Random physics

Alberto Verga, research notebook

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Lectures on statistical mechanics.

Solids: effective models

Solid is an ordered state of matter characterized by a crystal structure and electronic bands. Atoms are located on the sites of the crystal lattice, tightly bound to their neighbors, their vibrations give raise to phonons, the quantum of the acoustic modes; electrons, depending on the chemical properties of the atoms, can jump from one site to another, carry a magnetic moment (spin), or interact with the nucleus and crystal potentials through the spin-orbit coupling, giving rise to a wealth of different physical properties and effects, such thermal or electric conductivity, ferromagnetism, piezoelectric effects, quantum Hall conductance, giant magnetoresitence or topological effects. In spite of the complexity of the solid’s band structure, specific electronic properties can be described by relatively simple effective hamiltonians comprising a few parameters, in which the complex crystal structure is replaced by a regular lattice and the collective electron coulomb interactions, by short range neighbors interactions.

The simplest example is a free electron in a metal. The presence of the crystal periodic potential creates gaps in the continuum spectrum. A tight binding hamiltonian on a cubic lattice, in which the electron can jump from one site to one of its six neighbors, is enough to account for the band structure; the hoping energy \(t\) is the only parameter of the model:

$$H = - t \sum_{(\boldsymbol x, \boldsymbol y)} \big[c^\dagger({\boldsymbol x}) c({\boldsymbol y}) + c^\dagger({\boldsymbol y}) c({\boldsymbol x})\big]$$

where \(\boldsymbol y = \{ (x \pm 1,y,z), (x,y\pm1,z), (x,y,z\pm1) \}\), is a neighbor of the site \(\boldsymbol x \in \mathbb{Z}^3\) of the cubic lattice. The operator \(c({\boldsymbol x})\) annihilate the particle at site \(\boldsymbol x\):

$$c^\dagger(x+1,y,z) c(x,y,z) \ket{x,y,z} = \ket{x+1,y,z}\,,$$

its hermitian conjugate \(c^\dagger({\boldsymbol x})\), creates a particle at this site.

Many universal properties of magnetic systems are well described by the ising hamiltonian,

$$H = -J \sum_{(\boldsymbol x, \boldsymbol y)} s_{\boldsymbol x} s_{\boldsymbol y}\,,$$

where \(s_{\boldsymbol x} = \pm 1\) is a classical spin variable, \(J\) is the exchange energy constant, and \((\boldsymbol x, \boldsymbol y)\) are neighboring sites in a lattice. The ising model is a simplified version of the quantum heisenberg hamiltonian:

$$H = -J\sum_{(\boldsymbol x, \boldsymbol y)} \boldsymbol\sigma_{\boldsymbol x} \cdot \boldsymbol\sigma_{\boldsymbol y}\,,$$

where \(\sigma\) is the vector of pauli matrices. The ising and heisenberg hamiltonians, describe a magnetic material as a set of spins (classical or quantum), attached to the sites of a lattice (linear, square or cubic, depending on the dimension, for example), interacting with their nearest neighbors via the \(J\) exchange coupling (mimicking the overlap of electronic functions).

More generally, lattice models are useful in a variety of domains, ranging from condensed matter to biology or social sciences. They can describe statistical equilibrium properties, like phase transitions, or nonequilibrium processes, like crystal growth,1 population dynamics in a living system,3 or opinion propagation in a social network.2

Free particles in a lattice

We consider a gas of noninteracting, spinless bosons in a cubic lattice of step size \(a=1\), which we take as the length unit (fermions can be treated similarly). The system’s hamiltonian is,

$$H_N = \sum_{n=1}^N H_n\,, \quad H_n = - t \sum_{(\boldsymbol x, \boldsymbol y)} [a^\dagger_n({\boldsymbol x}) a_n({\boldsymbol y}) + a^\dagger_n({\boldsymbol y}) a_n({\boldsymbol x})]\,,$$

where \(N\) is the number of particles and the operators create or annihilate particle \(n\) at a lattice site \(\boldsymbol x \in \mathbb{Z}^3\). In order to write the grand potential we diagonalize the hamiltonian using a change of the position basis \(\boldsymbol x\), to the momentum \(\boldsymbol k\) basis, through the fourier transformation,

$$a_n({\boldsymbol x}) = \frac{V}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \, \E^{-\I \boldsymbol k \cdot \boldsymbol x} a_n({\boldsymbol k})\,,$$

where \(V=L^3\) is the number of sites in the cubic lattice of edge \(L\), the integration is over the first “brillouin zone” (here, the cube \(\mathrm{BZ} = (-\pi,\pi)^3\) in momentum space). Note that the inverse transformation gives,

$$c_n({\boldsymbol k}) = \frac{1}{V} \sum_{\boldsymbol x} \E^{\I \boldsymbol k \cdot \boldsymbol x} c_n({\boldsymbol x})\,,$$

and that,

$$\frac{V}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \, \E^{-\I \boldsymbol k \cdot (\boldsymbol x - \boldsymbol y)} = V \delta_{\boldsymbol x, \boldsymbol y}$$

is the kronecker delta. In the momentum basis the hamiltonian becomes EX,

$$H = \frac{V}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \, \epsilon(\boldsymbol k) a^\dagger(\boldsymbol k) a(\boldsymbol k)\,,$$

where

$$\epsilon(\boldsymbol k) = -2(\cos k_x + \cos k_y + \cos k_z)\,.$$

The grand potential \(\Phi(T,\mu,V)\) of the noninteracting hamiltonian, writes:

$$\Phi(T,\mu,V) = \frac{TV}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \, \ln\left( 1 - \E^{\mu/T} \E^{-\epsilon(\boldsymbol k)/T} \right)\,,$$

where \(H\) is the single particle hamiltonian. The number of particles and the energy are given by the integrals,

$$N = \frac{V}{(2\pi)^3} \int_{-\pi}^\pi \frac{\D \boldsymbol k} {\E^{\mu/T} \E^{\epsilon(\boldsymbol k)/T} - 1} \,,$$

and

$$E = \frac{V}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \, \frac{\epsilon(\boldsymbol k)}{\E^{\mu/T} \E^{\epsilon(\boldsymbol k)/T} - 1} \,.$$

To compare the behavior of the gas in a lattice with the ideal gas, we keep the first term in the fugacity (Boltzmann limit, which is similar for fermions and bosons). In this limit we have,

\begin{align*} \Phi & = - \frac{TV \E^{\mu/T}}{(2\pi)^3} \int_{-\pi}^\pi \D \boldsymbol k \,\E^{\epsilon(\boldsymbol k)/T} \\ & = - \frac{TV \E^{\mu/T}}{(2\pi)^3} \left[\int_{-\pi}^\pi \D k \, \E^{2t \cos k/T} \right]^3\\ & = - TV \E^{\mu/T} I_0(2t/T)^3\,, \end{align*}

where \(I_0\) is the modified bessel function of the first kind,

$$I_0(z) = \frac{1}{\pi} \int_0^\pi \D t \, \E^{z \cos t}\,,$$

and the particle number,

$$N = \frac{V \E^{\mu/T}}{(2\pi)^3} \left[\int_{-\pi}^\pi \D k \, \E^{2t \cos k/T} \right]^3 = V \E^{\mu/T} I_0(2t/T)^3\,,$$

which combined with the expression of \(\Phi= -PV\), one recovers

$$PV = NT$$

independent of the hoping energy \(t\). However, the lattice gas energy differs to the one of the ideal gas.

Indeed, the energy is given by,

$$E = -2t\frac{V \E^{\mu/T}}{(2\pi)^3} \left[\int_{-\pi}^\pi \D k \, \cos k \, \E^{2t \cos k/T} \right]^3 = -2TV \E^{\mu/T} I_1(2t/T)^3\,,$$

or, eliminating the chemical potential,

$$E = -2tN \left[\frac{I_1(2t/T)}{I_0(2t/T)} \right]^3\,.$$

The ideal gas energy is found in the continuous limit (\(a \rightarrow 0\)), which corresponds to \(t \sim \hbar^2/2ma^2 \rightarrow \infty\) EX,

$$E \approx -2tN \frac{1-9T/16t}{1+3T/t} = -2tN + \frac{3}{2}NT\,,$$

where, up to an irrelevant constant, gives the ideal gas result. We used the asymptotic expression,

$$I_n(z) \approx \frac{\E^z}{\sqrt{2\pi z}} \left( 1 - \frac{4n^2 - 1}{8 z} \right)\,.$$

In the opposite limit \(T \gg t\), the energy grows slowly with the temperature:

$$E \approx -2tN \left( \frac{t}{T} \right)^3\,.$$

(The small \(z\) limit of the \(I\) bessel functions is \(I_n(z) = (z/2)^n/n!\).)

One dimensional spin model

In the presence of an external field \(B\), the one dimensional ising hamiltonian is,

$$H = -J\sum_{x=1}^N s_x s_{x+1} - \sum_{x=1}^N s_x B$$

where we assumed periodic boundary conditions \(s_1 = s_{N+1}\). We choose units based on the lattice constant and the spin magnetic moment, \(a = g\mu_\text{B}=1\) (we also put the permeability \(\mu_0=1\)); in such units the magnetic field \(B\) has the dimensions of the energy.

The probability of a spin configuration \(s\),

$$s = \{s_1=\pm1, s_2=\pm1, \ldots, s_N=\pm 1\}$$

is,

$$P[s] = \frac{1}{Z} \prod_x \exp\left[\frac{J}{T} s_x s_{x+1} + \frac{B}{2T}(s_x + s_{x+1}) \right]$$

and the partition function is given by,

$$Z(T,B) = \sum_s \prod_x \exp\left[K s_x s_{x+1} + \frac{B}{2T} (s_x + s_{x+1}) \right] \,,$$

where \(K = J/T\). We observe that the thermodynamics of the present model depends on two nondimensional parameters, the interaction energy-temperature ratio \(K = J/T\) and the external field-temperature ratio \(h = B/T\). The idea to compute the partition function is to try (obviously this is not always possible) to factor it in a series of identical terms, like in the noninteracting case; we symmetrized the \(B\) term to this goal. We note first that the product appearing in \(Z\) can be written in the form,

$$\prod_x \exp\left[K s_x s_{x+1} + \frac{B}{2T} (s_x + s_{x+1})\right] = T_{s_1,s_2} T_{s_2,s_3}\ldots T_{s_N,s_1}\,,$$

where each term \(T_{s_x,s_{y}} = T_{s_y,s_x}\) can take different values according to the values of \(s_x = \pm 1\) and \(s_{y} = \pm 1\):

$$T_{s_x,s_y} = \{\E^{K \pm h}, \E^{-K}\}\,.$$

Therefore, the partition function becomes,

$$Z = \sum_{s_1} \sum_{s_2} \ldots \sum_{s_N} T_{s_1,s_2} T_{s_2,s_3}\ldots T_{s_N,s_1}\,.$$

A generic factor is,

$$\sum_{s_x} T_{s_{x-1},s_x} T_{s_x,s_{x+1}} = T_{s_{x-1},s_{x+1}}\,,$$

corresponding to the product of two matrices,

$$T = \begin{pmatrix} \E^{K+h} & \E^{-K} \\ \E^{-K} & \E^{K-h} \\ \end{pmatrix} \,.$$

Using the transfer matrix \(T\) to write the partition function,

$$Z = \mathrm{Tr}\, T^N\,.$$

Therefore, using the invariance of the trace under basis transformations EX,

$$Z = \lambda_+^N + \lambda_-^N$$

where \(\lambda_\pm\) are the eigenvalues of the transfer matrix \(T\):

$$\lambda_\pm = \E^K \cosh h \pm \sqrt{\E^{2K}\sinh^2h + \E^{-2K}}\,.$$

In the thermodynamic limit \(N\rightarrow\infty\), only the larger eigenvalue of the transfer matrix survives in \(Z\):

$$Z = \lambda_+^N+\lambda_-^N= \lambda_+^N \big[1 + (\lambda_-/\lambda_+)^N \big] \approx \lambda_+^N\,,$$

from which we compute the free energy,

$$F(T,B) = -N T \ln\left( \E^{J/T} \cosh(B/T) + \sqrt{\E^{2J/T}\sinh^2(B/T) + \E^{-2J/T}} \right)\,.$$

The mean magnetization per particle \(m=M/N \braket{s_x}\), is given by EX,

$$m = -\frac{1}{N}\frac{\partial F}{\partial B} = \frac{\E^{2J/T}\sinh(B/T)}{\sqrt{\E^{4J/T}\sinh^2(B/T) + 1}}\,.$$

In the limit of small \(B\) (and finite temperature), the magnetization is proportional to the field:

$$m \approx \chi B\,, \quad \chi = \frac{\E^{2J/T}}{T}\,,$$

a behavior of the paramagnetic type, with a susceptibility inversely proportional to the temperature, corrected with the factor \(\E^{J/T}\) depending on the interaction energy. However, for \(T=0\), te magnetization becomes,

$$m = \mathrm{sgn}(B)\,,$$

behavior reminiscent to a phase transition (at critical temperature \(T_c=0\)): indeed, at zero temperature the two values of the magnetization \(m=\pm1\) correspond to an ordered state with all spins pointing up or down.

Mean field spin model

Instead of considering the nearest neighbor interaction, we can investigate the case of long range interaction, where each spin on a lattice indexed by \(x=1,2,\ldots,N\) (\(N\) is the number of spins) can interact with all other spins:

$$H=-\frac{J}{2N} \sum_{x,y} s_x s_y - B \sum_x s_x\,,$$

where the factor two is to avoid counting twice each pair \((x,y)\), and the factor \(1/N\) is to ensure that the energy is extensive (there are \(\sim N^2\) terms of order one in the sum). This model is easily solved by noting that the hamiltonian corresponds to one spin in an effective field \(B_s\) created by the other spins, in addition to the real applied field:

$$B_s = \frac{J}{N}\sum_x s_x + B = Jm + B = \frac{J}{2}m + B\,,$$

where

$$m = \frac{1}{N}\sum_x s_x,$$

is the magnetization per site, or, inserting this expression into \(H\),

$$H = -B_s\sum_x s_x\,,$$

The partition function is then,

$$Z = [2\cosh(B_s/T)]^N\,.$$

the free energy is,

$$F(T,B) = -NT \ln\left( 2\cosh\frac{B_s}{T} \right)\,,$$

and the magnetization is given by the self-consitent equation,

$$m = \tanh\left( \frac{Jm/2 + B}{T} \right)\,.$$

At variance to the one dimensional case, the mean free model account for the existence of a non zero magnetization, even in the absence of an external magnetic field. Such a state is called ferromagnetic: it is a thermodynamic phase with ordered spins and spontaneous magnetization; it differs to the disordered paramagnetic phase, whose spontaneous magnetization vanishes. Therefore, the mean field model is able to describe a magnetic system undergoing, as a function of the temperature and magnetic field, a phase transition from a disordered state at high temperatures, to an ordered state at low temperatures. The critical temperature is given by \(T_c=J/2\). For \(T > T_c\), only the value \(m = m_P(T) = 0\) is solution, while for \(T,T_c\) two equilibrium values are found \(m = \pm m_F(T)\), corresponding to the paramagnetic and ferromagnetic phases respectively.

Let us compute the heat capacity and the susceptibility in the present framework. Calculating the derivative of the free energu with respect to the temperature we find the energy EX,

$$\frac{E}{N} = -mB - \frac{Jm^2}{2}\,, \quad m = m_P,\pm m_F\,,$$

in the form of an implicit function of the temperature, through the equilibrium value of the magnetization (\(m_P\) or \(\pm m_F\)). The derivative of \(E\) with respect to \(T\) gives the specific heat,

$$c_V = \frac{1}{T^2} \frac{\left(B + J m\right) \left(B + J m/2\right)}{ \cosh^{2}{\left (\frac{2 B + J m}{2 T} \right )} - \frac{J}{2T} }\,, \quad m = m_P,\pm m_F\,,$$

where we used the expression of the derivative,

$$\frac{\D m}{\D T} = -\frac{ \frac{\D g}{\D T} }{ \frac{\D g}{\D m} } \,,\quad g = m - \tanh\left( \frac{Jm/2 + B}{T} \right)\,.$$

The susceptibility is computed using also the derivative of the implicit function \(m=m(T)\),

$$\chi = \frac{1}{T} \left[\cosh^{2}{\left (\frac{J m}{2 T} \right ) - \frac{J}{2T}}\right]^{-1}\,, \quad m = m_P,\pm m_F\,,$$

at \(B=0\).

Magnetization and susceptibility as a function of the temperatue (\(B=0\)). A phase transition occurs at \(T=T_c\) between a hight temperature disordered state \(m=0\) (paramagnetic) and a low temperature ordered state \(m \ne 0\) (ferromagnetic); at the transition temperature the susceptibility diverges.

Specific heat capacity of the ferromagnet as a function of the temperature; at the critical temperature it has a jump of \(3\). In the paramagnetic state the specific heat of the magnet vanishes.

Notes


  1. Barabási, A. L. and Stanley, H. E., Fractal Concepts in Surface Growth (Cambridge, 1995) 

  2. Barrat, A., Barthélemy, M., and Vespignani, A., Dynamical Processes in Complex Networks (Cambridge, 2008) 

  3. Krapivsky, P., Redner, S., and Ben-Naim, E., A Kinetic View of Statistical Physics (Cambridge, 2010)