Self-consistent Hartree-Fock approach to Phase Transitions
Hartree-Fock approach to Phase Transitions
Introduction to self consistent Hartree-Fock theory
The self consistent Hartree-Fock method is a way to approximately find the ground state of a many body system made of n interacting fermions. We start by writing the many particle wavefunction such that it respects the antisymmetry property of fermions. This is achieved by taking Slater determinant of one particle wave functions. The one particle wave functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|i\right\rangle} can themselves be expressed in terms of the basis functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\chi_{k}\right\rangle} as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|i\right\rangle =\underset{k}{\sum}c_{ik}\left|\chi_{k}\right\rangle}
So, the Slater determinant is a function of the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ik}} .
In self consistent field theory, we make a guess for the form of the many electron ground state wave function and then use variation principle to calculate the best approximation for the ground state energy as a function of the coefficients. This leads to a set of non linear Hartree-Fock equations in terms of the coefficients. We then solve the equations iteratively to find the right values of the coefficients upto a certain accuracy. Thus, we obtain the self consistent solution for the ground state wave function of the n particle system and the ground state energy.
With the general introduction self consistent Hartree-Fock method mentioned above, let us now elaborate on the topic with more mathematical details. (WRITE REFERENCE HERE) Consider a system of n interacting fermions. The Hamiltonian is expressed as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\underset{\alpha\alpha'}{\sum}b_{\alpha}^{\dagger}\left\langle \alpha\left|H_{0}\right|\alpha'\right\rangle b_{\alpha'}+\frac{1}{2}\underset{\alpha\alpha'\beta\beta'}{\sum}b_{\alpha}^{\dagger}b_{\beta}^{\dagger}\left\langle \alpha\beta\left|V\right|\alpha'\beta'\right\rangle b_{\beta'}b_{\alpha'}}
The independent particle state can be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_{0}\right\rangle =a_{n}^{\dagger}a_{n-1}^{\dagger} ...a_{2}^{\dagger}a_{1}^{\dagger}\left|0\right\rangle}
In this basis,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\underset{i,j}{\sum}a_{i}^{\dagger}\left\langle i\left|H_{0}\right|j\right\rangle a_{j}+\frac{1}{2}\underset{qrst}{\sum a_{q}^{\dagger}a_{r}^{\dagger}\left\langle qr|V|ts\right\rangle a_{s}a_{t}}}
The variation of state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_{0}\right\rangle} is given by a linear combination
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\psi_{jk}=\varepsilon_{jk}a_{j}^{\dagger}a_{k}\left|\psi_{0}\right\rangle}
The “best” state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi_{0}\right\rangle} should minimise the ground state energy. Also, since H is Hermitian, therefore
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \psi_{0}|\psi_{0}\right\rangle \left\langle \delta\psi|H|\psi_{0}\right\rangle -\left\langle \psi_{0}|H|\psi_{0}\right\rangle \left\langle \delta\psi|\psi_{0}\right\rangle =0}
Further, since the variation preserves the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \psi_{0}|\psi_{0}\right\rangle =1} , this makes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \delta\psi_{0}|\psi_{0}\right\rangle =0} . Then the above equation reduces to
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \delta\psi|H|\psi_{0}\right\rangle =0}
Substituting the form of the variation and the Hamiltonian, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle j\left|H_{0}\right|k\right\rangle +\frac{1}{2}\underset{t}{\sum}[\left\langle jt\left|V\right|kt\right\rangle -\left\langle jt\left|V\right|tk\right\rangle ]=0}
Or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{0}\left|m\right\rangle +\underset{p}{\sum}\left|p\right\rangle \underset{t}{\sum}[\left\langle pt\left|V\right|mt\right\rangle -\left\langle pt\left|V\right|tm\right\rangle ]=\varepsilon_{m}\left|m\right\rangle}
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \alpha\beta\left|V\right|\alpha'\beta'\right\rangle =V_{\alpha\beta}\delta_{\alpha\alpha'}\delta_{\beta\beta'}} then, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{0}\left|m\right\rangle +\underset{t}{\sum}\underset{\alpha,\beta}{\sum}\left|\alpha\right\rangle \left\langle t\right|\left.\beta\right\rangle V_{\alpha\beta}[\left\langle \beta\right|\left.t\right\rangle \left\langle \alpha\right|\left.m\right\rangle -\left\langle \alpha\right|\left.t\right\rangle \left\langle \beta\right|\left.m\right\rangle ]=\varepsilon_{m}\left|m\right\rangle}
In matrix form, this can be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underset{\beta}{\sum}[\left\langle \alpha\right|H_{0}\left|\beta\right\rangle \left\langle \beta\right|\left.m\right\rangle +\underset{t}{\sum}\left\langle t\right|\left.\beta\right\rangle V_{\alpha\beta}[\left\langle \beta\right|\left.t\right\rangle \left\langle \alpha\right|\left.m\right\rangle -\left\langle \alpha\right|\left.t\right\rangle \left\langle \beta\right|\left.m\right\rangle ]]=\varepsilon_{k}\left\langle \alpha\right|\left.m\right\rangle}
This equation is known as Hartree-Fock equation. The first term in the summation over t on the left hand side of the above equation is known as the Hartree term or direct energy term. The last term in the sum is called the Fock term and it constitutes the exchange energy of the interaction between electrons.
The Hartree-Fock equations cannot be computed without knowing the n eigensolutions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|m\right\rangle} since the matrix elements of interaction term V depend on them. These equations are solved iteratively starting by guessing a set of one particle states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|m\right\rangle} to find the next set of eigensolutions. This set is compared with the previous set and if they don't match within error then the process is repeated till we get a self consistent set of eigensolutions.
Hartree-Fock in second quantized form
The interaction energy operator for a two body system with pairwise interactions is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = \frac{1}{2} \sum_{\sigma \sigma^{\prime}} \int d^{3}r d^{3}r^{\prime} v(\vec{r}-\vec{r}^{ \prime}) \psi_{\sigma}^{\dagger}(\vec{r}) \psi_{\sigma^{\prime}}^{\dagger}(\vec{r}^{ \prime}) \psi_{\sigma^{\prime}}(\vec{r}^{ \prime}) \psi_{\sigma}(\vec{r}) }
In the above operator, it is important to note the order of the operators. Including the kinetic energy term we can write the Hamiltonian for particles of mass m with pairwise interactions in second quantized formalism as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \sum_{\sigma \sigma^{\prime}} \int d^{3}r \frac{1}{2m} \nabla \psi_{\sigma}^{\dagger}(\vec{r}) \cdot \nabla \psi_{\sigma}(\vec{r}) + \frac{1}{2} \sum_{\sigma \sigma^{\prime}} \int d^{3}r d^{3}r^{\prime} v(\vec{r}-\vec{r}^{ \prime}) \psi_{\sigma}^{\dagger}(\vec{r}) \psi_{\sigma^{\prime}}^{\dagger}(\vec{r}^{ \prime}) \psi_{\sigma^{\prime}}(\vec{r}^{ \prime}) \psi_{\sigma}(\vec{r}) }
The first-order correction to the energy induced by the interaction is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{(1)} = \frac{1}{2}\int d^{3}rd^{3}r^{\prime}v(\vec{r} - \vec{r}^{\prime})\sum_{\sigma\sigma^{\prime}}<\Phi_{0}|\psi_{\sigma}^{\dagger}(\vec{r}) \psi_{\sigma^{\prime}}^{\dagger}(\vec{r}^{ \prime}) \psi_{\sigma^{\prime}}(\vec{r}^{ \prime}) \psi_{\sigma}(\vec{r})|\Phi_{0}>}
Define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{\sigma}(\vec{r}-\vec{r}^{\prime})= <\Phi_{0}|\psi_{\sigma}^{\dagger}(\vec{r})\psi_{\sigma}(\vec{r}^{\prime})|\Phi_{0}>}
Then, we find that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{(1)} = \frac{1}{2}\int d^{3}r d^{3}r^{\prime} v(\vec{r}-\vec{r}^{ \prime})[n^{2}-\sum_{\sigma}G_{\sigma}(\vec{r}-\vec{r}^{\prime})^{2}]}
(You can find intermediate steps in "Lectures on Quantum Mechanics" by Gordon Baym).
The first term in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{(1)}} is called the Hartree energy or direct energy. The second term is called the Fock term and it corresponds to the energy due to exchange interactions.
Examples
Electron Gas Phase Transition
Using Hartree-Fock theory, it can be shown that an electron gas will undergo a first-order phase transition at a critical value of the chemical potential. At this value, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{0}} , the particle density n experiences a finite discontinuity. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(k) = k^{2} - \frac{1}{\pi^{2}}\int d^{3}q \frac{1}{\left(\vec{q} - \vec{k}\right)^{2} + \lambda^{2}} \theta(\mu - \epsilon(\vec{q}))}
Lambda has been introduced to avoid any divergences. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu < 0} , then there is only one valid solution:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(k) = k^{2}\ \ ,\quad\mu<0}
Define
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(k,k_{0}) = k^{2} - \frac{1}{\pi^{2}}\int d^{3}q \frac{1}{\left(\vec{q} - \vec{k}\right)^{2} + \lambda^{2}} \theta(k_{0} - q))}
k0 is often called the radius of the Fermi sphere and is determined by the equation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(k_{0},k_{0}) = \mu}
Particle density function
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(\mu) = \frac{2}{(2\pi)^{3}} \int d^{3}k \theta(\mu - \epsilon(\vec{k})) = \frac{1}{3\pi^{2}}k_{0}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\mu) = -\int_{-\infty}^{\mu}n(\mu)d\mu = -\frac{1}{3\pi^{2}} \int_{0}^{\mu}k_{0}^{3}d\mu}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{0} = \frac{1}{\pi} + \left( \frac{1}{\pi^{2}} + \mu \right)^{\frac{1}{2}} }
N-electron atom
Briefly presented below is work by Pablo Serra and Sabre Kais using the Hartree-Fock theory to estimate the symmetry breaking in N-electron atoms at the large density (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D \rightarrow \infty} )limit.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}_{\infty }= \frac{1}{2}\underset{i=1}{\overset{N}{\Sigma }}\frac{1}{\rho _i{}^2}-Z\underset{i=1}{\overset{N}{\Sigma }}\frac{1}{\rho _i}+\underset{i=1}{\overset{N-1}{\Sigma }}\underset{j=i+1}{\overset{N}{\Sigma }}\frac{1}{\left(\rho _i{}^2+\rho _j{}^2\right){}^{1/2}}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\infty}(N,Z)=\underset{\left\{\rho _i\right\}}{min\mathcal{H}_{\infty}}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-1}{\rho _i{}^3}+\frac{Z}{\rho _i{}^2}-\rho _i\underset{j\neq i}{\Sigma }\frac{1}{\left(\rho _i{}^2+\rho _j{}^2\right){}^{3/2}}=0,\text{ }i=1,2,\text{...},N }
Symmetric solution with
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho _i=\rho =\frac{2^{2/3}}{\left(2^{2/3}Z-N+1\right)}}
with energy given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\infty }{}^{\text{sym}}(N,Z)=\frac{-N}{2}\left(Z-\frac{N-1}{2^{3/2}}\right)^2}
In the case of N-electrons, the symmetry breaking does not coincide with the stability limit of the symmetric phase point, as it would for N=2. For NFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq } 2, a region has been discovered with more than one local minima indicating a phase coexistence region in mean field theories. The first-order phase transition appears when the global minimum degenerates. The symmetric phase occurs when all electrons are equidistant from the nucleus. For small values of N, this symmetry may be broken by removing one electron to a much larger distance, leaving a core of N-1 electrons, equivalent to saying
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho _1=\rho _2\text{=.}\text{..}=\rho _{N-1}=\rho ,\text{ }\rho _N=(1+\eta )\rho (Phase A_{1})}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta \neq 0} . This is stable for small Z values, but there is a coexistence region with the symmetric solution with both being minimas of the variational equations. For fixed N,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\infty }{}^{\text{sym}}\left(N,Z_1\right)=E_{\infty }{}^{\left(A_1\right)}\left(N,Z_1\right)}
gives the first-order phase transition point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_1} with the right hand side giving the energy in phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} in the above equation is not restricted except that it Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta \neq 0} , with positive and negative values giving a different energy on the right hand side. N2 corresponds to the global minimum when .
