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arXiv:2312.00755

Ground-state entanglement spectrum of a generic model with nonlocal excitation-phonon coupling

Vladimir M. Stojanović

Published Dec 1, 2023• 0 citations
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Abstract

While the concept of the entanglement spectrum has heretofore been utilized to address various many-body systems, the models describing an itinerant spinless-fermion excitation coupled to zero-dimensional bosons (e.g. dispersionless phonons) have as yet not received much attention in this regard. To fill this gap, the ground-state entanglement spectrum of a model that includes two of the most common types of short-ranged, nonlocal excitation-phonon interaction -- the Peierls- and breathing-mode couplings -- is numerically evaluated here. This model displays a sharp, level-crossing transition at a critical coupling strength, which signifies the change from a nondegenerate ground state at the quasimomentum $K_{\textrm{gs}}=0$ to a twofold-degenerate one corresponding to a symmetric pair of nonzero quasimomenta. Another peculiarity of this model is that in the special case of equal Peierls- and breathing-mode coupling strengths the bare-excitation Bloch state with the quasimomentum $0$ or $π$ is its exact eigenstate. Moreover, below a critical coupling strength this state is the ground state of the model. Thus, the sharp transition between a bare excitation and a heavily phonon-dressed (polaronic) one can be thought of as a transition between vanishing and finite entanglement. It is demonstrated here that the smallest ground-state entanglement-spectrum eigenvalue to a large extent mimics the behavior of the entanglement entropy itself and vanishes in this special case of the model; by contrast, all the remaining eigenvalues diverge in this case. The implications of excitation-phonon entanglement for $W$-state engineering in superconducting and neutral-atom-based qubit arrays serving as analog simulators of this model are also discussed.

DOI
10.48550/arxiv.2312.00755

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While the concept of the entanglement spectrum has heretofore been utilized to address various many-body systems, the models describing an itinerant spinless-fermion excitation coupled to zero-dimensional bosons (e.g. dispersionless phonons) have as yet not received much attention in this regard. To fill this gap, the ground-state entanglement spectrum of a model that includes two of the most common types of short-ranged, nonlocal excitation-phonon interaction -- the Peierls- and breathing-mode couplings -- is numerically evaluated here. This model displays a sharp, level-crossing transition at a critical coupling strength, which signifies the change from a nondegenerate ground state at the quasimomentum Kgs=0K_{\textrm{gs}}=0Kgs​=0 to a twofold-degenerate one corresponding to a symmetric pair of nonzero quasimomenta. Another peculiarity of this model is that in the special case of equal Peierls- and breathing-mode coupling strengths the bare-excitation Bloch state with the quasimomentum 000 or πππ is its exact eigenstate. Moreover, below a critical coupling strength this state is the ground state of the model. Thus, the sharp transition between a bare excitation and a heavily phonon-dressed (polaronic) one can be thought of as a transition between vanishing and finite entanglement. It is demonstrated here that the smallest ground-state entanglement-spectrum eigenvalue to a large extent mimics the behavior of the entanglement entropy itself and vanishes in this special case of the model; by contrast, all the remaining eigenvalues diverge in this case. The implications of excitation-phonon entanglement for WWW-state engineering in superconducting and neutral-atom-based qubit arrays serving as analog simulators of this model are also discussed.
The polaron concept captures physical situations involving an itinerant quantum particle (excitation) that interacts strongly with bosonic degrees of freedom and becomes heavily boson-dressed. While the Gerlach-L\"owen theorem rules out the occurrence of nonanalyticities of ground-state-related quantities for a broad class of polaron models, examples were found in recent years of sharp transitions pertaining to strongly momentum-dependent interactions of an excitation with dispersionless (zero-dimensional) phonons. On the example of a lattice model with Peierls-type excitation-phonon interaction, such level-crossing-type small-polaron transitions are analyzed here through the prism of the entanglement spectrum of the excitation-phonon system. By evaluating this spectrum in a numerically exact fashion it is demonstrated that the behavior of the entanglement entropy in the vicinity of the critical excitation-phonon coupling strength chiefly originates from one specific entanglement-spectrum eigenvalue, namely the smallest one. While the discrete translational symmetry of the system implies that those eigenvalues can be labeled by the bare-excitation quasimomentum quantum numbers, here it is shown numerically that they are predominantly associated with the quasimomenta 0 and $\ensuremath{\pi}$, including cases where a transition between the two takes place deeply in the strong-coupling regime.
We investigate the spectral properties of polaronic excitations within the framework of an analog quantum simulator based on inductively coupled superconducting transmon qubits and microwave resonators. This system emulates a lattice model that describes a nonlocal coupling of an itinerant spinless-fermion excitation to dispersionless (Einstein-type) phonons through the Peierls and breathing-mode interaction mechanisms. The model is characterized by a sharp, level-crossing transition at a critical value of the effective excitation-phonon coupling strength; above the transition point, the ground state of this model corresponds to a heavily dressed (small-polaron) excitation. Using the kernel-polynomial method, we evaluate the momentum-frequency resolved spectral function of this system for a broad range of parameters. In particular, we underscore the ramifications of the fact that the zero-quasimomentum Bloch state of a bare excitation represents the exact eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon coupling strength. We also show that---based on the numerically evaluated spectral function and its well-known relation with the survival probability of the initial, bare-excitation Bloch state (the Loschmidt echo)---one can make predictions about the system dynamics following an excitation-phonon interaction quench. To make contact with anticipated experimental realizations, we utilize a previously proposed method for extracting dynamical-response functions in systems with local (single-qubit) addressability using the multiqubit (many-body) version of the Ramsey interference protocol.
This Letter unravels an interesting property of a one-dimensional lattice model that describes a single itinerant spinless fermion (excitation) coupled to zero-dimensional (dispersionless) bosons through two different nonlocal coupling mechanisms. Namely, below a critical value of the effective excitation-boson coupling strength, the exact ground state of this model is the zero-quasimomentum Bloch state of a bare (i.e., completely undressed) excitation. It is demonstrated here how this last property of the lattice model under consideration can be exploited for a fast, deterministic preparation of multipartite W states in a readily realizable system of inductively coupled superconducting qubits and microwave resonators.
We propose an analog superconducting quantum simulator for a one-dimensional model featuring momentum-dependent (nonlocal) electron-phonon couplings of Su-Schrieffer-Heeger and ``breathing-mode'' types. Because its corresponding coupling vertex function depends on both the electron and phonon quasimomenta, this model does not belong to the realm of validity of the Gerlach-L\"owen theorem that rules out any nonanalyticities in single-particle properties. The superconducting circuit behind the proposed simulator entails an array of transmon qubits and microwave resonators. By applying microwave driving fields to the qubits, a small-polaron Bloch state with an arbitrary quasimomentum can be prepared in this system within times several orders of magnitude shorter than the typical qubit decoherence times. We demonstrate that---by varying the externally tunable parameters---one can readily reach the critical coupling strength required for observing the sharp transition from a nondegenerate (single-particle) ground state corresponding to zero quasimomentum (Kgs=0{K}_{\mathrm{gs}}=0Kgs​=0) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta Kgs{K}_{\mathrm{gs}}Kgs​ and $\ensuremath{-}{K}_{\mathrm{gs}}$. Through exact numerical diagonalization of our effective Hamiltonian, we show how this nonanalyticity is reflected in the relevant single-particle properties (ground-state energy, quasiparticle residue, average number of phonons). We also show that the proposed setup provides an ideal testbed for studying the nonequilibrium dynamics of small-polaron formation in the presence of strongly momentum-dependent electron-phonon interactions.
Motivated by the compelling need to understand the nonequilibrium dynamics of small-polaron formation following an electron-phonon interaction quench, in this work we propose a digital quantum simulator of a one-dimensional lattice model describing an itinerant fermionic excitation (e.g. an electron) nonlocally coupled to zero-dimensional bosons (e.g. Einstein-type phonons). Quantum circuits implementing the dynamics of this model, which includes Peierls- and breathing-mode-type excitation-boson interactions, are designed here, their complexity scaling linearly with the system size. A circuit that generates the natural initial (pre-quench) state of this system -- a bare-excitation Bloch state, equivalent to a WWW state of a qubit register -- is also presented. To facilitate comparisons with the proposed simulator, once experimentally realized, the system dynamics are also evaluated numerically and characterized through the Loschmidt echo and various correlation functions.
Recent years have seen growing interest in sharp polaronic transitions in systems with strongly momentum-dependent interactions of an itinerant excitation (electron, hole, exciton) with dispersionless phonons. This work presents a scheme for investigating such phenomena in a controllable fashion within the framework of an analog quantum simulator based on an array of neutral atoms in optical tweezers. The envisioned analog simulator, in which the atoms interact through Rydberg-dressed resonant dipole-dipole interaction, allows one to study the rich interplay of Peierls- and breathing-mode-type excitation-phonon interactions. Based on a numerically-exact treatment of one special case of this system -- namely, the case with equal Peierls- and breathing-mode coupling strengths -- a sharp small-polaron transition was shown to take place for a critical value of the effective excitation-phonon coupling strength. This transition signifies the change from a completely bare (undressed by phonons) excitation below the transition point and a strongly phonon-dressed one (small polaron) above it. This work also highlights the comparative advantages of highly-controllable Rydberg-atom-based systems to other physical platforms for simulating polaronic phenomenology, which could be exploited to study the nonequilibrium dynamics of the small-polaron formation.