Droplet formation in a one-dimensional system of attractive spinless fermions

Vera V. Vyborova, Oleg Lychkovskiy, and Alexey N. Rubtsov

Phys. Rev. B 98, 235407 – Published 5 December 2018

Abstract

A translation invariant one-dimensional system of spinless fermions with a finite-range attraction experiences a quantum phase transition to a phase-separated state. While being a conventional Luttinger liquid for a small interaction strength, spinless fermions form a droplet with the size smaller than the available one-dimensional volume when the interaction strength exceeds some critical value. A particularly remarkable signature of the droplet formation is the change in the lower edge of the many-body excitation spectrum. In the homogeneous phase, it has a Luttinger-liquid shape of periodic arcs on top of the shallow parabolic dispersion of the center of mass. When the interaction strength is increased, the arcs disappear completely as soon as the droplet is formed. We perform an exact diagonalization study of this system with the focus on the signatures of the quantum phase transition and the droplet properties. The one-particle and density-density correlation functions, the pressure, the sound velocity, and the droplet density are examined. The value of the critical interaction strength obtained from numerical data reasonably agrees with a simple mean-field analytical estimate. Due to the boson-fermion correspondence valid in one dimension, our results also hold for hard-core bosons with a finite-range attraction.

Received 6 July 2018
Revised 16 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.235407

Physics Subject Headings (PhySH)

Phase separation Quantum phase transitions Spinless fermions

1-dimensional systems Quantum fluids & solids

Exact diagonalization Luttinger liquid model Mean field theory Nonperturbative methods

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Vera V. Vyborova^1,2,*, Oleg Lychkovskiy^3,4, and Alexey N. Rubtsov^1,2

¹Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia
²Department of Physics, Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia
³Skolkovo Institute of Science and Technology, Nobel Street 3, Moscow 121205, Russia
⁴Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia

^*vv.vyborova@physics.msu.ru

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 98, Iss. 23 — 15 December 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Quantum phase transition in a one-dimensional spinless fermionic system. (a) The ground-state energy $E_{P}$ as a function of the total momentum $P$ has a Luttinger-liquid quasiperiodic arc shape for $U < U_{c r}$ and follows the parabolic dispersion law of a free particle with the mass $N m$ for $U > U_{c r}$ . The number of fermions is $N = 11$ ; all curves are vertically shifted to start from the same origin; $U$ in the legend is quoted in units of $E_{F}$ . (b) Derivative $\frac{\partial E_{P}}{\partial P} |_{P \to 0}$ as a function of interaction strength $U$ for a different number of fermions and fixed average density. The solid line represents a dependence of the acoustic mode velocity $v_{a c}$ for $N = 13$ , which starts to deviate from $\frac{\partial E_{P}}{\partial P} |_{P \to 0}$ for $U > U_{c r}$ . (Inset) The critical interaction $U_{c r}$ as a function of $N$ (color markers) and its fit $U_{c r} = α_{1} + α_{2} / N$ (blue dotted line).
Reuse & Permissions
Figure 2
Correlation functions for free fermions (blue squares), the homogeneous phase $U < U_{c r}$ (yellow circles), and the droplet phase $U > U_{c r}$ (red triangles) for the state with zero total momentum and $N = 13$ (the finite-size value of $U_{c r}$ for this $N$ is $0.89 E_{F}$ ). Interaction strength $U$ in the legends is quoted in units of $E_{F}$ . (a) One-particle correlation function $G (x) = 〈 {\hat{c}}_{x}^{†} {\hat{c}}_{o} 〉$ . Its Fourier transform $G (q)$ is shown in the inset. Observe the decrease of the oscillation period in $G (x)$ in the droplet phase. (b) Density-density correlation function $C (x) = - 〈 {\hat{c}}_{x}^{†} {\hat{c}}_{o}^{†} {\hat{c}}_{x} {\hat{c}}_{o} 〉 - ρ^{2}$ . Its Fourier transform $C (q)$ is shown in the inset. Note the qualitative change in $C (x)$ and the emergence of a spike at $q = δ k$ in $C (q)$ in the droplet phase. The black solid line represents a fit by the correlation function of a classical droplet (see Sec. 2c for details).
Reuse & Permissions
Figure 3
Density of the droplet $ρ_{d}$ as a function of interaction strength between particles. The number of particles is $N = 13$ ; the finite-size value of $U_{c r}$ is $0.89 E_{F}$ . The blue curve represents a fit $ρ_{d} = ρ + α \sqrt{U - U_{c r}}$ .
Reuse & Permissions
Figure 4
Pressure in a 1D fermionic system for different values of $U$ (in units of $E_{F}$ for a reference system size $L_{0}$ ) for $N = 11$ . (Inset) Zoomed view of a negative pressure region for $U = 0.8 E_{F}$ and a various number of particles $N$ . The size of this region decreases with increasing $N$ .
Reuse & Permissions
Figure 5
Imaginary-time Feynman trajectories of the particles moving in a periodized system. (a) For a single particle the trajectory returns to an initial point as evolving from to $τ = 0$ to $β$ (thin black line) or performs a shift by an integer number of the quantization box lengths $L$ (thick red line shows a shift by $L$ ). (b) For several indistinguishable particles, there are cyclic shifts. (c) For the phase separation, cyclic shifts are suppressed, as one of the particles should leave the drop during the evolution (dashed red line).
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics