Path integrals and the canonical density matrix

In this section, path integral will be illustrated on the canonical density matrix.

We will give a more precise mathematical formulation and for simplicity of the demonstration we will focus on a single particle moving in one dimension. The generalisation to three dimension and many particles can be done easily.

In the canonical ensemble a state is defined by the fixed quantities N (number of particles), volumeV, temperature T. The total energy of the system is denoted E.

The Hamiltonian of the single particle is:

Hb= pb²

2m +V(x) =b Kb+Vb (2.1)

whereV(bx) is the potential,pbthe momentum andm the mass of the particle.

The definition of the quantum density operator is:

ρ(β) =b 1

Z(N, V, T)e^−βHb (2.2)

whereβ = _k¹

BT and Z(N, V, T) (denoted Z in the following) is the canonical parti-tion funcparti-tion and has the following definiparti-tion:

Z =T rh e^−βHbi

(2.3) whereT r[]represents the trace of the operator inside the squares.

If we consider a coordinate space matrix elements of the density matrix ρ(βb ):

ρ(x, x⁰;β) = 1

Zhx⁰|e^−βHb|xi (2.4) We can notice that Hb is the sum of two operators K(p)b and V(x)b which do not commute with each other. As a consequence:

e^{−β(K+b Vb)}6=e^−βKbe^−βVb (2.5)

To avoid this problem we have to find a way to split kinetic and potential oper-ators. We can do that in two steps. The first one is a "time" composition property:

e^−βHb =h

e^−PβHbiP

(2.6) The second is a Trotter decomposition and exploits the fact that if P is an integer suffienciently high, we can write:

e^{−β(K+b Vb)/P} ≈e^−βK/Pb e^{−βV /Pb} (2.7) e^{−β(K+b Vb)/P} ≈e^{−βV /2Pb} e^−βK/Pb e^{−βV /2Pb} (2.8) The first expression is an approximation of the second order in 1/P and the second one of the third order in1/P.

Then we can use the Trotter theorem (1958) which allows us to express our density operator as (considering only the second approximation but analogous with the first one):

e^{−β(K+b Vb)}= lim

P→∞

he^{−βV /2Pb} e^−βK/Pb e^{−βV /2Pb} iP

(2.9) The derivation of the Trotter theorem is technical and will not be shown in this thesis. However, there are several conditions to satisfy. Our operators have to be lower bounded and the operators T, V and T+V must be self-adjoint. In all cases considered, these conditions are satisfied.

Coming back to our density matrix element and inserting the Trotter theorem, we have:

ρ(x, x⁰;β) = 1 Z lim

P→∞hx⁰|h

e^{−βV /2Pb} e^−βK/Pb e^{−βV /2Pb} iP

|xi (2.10)

In order to simplify the notation we can define an operator bγ as:

bγ =e^{−βV /2Pb} e^−βK/Pb e^{−βV /2Pb} (2.11) ρ(x, x⁰;β) = 1

Z lim

P→∞hx⁰|bγ^P|xi (2.12) We then insert P −1 times the identity operator between theP factors of bγ. This will introduceP −1 integrations over coordinates label:

Ib= Z

dy|yihy| (2.13)

ρ(x, x⁰;β) = 1 Z lim

P→∞

Z

dx2· · ·dxP hx⁰|γb|xPihxP|bγ|xP−1i · · · hx2|bγ|xi (2.14)

The advantage of using the Trotter theorem is that we have to evaluate P −1 identical terms (only the index is different) simpler than the general one. Vb is only a function of the coordinates and as a consequence is diagonal in the coordinate basis:

hx_k+1|bγ|x_ki=hx_k+1|e^{−βV /2Pb} e^−βK/Pb e^{−βV /2Pb} |x_ki

hxk+1|bγ|xki=e^{−βV(xk+1)/2P}hxk+1|e^−βK/Pb |xkie^{−βV(xk)/2P} (2.15)

Since Kb is a momentum operator, the matrix element e^−βK/Pb |x_ki is slightly less trivial to evaluate. We have to insert the identity operator but in the momentum basis this time and use the fact that we known the form ofhx|pi:

Ib= Z

dp|pihp| (2.16)

hx|pi= 1

√2π~

e^ipx~ (2.17)

We insert the identity operator to obtain:

hxk+1|e^−βK/Pb |xki= Z

dphxk+1|e^−βK/Pb |pihp|xki (2.18)

The utility of introducing the identity in the momentum basis becomes clear because now we havee^−βK/Pb which is diagonal in the momentum basis and as a consequence we have:

hxk+1|e^−βK/Pb |xki= Z

dphxk+1|pihp|xkie^−βp2/2mP (2.19) hx_k+1|e^−βK/Pb |x_ki= 1

2π~ Z

dp e^{ip(xk+1−xk)/~}e^−βp2/2mP (2.20)

Then it is simply a Gaussian integral because we integratepbetween−∞and+∞.

Thus:

hx_k+1|e^−βK/Pb |x_ki=

mP 2πβ~²

1/2

exp

−mP

2β~²(x_k+1−x_k)²

(2.21)

which leads us to:

hx_k+1|bγ|x_ki=

mP 2πβ~²

1/2

e^{−2βmP~2(xk+1−xk)2}e^{−2Pβ (V(xk+1)+V(xk))} (2.22)

Using the result above, we obtain for the density matrix: In equation 2.23, the endpoints of the paths are at x1 and xP+1 and are fixed at the initiation and detection points so x and x⁰ respectively. We can see that the kinetic energy is now present in the form of a harmonic coupling between points (x1,· · · , xP+1) of consecutive indexes. We are here in the limit P → ∞ of a dis-cretized representation for the density matrix and the intermediate integrations over x₂· · ·x_P represent the sum over all the possible paths betweenx and x⁰. For finite P, which will be our case for simulations, the paths are lines between successive points as it is shown in Figure 2.1. The probability to follow a path depends of the value inside the exponential in the equation2.23.

We can connect the canonical density matrixρbto the quantum time propagatorUb. Indeed, we have:

ρ(β) =b Ub(−iβ~) Ub(t) =ρ(it/b ~) (2.25) This equivalence allows us to have easily a formal expression of the real time propago-tor using path integrals. We can also interpret obtain the density matrix as the time propagator at an imaginary timet=−iβ~and we can refer to the density matrix as an imaginary time propagator. Using this correspondance, we have a path integral expression for the coordinate-space matrix elements of the real time propagator:

U(x, x⁰;t) = lim

Following the steps to obtain the equation 2.23 which gives us the density matrix elements we can derive the expression of the canonical partition function which is necessary to calculate all the equilibrium properties in the canonical ensemble. Indeed, we have the following expression for our partition function:

Z =T rh

We need only to evaluate the diagonal elements of the density matrix in order to obtain the canonical partition function. As a consequence we only need to set x1 = x_P+1 = x in the equation 2.23 and then we integrate all these diagonal

The integration over cyclic paths in equation2.30is illustrated in Figure2.2. We can notice that in the classical limit (T → ∞), the harmonic spring constant connecting neighbor points becomes infinite and as a consequence all the points collapse in one point which corresponds to the classicle point particle. Thus, the high temperature limit of the path integral formalism is equivalent to the classical limit.

Analytical evaluation of path integrals is only possible for free particle (V(x) = 0) and quadratic potentials (V(x) = ¹₂kx²). However, the path integral picture is applicable for quantum statistical mechanical calculations, even for large system where wave function method are impossible to use due to the exponential scaling with the number of degree of freedom of the numerical cost. In order to do that, we use what we call the classical isomorphism of path integral which allows us to use what we call Path Integral Molecular Dynamics (PIMD) or Path Integral Monte Carlo (PIMC). Unfortunately, this simplification to discrete paths is not so easy with the time propagator due to the complex exponentials (see equation2.26). The latter causes numerical calculations to oscillate widly as different paths are sampled leading to a severe convergence problem known as the dynamical sign problem.

Thus, as we are going to see later in this chapter with the presentation of RPMD, CMD and the linearization approximation, the calculation of dynamical properties from path integrals remains one of the most challenging problem in computational science.

x

x’

x

β¯h/2 β¯h 0

1

Figure 2.1: Possible paths in the path sums of equation2.23usingP = 5slices.

x

x

β¯h/2 β¯h 0

1

Figure 2.2: Possible paths in the discrete path sums for the canonical function in equation 2.30using P = 5 slices.