Nilotpal Chakraborty – PhD student
Max Planck institute for physics of complex systems (MPIPKS), Dresden
Email: nilotpal@pks.mpg.de
Twitter: @Nilotpal_Chak
Roderich Moessner (MPIPKS) Benoit Doucot (LPTHE, Sorbonne)
arXiv:2304.13049
Riemann meets Goldstone: Magnon scattering off quantum Hall skyrmion crystals probes interplay of symmetry breaking and topology (and entanglement)
Encouraged to ask in questions
What is a Skyrmion?
Interaction induced (in absence of Zeeman term) quantum Hall ferromagnet at 𝛎 = 1
Spin polarized state acts as a coherent source of spin waves – magnons
Cheapest charged excitations:textured spins, (Skyrmions - see fig) cheaper due to exchange – first introduced in nuclear physics in the Skyrme model
Quantum Hall setting - Sondhi et al, PRB 1993
Also occur in chiral magnets, but due to DM interaction
Skyrmions - topological excitations - quantized topological charge- also carry electric charge
S.M. Girvin – arXiv 1999
Slightly away from 𝛎 = 1 - a ground state Wigner crystal – the skyrmion crystal – Brey,Fertig,Cote and Macdonald, PRL 1995
Magnon injection experiments
New wave of experiments which allow us to probe spin structure of ground states – a direct probe and a new paradigm of scattering experiments
Coherent source of spin waves, magnons, required – analogous to X-rays for crystals
Zhou,…..,Yacoby PNAS, 2021
Several groups
Wei et al, Science 2018
T. Zhou et al, PNAS 2021
Pierce et al, Nat. Phys. 2022
H. Zhou et al, Nat Phys. 2020
A. Assouline et al. Nat. Phys. 2021
H. Zhou et al. , PRX 2022
Experiments thus far:
Experimental evidence for Quantum Hall Skyrmion crystals
Previous experiments fall into three types:
All the above mostly provide indirect evidence via spin relaxation rates/ heat capacity etc.
New platform for unambiguous evidence of crystalline order and excitation spectra – Magnon transport in Quantum Hall junctions
H. Zhou,…, A.F. Young Nat. Phys. 2020
Experiments thus far:
Wei et al, Science 2018
T. Zhou et al, PNAS 2021
Pierce et al, Nat. Phys. 2022
H. Zhou et al, Nat Phys. 2020
A. Assouline et al. Nat. Phys. 2021
H. Zhou et al. , PRX 2022
Ferromagnet – Skyrmion crystal – Ferromagnet quantum Hall junction
FM
𝛎 = 1
FM
𝛎 = 1
SKC
𝛎 = 1 ± 𝜀
Experimental setup – “Easy experiment” – P. Roulleau, Jan 2023
Theory - Technically hard for the following reasons
Inject magnon
Measure transmitted signal
Unambiguous and direct signatures of crystalline order and excitation spectra
Magnon transport probes confluence of topology-symmetry breaking
Topology-symmetry breaking dichotomy results in –
Pinned to zero energy if one considers only exchange terms – consequence of holomorphic nature – dispersive Goldstone modes arise from this flat band
Landau level structure due to spatially modulating topological charge density of skyrmion crystal.
Direct signatures for a quantum Hall skyrmion crystal phase of matter!
First introduced in quantum hall physics by Haldane and Rezayi (PRB 1985) for Laughlin-Jastrow wavefunctions with periodic boundaries.
Methods: 1. Basis for quantum hall junction with skyrmion crystal – Truncated theta functions
Kovrizhn, Doucot, Moessner, PRL 2013
Zeros of theta functions correspond to skyrmion cores!
Truncated theta functions provide an analytically tractable model for ferromagnet-Skyrmion crystal-ferromagnet junctions!
Next question – how does an incoming ferromagnetic magnon scatter off of such a skyrmion lattice?
Acts as effective magnetic field – periodically modulating magnetic field with non-zero mean – possible connections to FCI physics (possible Bosonic analogue?)
Small oscillations about minimal energy configuration – shown on earlier slide
Intuitive picture – charged particle scattering off of a region of spatially varying magnetic field
Energy functional
How to discretize?
Methods 2. Discretize topological charge
Nearest neighbour tight binding model!
Methods: 3. Recursive transfer matrix approach
Tight-binding model
Transfer matrix for a slice at X
Rotate matrix to get scattering amplitudes
How to avoid numerical instabilities – At every step, instead of propagating the whole transfer matrix, propagate the reflection and transmission matrices of each slice using recursive relations from multiple scattering events
Magnon transport probes confluence of topology-symmetry breaking
Topology-symmetry breaking dichotomy results in –
Pinned to E = 0 if g = 0 – holomorphic constraint – dispersive Goldstone modes arise from this flat band limit
Landau level structure due to spatially modulating topological charge density of skyrmion crystal.
Unambiguous signatures of a quantum Hall skyrmion crystal phase of matter!
Warmup exercise – simplified model of FM/AFM/FM junction
Resolve problem of lattice mismatch
Motivation – Understand effects of kinetic mismatch and mismatch between number of goldstone modes (1 for FM, 2 for AFM)
Conventional scattering problem
Signatures:
Analysis done much better for more realistic scenarios by:
Wei, Huang, Macdonald – PRL, 2021
Atteia, Parmentier, Roulleau, Goerbig – arXiv 2022
Warmup exercise (contd.)
Step 1- Intuitive semiclassical picture based on cyclotron orbits due to constant magnetic field
Incoming charged particle
Fully reflected
Incoming charged particle
Fully transmitted but exiting with different velocity
Assume translational invariance along y – integrable problem
Effective potential
qy -conserved
Collection of 1D modes with Veff = (qy+eAy(x))2/(2m)
Depending on x and the value of qy there are 3 types of effective potentials
Similar phenomena for a magnon scattering off of a single skyrmion in chiral magnets – skew scattering of magnons and hall effect for magnons –
Iwasaki, Beekman, Nagaosa – PRB 2014
Schutte, Garst – PRB 2014
Energy threshold for transmission
How do modulations of the B-field affect this picture?
py
x
Tunneling
Classically allowed trajectories
Characteristic signature – non-monotonic dependence of transmission coefficient on channel number!
Angular spread of transmitted signal –some angles dominant!
Y variation introduces dispersion
X-variation pushes bands closer
Numerical results – Heuristic model
No transmission at all for constant B case!
Channel non-monotonicity – periodically modulated B
Fixed incoming momenta - certain outgoing channels have majority transmission!
Angular deviation – constant B
Outgoing magnon is deflected! – similar phenomena observed in magnon scattering off single skyrmions in metallic magnets!
Consequences of crystalline order
Lowest energy modes are pinned to zero for g = 0 – can deform spin vector in the space of holomorphic textures continuously without changing exchange energy!
Finite g induces dispersion in Goldstone modes in the Riemann-Goldstone Landau level!
Two heuristic models capture most qualitative physics
What about anisotropies?
Our model – isotropic SU(2) skyrmion crystal – 4 goldstone modes – 3 SU(2) generators + 1 for broken translational invariance
Anisotropies (such as a zeeman term) will gap out some of the goldstone modes, but probably not all (see R. Cote et al. PRB 2007).
Hence some signatures should remain, however, signatures of higher effective landau levels should be robust.
Cote et al, PRB 2007
Several anisotropies present in actual experiment – zeeman term, lattice scale anisotropie etc.
Outlook
Experiment - Unambiguous determination of the presence of a quantum Hall skyrmion crystal from magnon transport – such signatures are absent in any other possible competing phase in the quantum Hall phase diagram. Unveils the excitatioin spectrum to great detail.
Theory – Skyrmion crystal provides a fertile platform to probe the intersection of topology anf symmetry breaking which manifests in rich physics, c.f – Riemann Goldstone Landau level. Interaction of spin waves from collinear ferromagnets with non-collinear spinn structures such as skyrmion crystals - techniques transferrable to other settings such as chiral magnets, or situations with artificial gauge fields
Next step – SU(4) skyrmion crystals, especially configurations where spin and valley degrees of freedom are entangled (“Entanglement skyrmions” – Doucot, Goerbig, et al. PRB 2008)
Can one detect spin-valley entanglement from magnon transport signatures?
Spin-valley entanglement skyrmions – Mulitcomponent quantum Hall
X
Variational wavefunction at 𝛎 = 1
Coulomb interaction- approximate SU(4) symmetry (2 spin + 2 valley)
Schmidt decomposition of CP3 spinor – natural way to express entanglement
6 real parameters
Maximally entangled
Doucot et al, PRB 2008, Lian and Goerbig – PRB 2017
Spin and pseudospin vector vanishes at maximal entanglement!
Visualize via three Bloch spheres – Spin, Pseudospin and Entanglement
Doucot et al, PRB 2008, Lian and Goerbig – PRB 2017
Skyrmion crystal minimization problem – optimal textured configuration in presence of anisotropies
Question – Analytic ansatz for CP3 skyrmion crystals?
Use basis of four theta functions – p = 0,1,2,3
Interesting method question – How to construct an appropriate parametrization of an SU(4) matrix (in the basis of theta functions), whose first column is fixed?
Answer comes from Random Matrix theory literature
Result – a random 4*4 SU(4) matrix with 9 Euler angles
Minimize F over the integral of EA over the entire slab
Topological charge density much smoother than SU(2)
Strips of vanishing magnitude of spin vector!
Optimal configuration in presence of anisotropies
Strips of maximal entanglement – entanglement skyrmion crystal
Maximal entanglement – magnitude of spin vector vanishes along strips – maximal mismatch with incoming magnon
THANK YOU!
Summary
Magnon transport in quantum Hall heterojunctions in graphene -
First direct signatures of a skyrmion crystal!
Topology-symmetry breaking dichotomy in skyrmion crystal results in:
Magnon transport in such junction like setups allows one to probe crystalline order, dispersion and number of Goldstone modes, and width of higher Landau levels!
Email: nilotpal@pks.mpg.de
Twitter: @Nilotpal_Chak
arXiv:2304.13049