Recalling that a dynamical phase transition has been defined on the basis of real-time nonanalyticities the question remains which quantities are potentially able to show non- analyticities generically. The notion of generically in this context is supposed to mean as generic as there are equilibrium phase transitions. Below, it will be shown that the Loschmidt amplitude is such a quantity as has been demonstrated already for the specific case of magnetic field ramps in the Ising model [138]. Consider a generalized function
L(z) =hψ0|e−iHfz|ψ0i, z ∈C, (4.5)
by extending time into the whole complex plane. The basic observation is that on the
equilibrium boundary partition function
Z(τ) = L(−iτ) = hψ0|e−Hfτ|ψ0i (4.6)
of a system with boundaries separated by a distance τ with boundary conditions imple-
mented by the boundary states|ψ0i[105]. This identification is up to the present knowledge
of purely formal nature. The equilibrium boundary partition functionZ(τ) has no physical meaningful relationship to the nonequilibrium quench problem as it is known up to now.
The formal observation in Eq. (4.6) has a few important implications. First, it clarifies the appropriate scaling behavior in the thermodynamic limit. Note that the thermody- namic limit is of fundamental importance as already argued below Eq. (4.1). The free
energy is extensive in system size V such that the Loschmidt amplitude has to be of
large-deviation form [171]
L(z)V−→→∞eV f(z) (4.7)
withf(z) independent ofV. Concluding, the proper quantity in order to study the behavior
of the Loschmidt amplitude in the thermodynamic limit is the rate function f(z). This
is completely analogous to the equilibrium case where the appropriate quantity is the free energy density and not the partition function itself for large system sizes.
Letµbe a parameter of an equilibrium partition function Z of a system of sizeV such as magnetic field h or temperature T. Then Z and in particular the phase transitions of
the model are determined by the zeros in the extended complex parameter plane of µ. In
case of a magnetic field, i.e., µ =h, the corresponding partition function zeros are called Lee-Yang zeros [106], for µ = β = T−1 they are called Fisher zeros [46]. Denoting those byµr the partition function Z can be represented in product form
Z(µ) =e−G(µ)Y r 1− µ µr (4.8)
yielding the free energy F
F(µ) =−T X r log 1− µ µr +T g(µ). (4.9)
Here, Boltzmann’s constantkB = 1 has been set to one such that temperature is measured
in units of energy. The functions G(µ) and g(µ) = V−1G(µ) are smooth and analytic.
Concerning phase transition they will therefore be unimportant such that they will not be considered in what follows. If Z(µ) is just a finite polynomial in µ the factorization of the partition function is obvious and G(µ) is a constant. In case of an infinite polynomial the factorization is guaranteed by the Weierstrass factorization theorem as long as the expression in Eq. (4.8) converges. In the general case this cannot be proven but as it turns out the convergence is always given for all practical purposes. In the above representation in Eq. (4.8) all the details and properties of the physical system are encoded in the location of the zeros µ .
4.1 The concept of a dynamical quantum phase transition 81
Figure 4.1: Schematic illustration of the partition function zeros of a parameterµextended into the whole complex plane. The physical axis corresponds to real values of µ∈R. For a finite-size system the zeros are located at discrete and well-separated points as indicated by stars in the left picture. As the system size V increases the zeros accumulate on lines. At those points where such a line of zeros crosses the physicalµ∈Raxis - in this schematic picture this corresponds toµ=µc - the system undergoes a phase transition.
For a finite-size system the zeros are typically located at discrete points in the complex µplane away from the physicalµ∈Raxis. Increasing the system size the number of zeros increases and they typically accumulate on lines [46]. For a schematic picture see Fig. 4.1. Although for a finite-size system no zero lies directly on the realµaxis a line of accumulated zeros may cut the axis in the thermodynamic limit. In such case the partition function becomes nonanalytic and the system thus undergoes a phase transition. The character of this transition whether it is continuous or of first order as well as the respective exponents or the latent heat are all encoded in the line density of zeros at the transition point [79]. If the line density is a constant the phase transition is of first order and the respective constant is proportional to the latent heat. Provided the line density vanishes in a power- law fashion one obtains a continuous phase transition. For the case of Fisher zeros where
µ = β = T−1 is the inverse temperature the exponent of the algebraic behavior of the
density of zeros determines the critical exponent of the specific heat. Most importantly, for numerical simulations finite-size scaling of the zeros allows for the derivation of the latent heat or all the critical exponents [79].
As shown before the Loschmidt amplitude becomes an equilibrium partition function on the imaginary time axis such that it is possible to adopt the knowledge about the partition function zeros to the nonequilibrium quantum quench problem. As a function of time the Loschmidt amplitude is determined by its complex time zerostr such that it can be represented in the following way
L(t) =e−G(t)Y r 1− t tr (4.10)
where the prefactor determined by the smooth function G(t) will play no role in the fol- lowing considerations. The complex time zeros will be termed Fisher zeros in the following due to the formal similarity of imaginary time and inverse temperature. If in the thermo- dynamic limit such a line of Fisher zeros crosses the real-time axis at a specific pointt∗ the Loschmidt amplitude becomes nonanalytic att =t∗ defining the location of the dynamical phase transition.
Based on these general considerations about partition function zeros the Loschmidt amplitude can become nonanalytic generically in the sense of as generic as there are equi- librium phase transitions. It is therefore not just coincidence that the Loschmidt amplitude for magnetic field quenches in the transverse field Ising model studied below shows nonan- alyticities. One may thus expect that nonanalyticities appear also in Loschmidt echoes of other model systems. In case of the Ising model there is a direct connection between real- time nonanalyticities and the quantum critical point of the underlying equilibrium theory. A general proof of such a relationship, however, does not exist. In Sec. 4.2 below lines of Fisher zeros as well as the nonanalyticities of the Loschmidt amplitude are investigated for magnetic field quenches in the one-dimensional transverse field Ising model.
Note that even though there seems to be formal relationship between the Loschmidt amplitude and an equilibrium partition function, the Loschmidt amplitude should not be viewed as a “dynamical partition function” because it is not possible to deduce the time evolution of thermodynamic quantities from it such as generalized forces or the expectation values of local observables. What can be deduced from the Loschmidt amplitude is the average work performed Wav, compare Ch. 3, viaWav =idLdt(t)|t=0.