Localization measures of parity adapted U(D) -spin coherent states applied to the phase space analysis of the D -level Lipkin-Meshkov-Glick model

Abstract

We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N → ∞ limit. The D = 3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D = 2 atomic coherent states, to define the coherent state representation Q ψ (Husimi function) of a symmetric N-qudit state |ψ in the phase space CP D−1 (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N → ∞ limit, where the discrete parity symmetry Z D−1 2 is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2 D−1 different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.

Phase space analysis of the 3-level Lipkin-Meshkov-Glick model using parity adapted U(D)-spin coherent states

“…We shall follow the standard procedure to study quantum phase transitions (QPT) in the thermodynamic limit N → ∞ given by [12]. Firstly, we calculate the energy surface of the Hamiltonian (13), which is the coherent state expectation value of the Hamiltonian density in the thermodynamic limit,…”

“…, and dening new variational states as c-3CATs |z c evaluated in the z of the minimization (see reference [13] for a complete discussion).…”

“…where the numbers between parenthesis in the numerators are related with the number of humps of the Husimi function in phase space as we will see in the next section gures. The calculation of the thermodynamic limit of the ν-th moments of the c-DCAT, together with the Lieb conjecture proof for U(D) and discussion of the Wehrl entropy, can be found in the reference [13].…”

“…depending on the number of non-zero c-3CAT coordinates z (0) 0 (which is a function of λ, see eq. ( 16)), and on the number of non-zero components c L 0 of the parity string c L , which contains the elements of c whose indexes match with ones of the null components of z (0) (see reference [13] for more details).In phase III (bottom row in Figure 2), all the variational excited states has a similar structure to the GS, as it degenerates when increasing λ according to the energy spectrum in Figure 1.…”

Schmidt decomposition of parity adapted coherent states for symmetric multi-quDits

“…Coherent states (CSs), either of the harmonic oscillator (HO) or for a spin system, have many applications in Quantum Mechanics and Quantum Optics (among many other fields), and in particular parity adapted CS (a particular instance of a Schrödinger cat state) for the HO or for U(2) 4 are interesting since they are used in many protocols in Quantum Information Processing or appear as the lower energy states in some molecular or nuclear models like the Dicke [1], the vibron [2] or the Lipkin-Meshkov-Glick (LMG) models [3]. In all these models parity adapted CS have been used as variational states to approximate the ground state and some of the first excited states and to study their entanglement and localization properties across quantum phase transitions [4][5][6][7][8].…”