[Paper Review] Quantum incommensurate Skyrmion crystals and Commensurate to In-commensurate transitions in cold atoms and materials with spin orbit couplings in a Zeeman field

Fadi Sun, Jinwu Ye+1 more|arXiv (Cornell University)|Feb 18, 2015

Cold Atom Physics and Bose-Einstein Condensates39 references4 citations

TL;DR

This paper investigates quantum phases and phase transitions in spinor ultracold atoms with 2D anisotropic Rashba spin-orbit coupling (SOC) and a Zeeman field, revealing novel incommensurate Skyrmion crystal (IC-SkX) phases stabilized by SOC-Zeeman interplay. It identifies five commensurate phases (two collinear, two canted) surrounding the IC-SkX, computes their excitation spectra, and discovers three distinct quantum commensurate-to-incommensurate (C-IC) transitions, with critical temperatures tunable above current experimental limits via atomic filling N.

ABSTRACT

In this work, we study strongly interacting spinor atoms in a lattice subject to a 2 dimensional (2d) anisotropic Rashba type of spin orbital coupling (SOC) and an Zeeman field. We find the interplay between the Zeeman field and the SOC provides a new platform to host rich and novel classes of quantum commensurate and in-commensurate phases, excitations and phase transitions. These commensurate phases include two collinear states at low and high Zeeman field, two co-planar canted states at Mirror reflected SOC parameters respectively. Most importantly, there are non-coplanar incommensurate Skyrmion (IC-SkX) crystal phases surrounded by the 4 commensurate phases. New excitation spectra above all the 5 phases, especially on the IC-SKX phase are computed. Three different classes of quantum commensurate to in-commensurate transitions from the IC-SKX to its 4 neighboring commensurate phases are identified. Finite temperature behaviors and transitions are discussed. The critical temperatures of all the phases can be raised above that reachable by current cold atom cooling techniques simply by tuning the number of atoms $ N $ per site. In view of recent impressive experimental advances in generating 2d SOC for cold atoms in optical lattices, these new many-body phenomena can be explored in the current and near future cold atom experiments. Applications to various materials such as MnSi, Fe$_{0.5}$Co$_{0.5}$Si, especially the complex incommensurate magnetic ordering in Li$_2$IrO$_3$ are given.

Motivation & Objective

• To explore the interplay of strong interactions, 2D anisotropic Rashba spin-orbit coupling (SOC), and Zeeman fields in ultracold atoms.
• To identify new quantum phases beyond conventional collinear or coplanar commensurate states.
• To investigate the emergence of non-coplanar incommensurate Skyrmion crystal (IC-SkX) phases and their stability.
• To characterize quantum commensurate-to-incommensurate (C-IC) transitions and their excitation spectra.
• To demonstrate tunability of critical temperatures via atomic filling N, making phases accessible in current cold atom experiments.

Proposed method

• Formulates a spin-S Rotated Ferromagnetic Heisenberg model (RFHM) in the strong-coupling limit of interacting spinor bosons on a square lattice with 2D SOC and Zeeman field.
• Derives classical spin configurations using variational ansatz with sublattice-dependent angles and wavevectors, minimizing the energy to find ground states.
• Performs spin-wave analysis via Holstein-Feenberg bosonization and Bogoliubov transformation to compute excitation spectra, including Goldstone and roton modes.
• Identifies anisotropic dynamic exponents and Lifshitz transitions via long-wavelength expansions of the dispersion relations.
• Uses symmetry analysis (mirror reflection in SOC parameters) to relate phases and predict sign changes in velocity anisotropy terms.
• Tunes the number of atoms per site N to scale the spin S = N/2, thereby enhancing critical temperatures Tc ∼ N above current experimental limits.

Experimental results

Research questions

• RQ1 What quantum phases emerge from the interplay of strong interactions, 2D anisotropic SOC, and Zeeman fields in ultracold atoms?
• RQ2 Can incommensurate Skyrmion crystal (IC-SkX) phases be stabilized in such systems, and what are their spin textures?
• RQ3 How do the excitation spectra—especially Goldstone and roton modes—differ across commensurate and incommensurate phases?
• RQ4 What are the nature and universality class of the quantum commensurate-to-incommensurate (C-IC) transitions?
• RQ5 Can critical temperatures of these phases be experimentally accessible, and how can they be enhanced?

Key findings

• The system hosts five distinct commensurate phases: two collinear states at low and high Zeeman fields, and two coplanar canted states at mirror-reflected SOC parameters.
• Non-coplanar incommensurate Skyrmion crystal (IC-SkX) phases are stabilized between the five commensurate phases, forming a central quantum phase.
• The Goldstone mode in the IC-SkX phase exhibits anisotropic dispersion ωG(k) = √(v²ₓk²ₓ + v²ᵧk²ᵧ) − cGky, with cG changing sign at β = π/4, indicating a Lifshitz transition at h = hL or hR.
• The roton mode near (0,π) also acquires a linear kₐ term: ωR(q) = √(Δ²R + v²ₓq²ₓ + v²ᵧq²ᵧ) − cRqy, with cR changing sign at β = π/4.
• At h → hc1⁻ and h → hc2⁻, both Goldstone and roton modes reduce to the same k⁴ dispersion as in the Z-x and FM states, indicating duality between canted and IC-SkX phases.
• Critical temperatures scale as Tc/J ∼ N, enabling Tc > current experimental limits by tuning the number of atoms per site N.