Thermal density matrix

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

Thus, in order to compute any average over the ground state we need to know the thermal density matrix at large enough time . Obviously, its analytic form for any non-trivial many-body system is unknown. However, at short time (or high temperature) the system approaches its classical limit and we can obtain approximations. Let us first decompose the time interval t in M smaller time intervals, t =t/M... 

Since [H, P] = 0, the imaginary time evolution preserves the symmetry and the fermion thermal density matrix takes the form... 

By increasing pressure and/or decreasing temperature, ionic quantum effects can become relevant. Those effects are important for hydrogen at high pressure [7, 48]. Static properties of quantum systems at finite temperature can be obtained with the Path Integral Monte Carlo method (PIMC) [19]. We need to consider the ionic thermal density matrix rather than the classical Boltzmann distribution ... 

One significant feature of Eq. (3.2) is the factorization of the expression into the centroid density (i.e., the centroid statistical distribution) and the dynamical part, which depends on the centroid frequency u>. It is not obvious that such a factorization should occur in general. For example, a rather different factorization occurs when the conventional formalism for computing time correlation functions is used [i.e., a double integration in terms of the off-diagonal elements of the thermal density matrix and the Heisenberg operator q t) is obtained]. This result sheds light on the dynamieal role of the centroid variable in real-time correlation functions (cf. Section III.B) [4,8]. 

Keywords Path integral Monte Carlo Takahashi-lmada propagator Quantum virial coefficients Helium-4 Thermal density matrix... 

The thermal density matrix p plays a key role in Feynman s imaginary-time Path Integrals (PI) formalism and its application in Monte Carlo (MC) algorithms to compute physical properties of interest. In position space, it is given by [1-3] ... 

In this section, we will show how the thermal density matrix is used in PIMC to compute quantum viiial coefficients. Consider the Hamiltonian of a monatomic molecule like helium with mass m (Eq. 7). Using the primitive approximation (Eq. 4), Trotter formula (Eq. 5), and following the procedure outlined in Ref. [9], we can obtain the kinetic-energy operator matrix elements as ... 

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