Quantum phase transition with a simple variational ansatz

Abstract

We study the zero-temperature quantum phase transition between liquid and hcp solid 4 He. We use the variational method with a simple yet exchange-symmetric and fully explicit wave function. It is found that the optimized wave function undergoes spontaneous symmetry breaking and describes the quantum solidification of helium at 22 atm. The explicit form of the wave function allows us to consider various contributions to the phase transition. We find that the employed wave function is an excellent candidate for describing both a first-order quantum phase transition and the ground state of a Bose solid. Properties of solid 4 He have regained attention due to a host of unexpected physics discovered in the past decade [1][2][3][4][5][6][7][8]. Most of the new features occur close to absolute zero and are believed to be primarily driven by quantum effects. Consequently, the solidification of 4 He came under renewed scrutiny. The role of quantum statistics in the transition location has been recently revisited in Ref. [9]. At small but nonzero temperatures, indistinguishability of particles destabilizes the quantum solid. Distinguishable particles, on the other hand, would solidify even at low pressures, with the phase diagram reminiscent of the Pomeranchuk effect [10,11]. The feature was dubbed in [9] as thermocrystallization. A similar effect was seen numerically for the Wigner-crystallization of a two-dimensional Coulomb system [12]. The solidification of 4 He at zero temperature was revisited in Ref. [13] with the density functional theory (DFT). Results were improved compared with previous DFT studies.In this paper, we show that the quantum solidification of 4 He can be considered variationally, with a single explicit wave-function which selects the phase through optimization of the thermodynamic potential. Quite surprisingly, we find that the phase transition is predicted properly, given the relative simplicity of the wave function. While the variational treatment is used for quantum phase transitions at the mean-field level [14], it is relatively uncommon that a (discontinuous) transition can be described with a microscopic wave function. The finding is also interesting in light of growing interest to the first-order quantum phase transitions [15,16].At zero temperature, the phases of 4 He can be studied in an essentially exact form with a family of projector methods, including Green's-function Monte Carlo [17,18] [23,24], and describe the transition [25][26][27] and coexistence [28] between the two phases. The SWFs can be seen as representing a single step of a projection calculation [23]. The projection is carried out by performing the numerical integration of the shadow degrees of freedom. In this sense, the SWF is not fully explicit, as one cannot write down the result of such an integration. We consider SWF calculations as a class of their own, in between the exact projection methods and the simple and fully explicit wave function used here.Highly effective wave functions have been developed over the years...

Path Integral Monte Carlo and Bose-Einstein condensation in quantum fluids and solids

“…The possibility for solid 4 He to present vacancies in its ground state seems to be hindered by the energetic cost of these defects. According to several Quantum Monte Carlo results, the vacancy formation energy is estimated to be of the order of 10 K [85,86,87,88,89], in agreement with experimental measurements [90].…”

A coordinated wavefunction for the ground state of liquid helium-4

“…The inspiration for the coordinated wavefunction ψ JC comes from the symmetrized Bose-solid wavefunction proposed by Cazorla et al 34 . This symmetrical solid wavefunction does an excellent work describing quantum Bose solid, both variationally 35 and as a guiding function for importance sampling in quantum Monte Carlo simulations of Bose solids [36][37][38][39] . In fact, one will recognize that Eq.…”

“…As discussed in Ref. 35, factors that bind atoms to the lattice sites in the solid wavefunction of Ref. 34 can be seen as a generalized symmetrical form of the one-body factor; the coordination part of Eq.…”

Shadow wave function with a symmetric kernel

“…Although good agreement with experiment was obtained with this approach, it was at a cost of spoiling translational invariance and the Bose character of the wave function. In a relative recent effort, a variational ansatz have restored the Bose symmetry in the Nosanow-Jastrow description of 4 He and presented interesting results for the solid-liquid phase transition of this quantum system [5].…”