Wick's Theorem for Non-Interacting Ground States

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Hello! Welcome to the next lesson in our journey through many-body quantum mechanics.

In our last session, we successfully translated the familiar operators of quantum mechanics—like kinetic and potential energy—into the language of second quantization. We now have a compact and powerful way to write down the Hamiltonian for a system of many interacting particles, regardless of their number.

However, writing down the Hamiltonian is only the first step. The ultimate goal is to solve for the system's properties, which often requires calculating expectation values of operators. This typically involves wrestling with long, complicated strings of creation and annihilation operators. Today, we'll learn a systematic and powerful technique to tame this complexity.

Our learning outcome is to apply Wick's theorem to evaluate expectation values of operator products in a non-interacting ground state. This theorem is a cornerstone of many-body theory, providing a combinatorial recipe that drastically simplifies calculations. It's the engine that powers perturbation theory and the diagrammatic methods you'll encounter in both condensed matter and quantum field theory.

The Challenge: Evaluating Operator Strings

Suppose we have a many-body Hamiltonian $H^= H^_{0} + V^$ , where $H^_{0}$ is the non-interacting part and $V^$ is the interaction. A central task in perturbation theory is to calculate matrix elements of $V^$ between eigenstates of $H^_{0}$ . As we saw, $V^$ is composed of strings of four creation/annihilation operators.

Let's consider the simplest non-interacting ground state: the vacuum, $∣0 ⟩$ , which has no particles. How would we calculate the expectation value of a random string of operators, say $⟨ 0∣ c_{i} c_{j}^{†} c_{k} c_{l}^{†} ∣0 ⟩$ ?

We could manually use the anti-commutation relations to move all annihilation operators to the right, where they would annihilate the vacuum state $∣0 ⟩$ , and all creation operators to the left, where they would annihilate the bra $⟨ 0∣$ . For example, for a single pair of fermionic operators:

⟨ 0∣ c_{i} c_{j}^{†} ∣0 ⟩ = ⟨ 0∣ (- c_{j}^{†} c_{i} + {c_{i}, c_{j}^{†}}) ∣0 ⟩ = ⟨ 0∣ (- c_{j}^{†} c_{i} + δ_{ij}) ∣0 ⟩

Since $c_{i} ∣0 ⟩ = 0$ and $⟨ 0∣ c_{j}^{†} = 0$ , the first term vanishes, leaving us with:

⟨ 0∣ c_{i} c_{j}^{†} ∣0 ⟩ = δ_{ij}

For a string of four, six, or more operators, this manual process becomes extremely tedious and prone to sign errors. We need a more systematic approach.

Wick's Theorem: The Systematic Approach

Wick's theorem provides a general algorithm for this task. It relates an arbitrary product of operators to a sum of "neater" products that are easier to handle. The theorem is built on two key concepts: normal ordering and contraction.

Let's dive into the formal definitions and the statement of the theorem.

INTRODUCTION TO THE MANY-BODY PROBLEM

The following text from 'INTRODUCTION TO THE MANY-BODY PROBLEM' provides a clear and formal statement of Wick's theorem. It defines the building blocks of the theorem—normal ordering and contraction—and outlines the general result.

Please read section 1.6, 'Wick’s theorem' (pages 16-18). Pay close attention to: The definition of a normal-ordered product (: ... :), which shuffles all creation operators to the left of all annihilation operators. The definition of a contraction, which is simply the vacuum expectation value of a pair of operators. The general statement of the theorem (the first paragraph on page 17), which expresses any product as a sum of normal-ordered products with all possible contractions. You can skim the formal proof by induction, but make sure you understand the structure of the result from the examples for n=2 and n=4 operators.

Let's summarize the key ideas from the reading:

Normal Order, $: A_{1} A_{2} ... A_{n} :$ : The product is reordered so all creation operators are on the left and all annihilation operators are on the right. For fermions, a factor of $(- 1)^{P}$ is included, where $P$ is the number of permutations needed to achieve the order. The crucial property of a normal-ordered product is that its vacuum expectation value is always zero (unless it's just a number): $⟨ 0∣ : A_{1} ... A_{n} : ∣0 ⟩ = 0$ .
Contraction, $A_{i} A_{j}$ : This is just a shorthand for the vacuum expectation value of a pair of operators: $A_{i} A_{j} \equiv ⟨ 0∣ A_{i} A_{j} ∣0 ⟩$ . A contraction is just a number. As we saw, the only non-zero elementary contraction is $c_{i} c_{j}^{†} = δ_{ij}$ .

Wick's Theorem states: An ordinary product of creation and annihilation operators is equal to the sum of its normal-ordered product plus the normal-ordered products of the string with all possible single contractions, all possible double contractions, and so on, up to the fully contracted term.

For example, for four operators:

A_{1} A_{2} A_{3} A_{4} =: A_{1} A_{2} A_{3} A_{4} : + pairs \sum A_{i} A_{j} : A_{k} A_{l} : + double pairs \sum A_{i} A_{j} A_{k} A_{l}

Application: Simplifying Vacuum Expectation Values

The real power of Wick's theorem becomes apparent when we want to calculate the vacuum expectation value (VEV) of an operator string. If we take the VEV of the Wick-expanded expression, every term that contains a normal-ordered product gives zero.

⟨ 0∣ A_{1} A_{2} ... A_{n} ∣0 ⟩ = ⟨ 0∣ (fully contracted terms) ∣0 ⟩

This simplifies the theorem immensely for this specific application:

The vacuum expectation value of a product of operators is equal to the sum of all possible full contractions.

A "full contraction" means pairing up every operator in the string with another one. This immediately implies:

The VEV is zero if there's an odd number of operators.
The VEV is zero if the number of creation operators does not equal the number of annihilation operators.

Example: A Four-Operator String

Let's revisit our problem of calculating $⟨ 0∣ c_{i} c_{j} c_{k}^{†} c_{l}^{†} ∣0 ⟩$ for fermions. According to the theorem, we just need to sum all possible ways to fully contract these four operators. A contraction pairs an annihilation operator ( $c$ ) with a creation operator ( $c^{†}$ ).

The possible pairings are:

Pair $c_{i}$ with $c_{k}^{†}$ and $c_{j}$ with $c_{l}^{†}$ .
Pair $c_{i}$ with $c_{l}^{†}$ and $c_{j}$ with $c_{k}^{†}$ .

The expression is the sum of these pairings, with signs determined by the permutations.

⟨ 0∣ c_{i} c_{j} c_{k}^{†} c_{l}^{†} ∣0 ⟩ = c_{i} c_{j} c_{k}^{†} c_{l}^{†} + c_{i} c_{j} c_{k}^{†} c_{l}^{†} (visual representation)

Let's evaluate each term:

$c_{i} c_{k}^{†} c_{j} c_{l}^{†}$ : The operators are already in a contractible order. The sign is $+ 1$ . The value is $δ_{ik} δ_{j l}$ .
$c_{i} c_{l}^{†} c_{j} c_{k}^{†}$ : To bring the pair $(c_{j}, c_{k}^{†})$ together, we need to move $c_{j}$ past $c_{k}^{†}$ . This is one permutation of fermionic operators, so we get a sign of $- 1$ . The value is $- δ_{i l} δ_{j k}$ .

Summing them up, we get the final result:

⟨ 0∣ c_{i} c_{j} c_{k}^{†} c_{l}^{†} ∣0 ⟩ = δ_{ik} δ_{j l} - δ_{i l} δ_{j k}

This determinant-like structure is a characteristic signature of fermionic statistics emerging naturally from the theorem. For bosons, the sign would always be positive.

Generalizing to Non-Interacting Ground States

The concept is not limited to the true vacuum $∣0 ⟩$ . It works for any ground state of a non-interacting Hamiltonian, such as the Fermi sea $∣ FS ⟩$ , which is the ground state for a system of non-interacting fermions.
$∣ FS ⟩ = \prod_{∣ k ∣ \leq k_{F}} c_{k}^{†} ∣0 ⟩$
In this case, a "contraction" is defined with respect to this new ground state:
$A_{i} A_{j}_{FS} \equiv ⟨ FS ∣ A_{i} A_{j} ∣ FS ⟩$
The theorem's structure remains identical: the expectation value of a string is the sum of all full contractions. The only thing that changes is the value of the elementary contractions. For example:

$⟨ FS ∣ c_{k}^{†} c_{l} ∣ FS ⟩ = ⟨ FS ∣ n^_{k} ∣ FS ⟩ δ_{k l} = n_{k} δ_{k l}$ , where $n_{k} = 1$ if the state $k$ is occupied (i.e., $∣ k ∣ \leq k_{F}$ ) and 0 otherwise.
$⟨ FS ∣ c_{k} c_{l}^{†} ∣ FS ⟩ = ⟨ FS ∣ (1 - n^_{k}) ∣ FS ⟩ δ_{k l} = (1 - n_{k}) δ_{k l}$ .

This generalization is exactly what is needed to apply perturbation theory to realistic systems like the electron gas, which you've studied in the context of the Jellium model. The following resource shows this application in the context of Hartree-Fock theory.

A refresher in many-body theory

The power of Wick's theorem extends to expectation values with respect to any non-interacting ground state. These notes by A.-M. S. Tremblay demonstrate this in the context of the Hartree-Fock ground state, which is a determinant of single-particle orbitals.

Read section 1.4.3, 'Wick’s theorem' (pages 14-15). Observe how the expectation value of the four-operator string in Eq. (1.60) is factorized into a product of two-operator expectation values, as shown in Eqs. (1.63) and (1.65). This is precisely Wick's theorem being applied to the Fermi sea ground state.

Test your understanding!

Let's use Wick's theorem for a physically important calculation. Find the first-order correction to the energy of a non-interacting Fermi gas due to a two-body potential $V^$ . This correction is given by $E^{(1)} = ⟨ FS ∣ V^∣ FS ⟩$ .

The two-body operator is $V^= \frac{1}{2} \sum_{i, j, k, l} V_{ij k l} c_{i}^{†} c_{j}^{†} c_{l} c_{k}$ .
Your task is to evaluate $⟨ FS ∣ c_{i}^{†} c_{j}^{†} c_{l} c_{k} ∣ FS ⟩$ using Wick's theorem.

Show answer

We need to calculate $⟨ FS ∣ c_{i}^{†} c_{j}^{†} c_{l} c_{k} ∣ FS ⟩$ . According to Wick's theorem, this is the sum of all possible full contractions. The operators are two creation and two annihilation operators. Contractions must pair a creation with an annihilation operator.

The possible pairings are:

$c_{i}^{†} c_{l} c_{j}^{†} c_{k}$ : This requires one permutation (moving $c_{l}$ past $c_{j}^{†}$ ). So the sign is $(- 1)^{1} = - 1$ . The value is $- ⟨ c_{i}^{†} c_{l} ⟩ ⟨ c_{j}^{†} c_{k} ⟩$ .
$c_{i}^{†} c_{k} c_{j}^{†} c_{l}$ : This requires zero permutations to bring $c_{i}^{†}, c_{k}$ together and two permutations to bring $c_{j}^{†}, c_{l}$ together. The sign is $(- 1)^{0} = + 1$ . The value is $⟨ c_{i}^{†} c_{k} ⟩ ⟨ c_{j}^{†} c_{l} ⟩$ .
$c_{i}^{†} c_{j}^{†} c_{l} c_{k}$ : This term is zero because $⟨ FS ∣ c_{i}^{†} c_{j}^{†} ∣ FS ⟩ = 0$ and $⟨ FS ∣ c_{l} c_{k} ∣ FS ⟩ = 0$ . You cannot change the particle number of the ground state and get a non-zero inner product with the ground state itself.

Summing the non-zero terms:

⟨ FS ∣ c_{i}^{†} c_{j}^{†} c_{l} c_{k} ∣ FS ⟩ = ⟨ c_{i}^{†} c_{k} ⟩ ⟨ c_{j}^{†} c_{l} ⟩ - ⟨ c_{i}^{†} c_{l} ⟩ ⟨ c_{j}^{†} c_{k} ⟩

Using our rule $⟨ FS ∣ c_{p}^{†} c_{q} ∣ FS ⟩ = n_{q} δ_{pq}$ , we get:

(n_{k} δ_{ik}) (n_{l} δ_{j l}) - (n_{l} δ_{i l}) (n_{k} δ_{j k})

Plugging this back into the expression for $E^{(1)}$ :
\begin{align*}
E^{(1)} &= \frac{1}{2} \sum_{i,j,k,l} V_{ijkl} (n_k \delta_{ik} n_l \delta_{jl} - n_l \delta_{il} n_k \delta_{jk}) \
&= \frac{1}{2} \sum_{k,l} (V_{klkl} n_k n_l - V_{kllk} n_k n_l) \
&= \frac{1}{2} \sum_{k,l} (V_{klkl} - V_{kllk}) n_k n_l
\end{align*}
This is a classic result. The first term, involving $V_{k l k l}$ , is the Hartree (or direct) energy. The second term, involving $V_{k l l k}$ , is the Fock (or exchange) energy, a purely quantum mechanical effect arising from particle indistinguishability, which Wick's theorem correctly captured through the anti-commutation sign.

Conclusion

Today, we've added a crucial computational tool to our many-body physics arsenal. Wick's theorem demystifies the process of evaluating the long operator strings that arise when dealing with interactions.

Key Takeaways:

Wick's theorem provides a systematic recipe for expanding any product of creation and annihilation operators into a sum of more manageable normal-ordered terms.
For calculating expectation values with respect to a non-interacting ground state (like the vacuum or the Fermi sea), the theorem simplifies to a powerful rule: the expectation value is the sum of all possible full pairings (contractions).
The value of a contraction depends on the specific ground state used, but the combinatorial structure of the theorem remains the same.
This method is the engine behind many-body perturbation theory, allowing for the systematic calculation of interaction effects, such as the Hartree-Fock energy you derived in the exercise.

Preview of the Next Lesson:

Having established our formal toolkit for many-body systems, we will next turn to a concrete physical problem: the scattering of identical particles. We will explore how the symmetry requirements for bosonic and fermionic wavefunctions fundamentally alter scattering cross-sections compared to the scattering of distinguishable particles. This will be a great application of our understanding of identical particles and a bridge to the more general scattering theory we will tackle in a future module.