## 1 Introduction  
The phenomena of photoinduced electronic phase transitions (PIPT) have been attracting much attention for decades [1; 2; 3; 4; 5; 6]. In particular, their ultrafast aspects are important [7; 8; 9; 10], because the dynamics are considered to be mainly related to the electronic degrees of freedom. Their time scales are of the order of several femto to several ten femto seconds, which are roughly the inverse of electron transfer energies. The actual materials are, for instance, tetrathiafulvalene-p-chloranil (TTF-CA) as a neutral-ionic transition system [7; 11], one of the MX chains, [Pd(chxn)\({}_{2}\)Br]Br\({}_{2}\) with M=Pd and X=Br [9; 10], which are both quasi-one-dimensional systems, and \(a\)-(BEDT-TTF)\({}_{2}\)I\({}_{3}\) (BEDT-TTF: bis(ethylenedithio)-tetrathiafulvalene) as a quasi-two-dimensional molecular solid [8]. To describe such systems, we assume a model, which consists of only electronic degrees of freedom. The actual examples are Hubbard models, extended Hubbard models, and other models that are appropriately modified depending on each system [12; 13; 14; 15; 16; 17; 18]. What is crucial here is that the Hilbert space associated with the model should have some special states originating from another phase embedded in a low-energy region that can be accessed by photoexcitation. We call it _Hilbert space anomaly_. In our scenario, we think that such states are excited by light and the system is inevitably changed into another phase. Here, we must be cautious about the following point. In Figure 1a, we draw a situation very schematically. The state \(|g\rangle\) is the ground state from which we start a photodynamics, while \(|g^{\prime}\rangle\) is that for another phase. In a finite-size cluster, their energies can be close to each other as drawn here. However, in the thermodynamic limit, the energy difference becomes infinite, which means that the photoexcitation with modest photon energy from \(|g\rangle\) to \(|g^{\prime}\rangle\) never occurs. Furthermore, the matrix element of photoexcitation, \(\langle g^{\prime}|\hat{J}|g\rangle\), where \(\hat{J}\) is the electric current operator, is considered to be very small, assuming that they have completely different electronic configurations. Note that \(\hat{J}\) is assumed to have a short-range nature, like the model itself. For these reasons, the direct transition from \(|g\rangle\) to \(|g^{\prime}\rangle\) is strictly prohibited in typical situations, and we think of another situation as illustrated in Figure 1b. Again, we assume a model such as a Hubbard model or an extended Hubbard model with short-range interactions. In such models, the photoexcitation is described as a charge transfer over only a few sites. From here on, we restrict ourselves to the simplest case where only one site transfer is allowed, because the generalization is straightforward. Such a situation is realized with a model in which only the nearest-neighbor electron hopping is assumed for the Hamiltonian and the electric current operator. The excited electron and the hole are located at neighboring sites just after photoexcitation. We consider this pair as a transition core of PIPT and think that this core evolves into a spatially extended region, that is, a domain, as shown in Figure 1b. We emphasize that this domain can be located anywhere, which property is nothing but a translational invariance. The true state is a linear combination of individual states at different positions and keeps quantum coherence. So far, we discussed the growth of such quantum domain [13; 14]. Particular attention was paid to the cases in which the initial state is in the vicinity of the phase boundary. In those cases, the domain grows most effectively once it is created with sufficient energy. Such growth is associated with one of its internal degrees of freedom, that is, the spatial size. In one dimension, this size is identical to the distance between the two domain walls on both sides.