Real Time Quantum Dynamics of Spontaneous Translational Symmetry Breakage in the Early Stage of Photo-induced Structural Phase Transitions

Real time quantum dynamics of the spontaneous translational symmetry breakage in the early stage of photoinduced structural phase transitions is reviewed and supplementally explained, under the guide of the Toyozawa theory, which is exactly in compliance with the conservation laws of the total momentum and energy. At the Franck Condon state, an electronic excitation just created by a visible light, is in a plane wave state, extended all over the crystal. While, after the lattice relaxation having been completed, it is localized around a certain lattice site of the crystal, as a new excitation. Is there a sudden shrinkage of the excitation wave function, in between. The wave function never shrinks, but only the spatial, or inter lattice site quantum coherence, interference of the excitation disappears, as the lattice relaxation proceeds. This is nothing but the spontaneous breakage of translational symmetry.


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The spontaneous symmetry breakage is one of the most important problems of great interests in 29 the solid state physics for these several decades. As already well-known, this problem is closely 30 related, not only to the various mechanisms of crystalline magnets, but also to the BCS mechanism of 31 the superconductivity, and even to the Higgs mechanism of the elementary particle physics [1].

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The mechanism for the ferromagnetism of itinerant electrons in a conductive crystal within the 33 mean field approximation [2], is most easy for us to understand the spontaneous symmetry breakage.

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At first, we start from a hypothetical paramagnetic state of itinerant electrons. It is perfectly 35 symmetric, in the sense that un-spin electrons and down-spin ones equally occupy all the lattice sites 36 of the crystal, resulting in no macroscopic magnetic (spin) moment, without an externally applied 37 magnetic field. In the next, we hypothetically assume a spatially uniform but finite unequal 38 occupation. Under this condition, we estimate the total free energy of the system within the mean 39 field theory. Finally, we determine the real value of this hypothetical finite unequal occupation, so 40 Appl. Sci. 2017, 7, x FOR PEER REVIEW 2 of 10 that it will give the lowest free energy. If this lowest energy is even lower than the starting 41 paramagnetic state, without an externally applied magnetic field, we can get a ferromagnetic state 42 which has a spontaneous and macroscopic magnetic (spin) moment. Thus, we can get the symmetry 43 breaking in the space of the electron spin.

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It should be noted that, during this symmetry breaking transition from the paramagnetic state 45 to the ferromagnetic one, the whole system is assumed to be always in the thermal equilibrium, and 46 hence, the speed of the transition has to be infinitely slow, according to the principle of the 47 thermodynamics.

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Keeping this point in mind, let us now proceed to the optical region spectroscopy of insulating 49 crystalline solids. In this research field, according to the rapid progress of time resolved laser 50 techniques, real time quantum dynamics of optically created electronic excitations is gradually 51 clarified in detail up to a pico-or femto-second time scale. This advantageous experimental 52 technology has also been intensively applied even to the present spontaneous symmetry breaking 53 problem. As a result, experimental and theoretical studies for this problem have been intensively 54 developed, although it is quite different way than mentioned above. That is, the real time quantum 55 dynamics of the symmetry breakage.

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One of its typical results is the spontaneous ( self-) localization of an exciton in insulating 57 crystals. The exciton is already well known to be the most elementary optical excitation across the 58 energy gap of insulating crystalline solids [3,4]. Just after the optical excitation, the exciton is always 59 in a plane-wave state extending all over the crystal. After the lattice relaxation having been completed, 60 however, it is in a localized state, being trapped by the self-induced local lattice distortion around it, 61 provided that the exciton-phonon coupling is short ranged and sufficiently strong. This concept was 62 initiated by Rashba [5] and Toyozawa [6] independently, and also developed afterwards rather 63 independently [4,7].

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This localization is intrinsic in the sense that it occurs without extrinsic trapping potentials, 65 say, due to impurities in the crystal [7]. Thus, it is nothing but the spontaneous translational 66 symmetry breakage. Usually, this self-localized exciton still remains within the energy gap of the 67 original insulating crystal, and is luminescent. Hence, it finally disappears after radiating another 68 photon whose energy is a little smaller than that used for the initial excitation

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One can now say, it is a tiny photo-induced structural phase transition (PISPT). As already 72 well known, there discovered a new class of many solids, which, being shone only by visible photons,

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The purpose of the present paper is to review and supplementally explain this spontaneous  As shown by Toyozawa [9], the PISPT phenomenon is closely related to the aforementioned self-85 localization of an exciton in an insulating crystal. It can be simply described by the following model where Ψ is omitted in the averages ⋯ , for simplicity. We should note that this Equation (3) holds 121 only at local minimum ( or extremum ) points in the multi-dimentional coordinate space spanned by Q , 122 since it is obtained by using Equation (2).

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When the exciton-phonon coupling is sufficiently strong, 6T ω S /2, according to

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The second local minimum is F F 1/N, where N denotes the total number of the lattice 132 sites in the crystal. This is the plane-wave state of the exciton whose wave-vector ≡ k is zero, k = 0,

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of which energy is a little smaller that the exciting one, as shown in Figure 1(a). This is the ordinary 145 situation widely realized in luminescent insulators [8]. 146 As shown in Figure 1(b), however, if the exciton-phonon coupling is so large as to relax down 147 even lower than the ground state at this largely displaced lattice configuration,

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Thus, we have seen the spontaneous translational symmetry beakage. Similar to the above Stoner 155 theory [2], its mechanism is also a sufficient energy lowering from the perfectly symmetric state.
According to the adiabatic principle, however, the speed of this symmetry breaking transition is also 157 infinitely slow.

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Incidentally, within the framework of the present theory, we can formally encouter an extremely 159 strong coupling case that the enegry of the STE becomes even lower than the starting ground state itself; 160 E 6T ω S /2 0. We can not use Equation (1) for a such contradicting case.

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The above arguments related with Figure 2 for the exciton self-localization, however, are quite Wannier radius. It is in the region from 10 ω to 10 ω (, ω ≡ the averaged acoustic phonon 256 energy), being more probable than the ordinary radiative decay rate of an exciton, in good 257 agreements with the experimental results in alkali iodides and rare gas solids.

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As for the PISPT, the self-localization is not the finall distination, but the exciton further

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This early stage dynamics is a purely quantum mechanical one, and hence, it has to be in 266 compliance with the conservation laws of the total momentum and the total energy, exactly. As for the total momentum, being zero from the beginning, can be easily seen to be conserved from Equation although it was not written explicitly in the stage of the section 1.

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By this interaction, the energy difference between the FC state and the STE ( shown in Figure 1) are such s as shown in Figure 3(a), and vice verse, as shown in Figure 3(b). That is, the radiation 295 occurs only in the relative ( or internal ) lattice space, whose central lattice site is occupied by this STE. This is an irreversible process, since this relative space is also infinitely large.

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Consequently, a superposition of states realized in each relative space can be possible, just like 302 Equation (8), even though we have included this irreversible phonon radiation. Since this Equation

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(14) does not change the total energy, but only makes phonons to move, we can now see the total 304 energy, as well as the total momentum, are well conserved.

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We can now think of the usual master equation method to describe the lattice relaxation [16]. By 306 this method, however, from the beginning, the whole system is clearly divided into two; a relevant 307 system on which we focus, and a heat reservoir which instantaneously absorbs energies released from 308 the relevant system. By tracing out the reservoir variables, we can thus describe the relaxation 309 dynamics of the relevant system. In the electron-phonon coupled systems, the electronic part is often 310 regarded to be the system, while the phonon is regarded to be the reservoir. As we can easily infer 311 from Figure 3, however, such a priori division is impossible in the present problem. The phonons at 312 infinitely distant lattice sites from the STE may be the heat reservoir, but the central SET site as well 313 as these distant sites are all in the relative space, being not fixed in the real lattice at all.

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Incidentally, long after this quantum and spontaneous localization, thus, having been 315 completed, an ordinary classical localization may also occur, since the localized exciton can also 316 slowly and diffusively move and will be trapped at dislocation or rare impurity sites, which 317 unavoidably exist in the ubiquitous crystal.