambda x, \lambda y)= \lambda ^{\gamma }\psi (x,y)$. Therefore, if one knows the value of the function at the point $x_0, y_0$ and the degree of homogeneity $\gamma$ as well, then the function is known everywhere. This is the condition of scale invariance analogous to the one described by the equation of correspondent states: the equation of state does not change (it has the same functional form) when the thermodynamic variables are scaled by any given quantity $\lambda$ to a certain exponent. Thus, Eq. A.17.2.5 indeed takes universal form, being invariant under the scale transformations

$$ \varepsilon \rightarrow \lambda \varepsilon ,\,\,\,\, m\rightarrow \lambda ^{1/\beta } m\,\,\,\, \mathrm { and} \,\,\,\, h\rightarrow \lambda ^{\gamma + 1/\beta } h . $$

In this way the critical behaviour appears to be controlled by two parameters only, $\beta$ and $\gamma$. Again in analogy to the law of correspondent states, according to the scaling hypothesis (which has been supported by more advanced theories as the renormalization group, by solutions of exact models and particularly by experiments) one can speculate that for a certain class of materials (for instance having the same dimensionality D of the lattice and d of the order parameter, see Sect. 17.1) universal behaviour occurs, with a minimum number of free parameters. In particular, all the critical exponents should be related to the two, $\beta$ and $\gamma$, defined above.

Now we are going to show that $\beta$ and $\gamma$ indeed are critical exponents, namely the same phenomenological exponents introduced at Sect. 15.1 and already derived in the MFA scenarios for ferroelectrics and for magnetic systems. In fact, the spontaneous magnetization $m_s$ is solution of the implicit equation

$$ h(\varepsilon , m_s)= 0 $$

(return for similarities at Sect. 16.4 and Eqs. (17.6)–(17.10)).Then from Eq. (A.17.2.5) we set

$$ \psi (\varepsilon , m_s^{1/\beta }) =0 $$

$\psi \,\,$ being homogeneous, i.e. $\psi (\varepsilon , m^{1/\beta })= \lambda ^{-\gamma }\psi (\lambda \varepsilon , \lambda m^{1/\beta })$, so that $\psi (\lambda \varepsilon , \lambda m^{1/\beta })=0$. The solution being a function of $\varepsilon$ only, one can write

$$ m_s^{1/\beta }= g(\varepsilon ) $$

and in analogous way for the scaled equation $\lambda m_s^{1/\beta }= g(\lambda \varepsilon )$. Therefore g must be homogeneous of degree 1 and thus

$$ m_s\propto (\varepsilon )^{\beta }. $$

Analogous demonstration can be carried out for the critical exponent $\gamma$, that controls the temperature dependence of the isothermal susceptibility for evanescent field. From Eqs. A.17.2.3 and A.17.2.5, the derivative of h with respect to m yields the inverse of the susceptibility:

$$\begin{aligned} \chi ^{-1}(h\rightarrow 0)= \biggl ( \frac{\partial h}{\partial m}\biggr )_T= \biggl [ \psi (\varepsilon , m^{1/\beta }) + m \biggl ( \frac{\partial \psi (\varepsilon , m^{1/\beta })}{\partial m} \biggr ) \biggr ]_{h\rightarrow 0} \end{aligned}$$

(A.17.2.6)

For $T> T_c$ $m= 0$ and then

$$ \chi ^{-1}(h\rightarrow 0)= \psi (\varepsilon , 0)= \lambda ^{-\gamma } \psi (\lambda \varepsilon , 0) . $$

By setting the scaling factor $\lambda = \varepsilon ^{-1}$ one derives

$$ \chi ^{-1}(h\rightarrow 0)= \varepsilon ^{\gamma } \psi (1, 0) $$

and since $\psi (1,0)$ is temperature-independent, $\gamma$ is the critical exponent for $\chi$.

Below $T_c$ it can be proved, by starting from Eq. (A.17.2.6), that the same critical exponent controls the temperature dependence of $\chi$ for $T\rightarrow T_c^-$.

A comprehensive presentation of the scaling theory and of the related consequences on the critical behaviour at the phase transitions (in particular in regards of the relationships among the various critical exponents) can be found in the book by Stanley.

Problems

Problem 17.1

Derive Eq. (17.25).

Solution: The expansion of the Brillouin function in the mean field approximation reads

$$ M_A= \frac{N\mu _J}{2}\biggl [ B_J(x_0) + \frac{\mu _J}{k_BT} \biggl (H_{ext}+ \lambda (M_B- M_0)- \alpha (M_A-M_0)\biggr ) |\frac{dB_J}{dx}|_{x_0}\biggr ] $$

and in analogous way

$$ M_B= \frac{N\mu _J}{2}\biggl [ B_J(x_0) + \frac{\mu _J}{k_BT} \biggl ( -H_{ext}+ \lambda (M_A- M_0)- \alpha (M_B-M_0)\biggr ) |\frac{dB_J}{dx}|_{x_0}\biggr ] $$

with $M_0$ the magnetization in zero field. Being the magnetization $M= (M_A- M_B)$, the susceptibility $(M/H_{ext})$ turns out as in Eq. (17.25).

Problem 17.2

By means of quantum mechanical procedure re-derive Eq. (17.31).

Solution: The ground state of a ferromagnet consists of all the spins ($S=1/2$) along the z-direction and the Heisenberg Hamiltonian reads

$$ \mathcal {H}= -2J \sum _i \biggl [ S_z^i S_z^{i+1} + \frac{1}{2}\biggl (S_+^i S_-^{i+1} + S_-^i S_+^{i+1}\biggr ) \biggr ] . $$

The eigenvalue for the ground state is $-NS^2 J$. When an excitation arises a given spin at the site j is flipped and as a consequence the total spin of the system is changed by $1/2 - (-1/2)= 1$. By applying the Hamiltonian one has

$$ \mathcal {H}|j>= 2\biggl [( -NS^2J+ 2SJ)|j>- SJ|j+1>- SJ|j-1>\biggr ] , $$

showing that it is not an eigenstate.

The Hamiltonian can be diagonalized by looking for plane wave solutions

$$ |p>= \frac{1}{\sqrt{N}} \sum _j |j> e^{i \mathbf {q}\cdot \mathbf {R}_j} $$

the total spin of the state $|p>$ being $(NS-1)$. Thus $\mathcal {H} |p>= E(p)|p>$ with $E(p)= -2NS^2 J + 4JS[1-cos(qa)]$, implying energy of the excitation as in Eq. (17.31).

The present treatment follows the one by Blundell. A comprehensive description of the spin waves in terms of response to the time-dependent and space-dependent external field can be found at Chap. 6 of the book by White.

Problem 17.3

Derive the q-dependent static magnetic susceptibility $\chi (\mathbf {q},0)$ for delocalized, non-interacting electrons (the Fermi gas) in lattice dimensions $D=3, 2$ and 1. In this latter case comment how the divergence of $\chi (q,0)$ at $q= 2k_F$ implies a spin density wave instability (Kohn anomaly).

Solution: $\chi (\mathbf {q}, 0)$ is the response function to a static spatially varying magnetic field yielding a perturbation $\mathcal {H}_P= g\mu _B \mathbf {S} \cdot \mathbf {H} cos(\mathbf {q}\cdot \mathbf {r})$. Since the eigenstates of the Fermi gas Hamiltonian are $|\mathbf {k}>= exp(i \mathbf {k} \cdot \mathbf {r})$, one immediately realizes that for any $q\ne 0$ $\mathcal {H}_P$ does not give any first order correction. In fact, the cosine term yields non-zero matrix elements of the form $<\mathbf {k}|\mathcal {H}_P|\mathbf {k}\pm \mathbf {q}>$, connecting an initial state $|\mathbf {k}>$ with a final state $|\mathbf {k}\pm \mathbf {q}>$. Then, from second order perturbation theory, by weighting for the probability that the initial state is occupied with Fermi-Dirac distribution function $f_{\mathbf {k}}$ and that the final state must be empty, one derives

$$ \varDelta E= -\frac{\mu _B^2 H^2}{2} \sum _{\mathbf {k}} \biggl [ \frac{f_{\mathbf {k}}(1- f_{\mathbf {k}+ \mathbf {q}})}{E_{\mathbf {k}+ \mathbf {q}}- E_{\mathbf {k}}} + \frac{f_{\mathbf {k}}(1- f_{\mathbf {k}- \mathbf {q}})}{E_{\mathbf {k}- \mathbf {q}}- E_{\mathbf {k}}} \biggr ], $$

with $E_{\mathbf {k}}$ the free-electron dispersion curve. Since the sum runs over all $|\mathbf {k}>$ states it is possible to replace $\mathbf {k}$ with $\mathbf {k}+ \mathbf {q}$ in the second term and taking the second derivative of the energy with respect to the magnetic field one derives

$$ \chi (\mathbf {q}, 0)= {\mu _B^2} \sum _{\mathbf {k}} \frac{f_{\mathbf {k}}- f_{\mathbf {k}+ \mathbf {q}}}{E_{\mathbf {k}+ \mathbf {q}}- E_{\mathbf {k}}} $$

(for the derivation of Eq. (17.13) see the book by White). The sum depends on the lattice dimensionality. The above equation shows that one should expect a large contribution to $\chi (\mathbf {q}, 0)$ at those wave-vectors $\mathbf {q}$ connecting a large number of initial filled states with quasi-degenerate empty final states. This happens if $\mathbf {q}$ connects large portions of the Fermi surface or, in other terms, if $\mathbf {q}$ is a nesting wave-vector. In particular, this occurs in quasi-1D systems for $q= 2k_F$, where a divergence is present, as shown below.

Problem 17.4

Starting from the Hubbard Hamiltonian (Eq. 17.3) show that for half-filled band in the $U\gg t$ limit one derives an effective antiferromagnetic exchange coupling as in Eq. (17.4).

Solution: For $U\gg t$ the hopping term $\mathcal {H}_t$ of the Hamiltonian in Eq. (17.3) is a perturbation of the term $\mathcal {H}_U$ involving U. The energies of the triplet and singlet configurations for two electrons on neighbouring sites i and j are

$$ |S=1, M_S=1>\,= |+_i +_j> , \,\,\, |1,-1>\,= |-_i -_j> , \,\, $$

$$ |1,0>\,= \frac{1}{\sqrt{2}}\biggl ( |+_i -_j> + |-_i +_j>\biggr ) $$

and

$$ |0,0>\,= \frac{1}{\sqrt{2}}\biggl ( |+_i -_j> - |-_i +_j>\biggr ) . $$

These four states are characterized by single occupancy and are eigenstates of $\mathcal {H}_U$ with eigenvalue $E^0_n=0$. Now consider the effect of $\mathcal {H}_t$ on these four states. Since the electron hopping driven by $\mathcal {H}_t$ leads to double occupancy it will generate states orthogonal to these four states and no first order correction. At the second order, i.e.

$$ \varDelta E^{(2)}_n= \sum _{m\ne n} \frac{|<m|\mathcal {H}_t|n>|^2}{E^0_n- E^0_m}, $$

$\mathcal {H}_t$ applied to $|1, \pm 1>$ triplet states yields zero correction since it would lead to double occupancy of a given site with same spin orientation. On the other hand

$$ \mathcal {H}_t|+_i -_j>= t(|+_i -_i> + |+_j -_j>), \mathrm {while} $$

$$ \mathcal {H}_t|-_i +_j>= -t(|+_i -_i> + |+_j -_j>) , $$

where the change of sign is associated with the different order of the spin orientations. By taking into account of the sign reversal one realizes that $<m|\mathcal {H}_t|1,0>\,= 0$, thus none of the triplet states show second order correction. For the singlet $|0,0>$ state one has

$$ \mathcal {H}_t|0,0>= \frac{2t}{\sqrt{2}} (|+_i -_i> + |+_j -_j>). $$

Since the doubly occupied states in parentheses are eigenstates of $\mathcal {H}_U$ with eigenvalue U it turns out

$$ \varDelta E^{(2)}_{|0,0>}= \frac{|<+_i -_i |\mathcal {H}_t|0,0>|^2}{0 - U} + \frac{|<+_j -_j |\mathcal {H}_t|0,0>|^2}{0 - U} = - \frac{4|t|^2}{U} , $$

showing that the energy of the singlet state is lowered by $4|t|^2/U$ with respect to the one of the triplet state. So, for $U\gg t$ Eq. (17.4) holds.

Problem 17.5

From the dispersion curve for the magnons in isotropic (Heisenberg) ferromagnet (Eq. 17.31) show that no magnetic order can be present in two-dimensions (2D), while in the presence of single ion anisotropy magnetic order can exist.

Solution: From Eq. (17.32) the number of magnons excited at a given temperature is given by

$$ n_m= \int _{0}^{\infty } \frac{D(\omega )}{e^{\hbar \omega /k_BT}-1} d\omega , $$

with $D(\omega )$ the density of states. The dispersion relation for a ferromagnet in the $q\rightarrow 0$ limit is quadratic in q, in 2D $D(\omega )$ is constant. Hence

$$ n_m\propto \frac{k_BT}{\hbar } \int _{0}^{\infty } \frac{1}{e^{x}-1} dx $$

with $x=\hbar \omega /k_BT$. For $x\rightarrow 0$ the integrand is 1 / x, giving rise to a logarithmic divergence in the number of excited magnons and then to the disruption of the magnetization. On the other hand, if anisotropy is present there will be a minimum energy cost in exciting a spin wave and an energy gap $\varDelta$ in the dispersion curve arises. Then the previous equation can be written

$$ n_m\propto \frac{k_BT}{\hbar } \int _{\varDelta /k_BT}^{\infty } \frac{1}{e^{x}-1} dx , $$

the logarithmic divergence is truncated and a magnetic order can set in.

Problem 17.6

Consider a domain wall in a ferromagnet and show that the size is related to the magnetic anisotropy and to the exchange coupling.

Solution: Consider a $\pi$ rotation of the magnetic moments across the domain wall. Then, for a domain wall of N lattice steps, with $N\gg 1$, the cost in exchange energy is

$$ E_{exc}= N J S(S+1) \biggl ( \frac{\pi }{N} \biggr )^2 . $$

The cost for the anisotropy energy is given by $E_{an}= K_{an} N$, with $K_{an}$ a phenomenological constant accounting for the magnetic anisotropy. From the derivative of $E_{exc}+ E_{an}$ with respect to N one derives

$$ N= \biggl ( \frac{\pi ^2 J S(S+1)}{K_{an}} \biggr )^2 . $$

Thus the thickness of the domain wall increases with J and decreases with the magnetic anisotropy.

Problem 17.7

For a one-dimensional antiferromagnet with elastic coupling among the nearest neighbour magnetic ions, show that the spatial dependence of the exchange coupling favours a lattice distortion.

Solution: By including in the Heisenberg Hamiltonian an elastic coupling characterized by a constant $k_{el}$, for small displacements from the equilibrium configuration, the energy variation for a pair of adjacent spins turns out

$$ \varDelta E= \frac{1}{2} k_{el} \varDelta x^2 - \biggl ( \frac{\partial J }{\partial x} \biggr )_0 S(S+1) \varDelta x . $$

$\varDelta E$ is minimized for

$$ \varDelta x= \frac{\biggl ( \frac{\partial J }{\partial x} \biggr )_0 S(S+1)}{k_{el}} , $$

namely for any finite change of the exchange coupling with the distance. In quasi-one-dimensional antiferromagnets (e.g. CuGeO$_3$) a structural dimerization is present at the so-called Spin-Peierls transition.

Problem 17.8

Show that in antiferromagnet the static uniform susceptibility for magnetic field perpendicular to the sublattice magnetization is given by Eq. (17.26).

Solution: The magnetic field yields the canting of the sublattice magnetizations as shown below (return to Fig. 17.4):

One writes the energy as

$$ E= \lambda \mathbf {M}_A\cdot \mathbf {M}_B + \alpha \mathbf {M}_B\cdot \mathbf {M}_B + \alpha \mathbf {M}_A\cdot \mathbf {M}_A - \mathbf {H}_{ext}\cdot (\mathbf {M}_A + \mathbf {M}_B) $$

with $\lambda , \alpha > 0$. For small $\theta$ values, by indicating with M the absolute value of the sublattice magnetizations, one has

$$ E= -\lambda M^2 \biggl [1- \frac{1}{2} ( 2\theta )^2 \biggr ] + 2\alpha M^2 - 2M H_{ext} \theta , $$

which is minimized for $\theta = H_{ext}/(2\lambda M)$, $\alpha$-independent. The transverse susceptibility is $\chi _{\perp }= M_{eff}/H_{ext}= 2M \theta / H_{ext}$ and Eq. (17.26) follows.