Rigorous stat-phys@Kyoto Univ. (2019/11/21)

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Ferromagnetism in the SU(n)Hubbard model witha nearly flat band
Hosho Katsura (桂 法称)
Department of Physics, UTokyo

Collaborator:
Kensuke Tamura (UTokyo)
 K. Tamura and H. Katsura, arXiv:1908.06286

Outline
1. Introduction• Origin of magnetism• Hubbard models (SU(2) and SU(n))• Frustration-free systems

2. SU(n) Hubbard model
3. 1D model and results
Summary

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Origin of magnetism
What’s the mechanism behind ferromagnetism?
Macroscopic number of spins(carried by electrons) arealigned in the same direction.

But why?

 Coupling between spins• Dipole-dipole interaction

Usually, too small (< 1K) to explain transition temperatures…
• Exchange interaction( Si: spin at site i )Direct exchange: J < 0  ferromagnetic (FM)Super-exchange: J > 0  antiferromagnetic (AFM)
Heisenberg

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Magnetism in ionic crystals
 Kanamori-Goodenough rules
Magnetic ions (cations) do notdirectly couple each other.
They interact via anions.

[From Wikipedia (perovskite)]• Simple examples (neglect orbital order)d-orbitalpdAntiferromagnetic int.
between mag. ionspdd

Ferromagnetic(except for d5.)

Symmetry argument… Is there a more rigorous approach?

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Hubbard model
Paradigmatic model of correlated electrons in solids• Operators: Creation (annihilation) op. of electron with spin σ=↑ or ↓ at site xΛ : Finite lattice: Number op.
Hamiltonian
Hopping term
On-site repulsion

Hubbard (1963)
Kanamori, GutzwillerU

http://theor.jinr.ru/~kuzemsky/jhbio.html

Manifestly SU(2) inv.
U

Hopping and interaction
terms do not commute…

6/25(Crude) derivation of super-exchange 2-site Hubbard modelU• Hamiltonian12• 2nd order perturbation at half-filling,
Basis states:

Pauli’s exclusion

Origin of exchange int. = electron correlation!
Can explain antiferromagnetic int. What about ferromagnetism?

↑↑
↑↓
↓↑
↓↓

Scarcity of exact/rigorous results

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Hubbard model lacks general solutions. Numerically demanding…

• Nagaoka ferromagnetism (Infinite-U, 1 hole)
Nagaoka, Phys. Rev. 147 (1966); Tasaki, PRB 40 (1989)• 1D Hubbard chain (Bethe ansatz)
Lieb-Wu, PRL, 20 (1968), “Absence of Mott transition …”• Ferrimagnetism (spin-reflection positivity)
Lieb, PRL, 62; Erratum PRL 62 (1989).
G.S. on a bipartite at half-filling has Stot = ||A|-|B||/2.
Recent extensions by Miyao, arXiv:1712.05529• Brandt-Giesekus, PRL 68 (1992)

Infinite-U Hubbard, RVB• Flat-band ferromagnetism (Frustration-free)
Mielke, JPA 24, L73, 3311 (1991); Tasaki, PRL 69, 1608 (1992).
Review: H.Tasaki, Prog. Theor. Phys. 99, 489 (1998).
Ferromagnetic states minimize Hhop & Hint simultaneously!

Multi-component generalization

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 SU(n) Hubbard model• Fermions carry flavor (α=1, …, n)• Realization in cold-atom systems
Taie et al., Nat. Phys. 8 (2012).

 Rigorous results• Nagaoka ferromagnetism
in SU(n) Hubbard model
H.K. and A. Tanaka,
Phys. Rev. A 87, 013617 (2013).

Underlying mechanism is the same as Puzzle & Dragons!
• Ferromagnetism in a model withcompletely or nearly flat band
H.K. & K. Tamura, arXiv:1908.06286[See 理学部ニュース 2019]

A crash course in inequalities

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 Positive semidefinite operators
Appendix in H.Tasaki, Prog. Theor. Phys. 99, 489 (1998).
: finite-dimensional Hilbert space.
A, B: Hermitian operators on• Definition 1. We writeand say A ispositive semidefinite (p.s.d.) if• Definition 2. We writeif Important lemmas• Lemma 1.

iff all the eigenvalues of A are nonnegative.

• Lemma 2. Let be an arbitrary matrix on . Then
Cor. A projection operator
is p.s.d.
• Lemma 3. If

and, we have

10/25Frustration-free systems
 Anderson’s bound (Phys. Rev. 83, 1260 (1951).)• Total Hamiltonian:• Sub-Hamiltonian: hj that satisfies(is the lowest eigenvalue of hj )(The g.s. energy of H) =:Used to obtain a lower
bound on the g.s. energyof AFM Heisenberg model Frustration-free Hamiltonian
The case where the equality holds.

Definition.
if there exists a stateis said to be frustration-freesuch thatfor all j.

Ex.) S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT), toric code, …

Flat-band ferro.
is another example.

Outline
1. Introduction
2. SU(n) Hubbard model• Hamiltonian and symmetry• What are flat bands?

• Frustration-free case

3. 1D model and results
Summary

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SU(n) Hubbard model Operators and Fock space• Finite lattice:• Creation & annihilation operators at site x with color α• Number operator:• Vacuum:• Many-particle states:

 Hamiltonian• Hopping term

• Interaction term

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Symmetry of the model Generators• Total fermion number• Color operators

Denote their eigenvalues by Nα.

• Color raising & lowring operators

They commute with the Hamiltonian.
NOTE) SU(n) symmetry for fixed Nf Subspaces• Hamiltonian is block-diagonal w.r.t.

• Degenerate eigenstates in different subspaces arerelated to one another by.

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14/25Hopping term
 Diagonalization
Boils down to the diagonalization of T=(tx,y)• Eigen-operators
Let v be an eigenvector of T with eigenvalue ε.
Then, the operator
satisfies

Acting withon an eigenstate ofraised energy by ε.

• Eigenstates
is an eigenstate ofwith energy 0.

General eigenstates take the form:
where15/25

Interaction Term
 Diagonalization
Already diagonal in the number basis!

• Eigenstates
is an eignstate ofFor example,

has energy 2U.

What about the full Hamiltonian?
• Hopping and interaction terms do not commute!
• Not even frustration-free in general…

But for a hopping term with a flat band (at the bottom),the full Hamiltonian becomes frustration-free!

16/25What are flat bands?
 Single-particle eigenstates of Hhop

• Energy bands
In systems with translation symmetry, wave-num.
k is a good quantum number.

• Flat bandSingle-particle energy

is independent of k.

 1D example (Tasaki lattice)17/25

Why frustration-free• Positive-semi-definite Hopping matrix• Kernel of T spanned by orthonormal• Zero-energy eigen-operators• Interaction term is p.s.d.
• Many-body zero-energy state(for fermion num. = D0)
Because of the Pauli principle,Frustration-free!

Are they unique (up to trivial degeneracy)?
• In the SU(2) case, Mielke established a necessary and sufficientcondition for the uniqueness [Mielke, Phys. Lett. A 174, 443 (1993)]• Related to irreducibility ofOutline
1. Introduction
2. SU(n) Hubbard model
3. 1D Model and results• Lattice and Hamiltonian• Model with completely flat band
• Model with nearly flat bandSummary

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19/25Model on 1D Tasaki lattice31

 Lattice and hopping term
• Lattice:• Periodic boundary conditions:
Identify site j with j+2M.

2M• Hopping term Localized eigen-operators of Hhop

The flat band is spanned by a-operators.

2

2M-1

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Model on 1D Tasaki lattice31

 Lattice and hopping term
• Lattice:• Periodic boundary conditions:
Identify site j with j+2M.

2M• Hopping term Localized eigen-operators of Hhop

The flat band is spanned by a-operators.

22M-1

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Model on 1D Tasaki lattice31

 Lattice and hopping term
• Lattice:• Periodic boundary conditions:
Identify site j with j+2M.

2M• Hopping term Localized eigen-operators of Hhop

The flat band is spanned by a-operators.

22M-1

20/25Flat-band ferromagnetism SU(n) Ferromagnetic (FM) states• Fix total fermion number:• Fully polarized states
are ground states of H=Hhop+Hint
as it makes both Hhop and Hint vanish.

Frustration-free!

• Other FM ground states:• Total number of FM states: Theorem 1 (uniqueness of the FM ground states)
Consider the Hubbard Hamiltonian H with the total fermion
number Nf=M. For arbitrary t >0 and U >0, the ground statesof the Hamiltonian are SU(n) ferromagnetic states and uniqueapart from trivial degeneracy due to the SU(n) symmetry.
A slight generalization of R.-J. Liu et al., arXiv:1901.07004.

Outline of proof

21/25• HamiltonianSince Hhop≧0 and Hint≧0, any ground state of H must be
annihilated by Hhop and Hint simultaneously. This further meansb’s do not appear in g.s.
• Multiple occupancy of a’s are prohibited• Examining the 2nd condition on top sites, we have
• In a subspace labeled by, the g.s. is an equal weightsuperposition of,(set of possible permutations). This state is equivalent to a FM state.

Model with nearly flat band

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 Lattice and hopping term
• Hopping term

• Total Hamiltonian Theorem 2 (uniqueness of the FM ground states)
Consider the Hubbard Hamiltonian H with the total fermion
number Nf=M. For sufficiently large t/s >0 and U/s >0, theground states of H are SU(n) ferromagnetic states and uniqueapart from trivial degeneracy due to the SU(n) symmetry.

A natural SU(n) generalization of H. Tasaki, PRL 75, 4678 (1995).

23/25Outline of proof (1) Decoupling of the Hamiltonian• Flat partFlat-band Hamiltonian(t=U=1)• Local term

 Lemma 1
If each local Hamiltonian hx is positive semi-definite, thenthe ground states of H are the same as those of Hflat.

24/25Outline of proof (2) Proof of Lemma 1• Frustration-free?
If each hx is p.s.d., any state annihilated by Hflat and all hx
is a ground states of H.
• Fully polarized statesannihilated by Hflat and all hx.
 They are eigenstates of H withare• Uniqueness
Uniqueness of these ground states just follows from Theorem 1.

 Positive semi-definiteness of hx• Computer-assisted proof
By numerially diagonalizing hx(5-site Hamiltonian), one can identify
the region in which hx is p.s.d.

 Lemma 2
Suppose that t, U are infinitely large and 0 < κ <1.
Then, hx is positive semi-definite.
Proof.) Based on the analysis of projected Hamiltonian(Projected onto the space of finite-energy states.)
Remark. Lemma 2 ensures finite thresholds for t/s and U/s,
above which hx is positive semi-definite.

Summary• Reviewed rigorous results for Hubbard models• Introduced SU(n) Hubbard model on 1D Tasaki lattice• Ferromagnetism in the model with a completely flat band

• Ferromagnetism in the model with a nearly flat band
Established rigorous example in a non-singular situation!

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