Lecture_Fermion_Determinant

In this lecture, we study Monte Carlo simulations for the model of interacting fermions known as the Hubbard model, for particles with spin 1/2. The story that we will develop is how one of the more complicated models of interacting quantum particles turns out to be ... simply a slightly more involved-than-usual classical Ising model

Hubbard model - hamiltonian
math H = - t \sum_{\langle i,j\rangle}c_i^+ c_j + \text{h.c.} + U \sum_i n_i^{\uparrow} n_i ^{\downarrow} math

Hirsch decoupling (Gaussian decoupling formula for a binary variable)
These days, Jorge Hirsch is uniformally famous for a bibliometric index, the //h//-index. For computational physicists, he is the author of the first simulations on the Hubbard model, of the famous Hirsch decoupling, and of the incredible Hirsch-Fye algorithm. Here we describe the decoupling.

In the following table, we set math X = \exp(-\Delta \tau U n^{\uparrow} n^{\downarrow}) math , and we compute the action of the operator //X// on the four states with math n^{\uparrow} n^{\downarrow} = \pm 1 math

math \begin{array}{c|cc|c} X&0&1& n^{\uparrow}\\ \hline 0&1&1\\ 1&1& \exp (-\Delta \tau U) \\ n^{\downarrow}\\ \end{array} math

In the following table, we set math Y = \frac{1}{2}\left[ \exp(\lambda [ n^{\uparrow}- n^{\downarrow} ] -\frac{\Delta \tau}{2} U [n^{\uparrow} + n^{\downarrow}] ) + \exp(-\lambda [ n^{\uparrow}- n^{\downarrow} ] -\frac{\Delta \tau}{2} U [n^{\uparrow} + n^{\downarrow}] ) \right] math , and likewise compute the action of the operator //X// on the four states with math n^{\uparrow} n^{\downarrow} = \pm 1 math

math \begin{array}{c|cc|c} Y&0&1& n^{\uparrow}\\ \hline 0&1&\exp(-\frac{\Delta \tau}{2} U) \left[\frac{\exp (\lambda)}{\exp( -\lambda)}\right]\\ 1&1& \exp (-\Delta \tau U) \\ n^{\downarrow}\\ \end{array} math

which leads to the condition math \cosh \lambda = \exp(-\frac{\Delta \tau}{2} U) math

and to the final result: math \exp(-\Delta \tau U n^{\uparrow} n^{\downarrow}) = \text{Tr}_{\sigma = \pm 1} \exp\left[ \lambda \sigma (n^{\uparrow} - n^{\downarrow}) - \frac{\Delta \tau}{2} U (n^{\uparrow} + n^{\downarrow}) \right] math

The trace in this expression is the sum over the values +/- 1 of a simple Ising variable.

=The Blankenbecler-Scalapino-Sugar (BSS) determinant formula=

The point of the BSS formula is that the fermion trace over the exponential of a bilinear operator expression can be done easily. math \text{Tr} \left[ \exp (-c_i A_{ij}c_j) \exp( - c_i B_{ij} c_j\right] = = \det \left[ 1 + \exp(-A) \exp(-B) \right] math

This formula can be derived using Grassmann algebras. A direct calculation was provided by J. E. Hirsch in the appendix of :

math Z = \sum_{\text{states} \alpha } \langle \alpha |\exp(- \beta H) | \alpha \rangle math

We first show how to treat bilinear terms, such as

math \text{Tr} \exp( - c_i^+ B_{ij} c_j) = \text{Tr} \prod_\mu \exp( - c_\mu^+ b_\mu c_\mu) math

For this simple case, we can expand the exponential into

math \exp( - c_\mu^+ b_\mu c_\mu) = 1 - c_\mu^+ b_\mu c_\mu + \frac{1}{2}c_\mu^+ b_\mu c_\mu c_\mu^+ b_\mu c_\mu - \frac{1}{3!}..... math

Using math c_\mu^+ c_\mu + c_\mu c_\mu^+ = 1 math we reach

math \exp( - c_\mu^+ b_\mu c_\mu) = 1 [+ c_\mu^+ b_\mu c_\mu ] + \frac{1}{2}c_\mu^+ b_\mu^2 c_\mu - \frac{1}{3!}c_\mu^+ b_\mu^3 c_\mu .... [-c_\mu^+ b_\mu c_\mu ] = \prod_\mu [ 1 + ( \exp (- b_\mu) - 1) c_\mu^+ c_\mu math

This expression is easily evaluated. It yields the expression math \det[1 + \exp(-B)] math

More generally,, one finds:

math \text{Tr} \exp( - c_i^+ A_{ij} c_j)\exp( - c_i^+ B_{ij} c_j) = \det[1 + \exp(-A) \exp(-B) ] math

This non-trivial formula is the key to determinantal fermion methods. We note that in it, the trace is over the fermion variables, that is we have integrated out the quantum variables. Notice that, at a difference with what we did in the last lecture, the fermion trace is over all occupations of fermions: we must introduce a chemical potential to fix the density of electrons.

Fermion algorithm
In the following, we put all pieces together. For simplicity, we imagine a one-dimensional Hubbard model:

math H = -t \sum_i c_i^+ c_j + U \sum_i n_i^{\uparrow} n_i^{\downarrow} = H_0 + H_1 math

math \exp( -\beta H) = \prod \exp( -\tau H) \exp( -\tau H) \exp( -\tau H) \exp( -\tau H) ... math

Using the Hirsch decoupling discussed just above, we have that the partition function is

math Z = \sum_{\sigma_1 = \pm 1} .... \sum_{\sigma_L = \pm 1} \left[ \exp (-\tau H_0) \exp(\lambda \sigma_1 (n^{\uparrow} - n^{\downarrow} - \frac{\tau}{2} U (n^{\uparrow} + n^{\downarrow}) \right] \times \times \times math We can now separate the (quantum) spin up from the (quantum) spin down and arrive, at each slice, arrive at an expression in terms of a hopping matrix K which connects nearest neighbors and a diagonal matrix V, which depends on the Ising spin.

math \exp ( - c_i^{+ \uparrow} (\tau K_{ij}) c_i^{ \uparrow}) \exp ( - c_i^{+ \uparrow} V(\sigma) c_j ) math

As we can use the BSS formula for performing the trace over the fermions, we are left with

math Z = \text{Tr}_{\sigma} \prod_{\alpha = \pm 1} \det [1 + B_L(\alpha) B_{L-1}(\alpha) \cdots B_1(\alpha)] = \text{Tr}_{\sigma} \det O_{\uparrow} O_{\downarrow} math

We have reached the final point of our discussion, namely the representation of an interacting fermion problem in terms of an Ising model with a strange interaction, namely a product of two determinants. This expression is due to Blankenbecler, Scalapino and Sugar, and the first simulations of the Hubbard model were done by Hirsch. Both are magnificent achievements in theoretical physics. Much of the complexity has still been hidden, for example concerning the calculation of Greens functions and other observables.

=References=