統計力学セミナー@本郷（2015/6/23）

Exact ground states and topologicalorder in interacting Kitaev chains

Hosho Katsura (U. Tokyo)
Collaborators:
Masahiro Takahashi (Gakushuin Univ.)
Dirk Schuricht (Utrecht Univ.)Outline
1. Introduction & Motivation•

Majorana fermions in particle and cond-mat.

•

Majorana fermions in stat. mech.

2. Kitaev/Majorana chain•

Trivial and topological phases, edge zero modes•

Mapping to quantum Ising chain

3. Interacting Kitaev/Majorana chain•

Operator inequalities, frustration-free Hamiltonian•

Exact ground states, solvable line in the phase diagram

4. Energy gap, edge zero modes & topo. order• Poor man’s definition of topo. order• Proof of the gap, exact edge zero modes

Much ado about Majorana fermions What are Majorana fermions?
“Majorana returns”, F. Wilczek, Nat. Phys. 5 (2009).
Particles that are own antiparticles
Real solutions of Dirac eq.  Electrically neutral
Neutrinos might be Majoranas (seems unlikely…)

 Emergent Majoranas in condensed matter“Unpaired Majorana fermions in quantum wires”,
A. Kitaev, Phys. Usp., cond-mat/0010440 (2000).
Majorana fermions = edge zero modes

qubit?

Experimental Signatures
Mourik et al., Science 336, 1003 (2012).
Nadj-Perge et al., Science 346, 602 (2014).

Majorana fermions in Stat. Mech.
 Kitaev chain = Quantum Ising chain
P. Fendley, J. Stat. Mech. P11020 (2012).
Surprisingly, the word “Ising” does not appear in Kitaev's paper(except inside “Surprisingly”) nor in the excellent Alicea’s review
article; often this chain is now referred to as the “Kitaev chain”.

 History of the 2d Ising model“Romance of the Ising model”, B. M. McCoy, arXiv:1111.7006.
1941: Kramers-Wannier, duality  critical temperature
1941-1944: Onsager, exact free energy, Onsager algebra
1949: Kaufman, Majorana-fermion technology, Pffafian
1949-1952: Onsager, Yang, exact magnetization
1970~: McCoy-Wu, Jimbo-Miwa-Sato, …, still ongoing!
Spontaneous magnetization

Tower of Babel
Exotic “particles” appear in several seemingly-unrelated fields.
People in these fields use different languages. Dictionary reads
Topological cond-mat.

2d stat-mech.

Math-phys.

2d Ising modelOnsager algebra

Chiral Potts model

superintegrable case

Yang-Baxter w/odifference property
PLUS Onsager alg.

RSOS (ABF) model

Temperley-Lieb alg.

Majorana fermions

ParafermionsFibonacci anyons

Their critical points are described by 2d CFTs,
correlators of which give FQHE wavefunctions.

What people are doing these days are just a rephrasingof what have been known for many decades…?

Interacting Majorana fermions Motivation
Do what no one else can do! Let’s solve interacting Majorana
fermions exactly. What is topo. order in interacting systems?
If that’s not motivation enough…
1. Interactions narrow or expand the topo. phase?
Narrow) S. Gangadharaiah et al., PRL 107, 036801 (2011).
Expand) E.M. Stoudenmire et al., PRB 84, 014503 (2011).
2. Physical realization of coupled Majorana wires
Array of Josephson junctions
F. Hassler & D. Schuricht, New. J. Phys. 14 (2012).

 Dictionary again…
Via Jordan-Wigner transformation,
Kitaev chain

Quantum Ising chain

Interacting Kitaev chain

XYZ chain in a field E-mail discussion
On May 6th, 2014:
H.K.: I found that the interacting Kitaev/Majorana chain isexactly solvable if the chemical potential is tuned toa particular function of the other parameters (t, Δ, U).
D.S.: That sounds very interesting. But did you know thePeschel-Emery line? [Z. Phys. B, 43, 241, (1981).]
H.K.: No, I didn’t. But I will look into their paper...

It turned out that I’m also the guy who is justrephrasing the known results in the literature…

Outline
1. Introduction & Motivation•

Majorana fermions in particle and cond-mat.

•Majorana fermions in stat. mech.

2. Kitaev/Majorana chain•

Trivial and topological phases, edge zero modes•Mapping to quantum Ising chain

3. Interacting Kitaev/Majorana chain•

Operator inequalities, frustration-free Hamiltonian•

Exact ground states, solvable line in the phase diagram

4. Energy gap, edge zero modes & topo. order• Poor man’s definition of topo. order• Proof of the gap, exact edge zero modes

Kitaev/Majorana chain (non-interacting) Hamiltonian (complex fermions, with OBC)
site

j-1

jhopping

j+1

j+2

A. Kitaev, Phys. Usp. (2001).

j+3

pairing
Chemical potential

 Complex fermion = pair of real (Majorana) fermionssite jDefiningrelations

Which particles are fundamental and which are secondary may
be a matter of interpretation. (“Nuclear democracy”, G.F. Chew)

Phases in the Kitaev/Majorana chain
 Hamiltonian (Majorana fermions, with OBC) Trivial phase
The fully filled state is
the unique g.s. (μ>0)

 Topological phase

New, non-local fermion:

Unpaired
Majoranapresence/absence of f ⇔ two-fold degenerate g.s.
Non-local zero mode commuting with H0 exists as long as w>μ.
Topological order! Quantum phase transition occurs at w=μ.

Unpaired
MajoranaKitaev chain = Quantum Ising chain
 Hamiltonian in terms of spins
Jordan-Wigner transformation
Ising model!

 Ground states of the spin model (Jx >> |h|)2-fold degenerate g.s.:

Ferromagneticallyordered in x direction

Order parameter
Local in spin variables,
but non-local in fermions!

Topological order and Majorana edge zero modes
 Fermionic (Z2) Parity
H0 conserves fermion number mod 2,
which means H0 commutes with

 Characteristic of topological orderi) Nonvanishing energy gap, ii) g.s. degeneracy (OBC), andiii) locally indistinguishable g.s. (in the fermionic basis)

PLUS iv) existence of Majorana edge zero modes such that••••

localized near the edge, and normalizableaseven in the infinite-size limit.

NOTE) i), …, iv) are not totally independent.

Parity=＋ーOutline
1. Introduction & Motivation•

Majorana fermions in particle and cond-mat.

•

Majorana fermions in stat. mech.

2. Kitaev/Majorana chain•

Trivial and topological phases, edge zero modes•

Mapping to quantum Ising chain

3. Interacting Kitaev/Majorana chain•

Operator inequalities, frustration-free Hamiltonian•Exact ground states, solvable line in the phase diagram

4. Energy gap, edge zero modes & topo. order• Poor man’s definition of topo. order• Proof of the gap, exact edge zero modes

Interacting Kitaev/Majorana chain
 Hamiltonian (complex fermions, with OBC)

4-Majorana int.

H  H0 when t =Δ=w, μj =μ, and U=0. μj may depend on j.

 Spin Hamiltonian = XYZ chain in a magnetic field

Symmetries: H commutes with fermionic parity
Integrable when μj =0 for all j (Baxter, 1971).
Easily solvable for a particular set of μj. (Frustration free!!)A crash course in inequalities
 Positive semidefinite operators (H. Tasaki, PTP 99, 489 (1998).)
Let be a finite dimensional Hilbert space.
Definition 1. For a hermitian matrix on , we write
and
say A is positive semidefinite if we have
Definition 2. For two hermitian matrices A and B on ,we write
ifLemma 1.

iff all the eigenvalues of A are nonnegative.

Lemma 2. Let C be an arbitrary matrix on
Lemma 3. Ifand. Then, we have

 Min-max theorem (Courant-Fischer-Weyl)

Theorem. Let A and B be two hermitian matrices on , andlet ai and bi be the i-th eigenvalues of A and B, respectively.
ai and bi are arranged so thatIf, then we haveEx.) For i=1, Theorem simply implies the variational principle.

Frustration-free Hamiltonian Anderson’s bound (Phys. Rev. 83 , 1260 (1951).
Suppose Hamiltonian takes the form
where each local hj satisfies(is the lowest eigenvalue of hj .) Then we have

(The g.s. energy of H) =:Gives a lower bound on
the g.s. energy of AFM
Heisenberg model.

 Frustration-free Hamiltonian
The case where the equality holds.
(Pseudo-)Definition. A Hamiltonianis said to be
frustration-free when the ground state is obtained by minimizing
each term independently.
Examples include Majumdar-Ghosh, AKLT, Kitaev’s toric code,
Shastry-Sutherland, Rokhsar-Kivelson (quantum dimer), …
A similar concept, Bogomolnyi’s bound appears in field theories.

From two to many (1)Cook up a (frustration free) toy model from 2 sites1

 Even and odd Hamiltonians
h1 commutes with fermion parity.  Even and odd sectors.
Even subspace:g.s.1:

Odd subspace:g.s.2:

g.s.1 and g.s.2 become degenerate if2

From two to many (2) Product ground states
When μ=μ*, the g.s. of h1 can be expressed asg.s.1:

g.s.2:

Disentangled (product) states are the ground states!
L-site Hamiltonianis frustration free if μ=μ*, in which case the unique g. s. areNOTE) The boundary chemical potential is half the bulk one.

A fermionic rephrasing of the known results in the XYZ spin chain.
Barouch-McCoy(1971), Peschel-Emery (1981), Mueller-Schrock (1985), …

Solvable line in the phase diagram
 Phase diagram (t=Δ)
 Quantum ANNNI model
Beccaria et al., PRB (2007); Sela & Pereira, PRB (2011);
Hassler and Schuricht, New. J. Phys. 14 (2012).

Solvable line:
Exact g.s.:c=1/2(TFI)

Gomez-Santos,
PRL (1993)  c=1
1.0
0.8
0.6

c=1(XY)

0.4
0.2
0.0

0.5

1.0

1.5

U
2.0 t

Outline
1. Introduction & Motivation•

Majorana fermions in particle and cond-mat.

•Majorana fermions in stat. mech.

2. Kitaev/Majorana chain•

Trivial and topological phases, edge zero modes•

Mapping to quantum Ising chain

3. Interacting Kitaev/Majorana chain•

Operator inequalities, frustration-free Hamiltonian•

Exact ground states, solvable line in the phase diagram

4. Energy gap, edge zero modes & topo. order• Poor man’s definition of topo. order• Proof of the gap, exact edge zero modesA poor man’s definition of topological order Topo. order in 1d interacting Majorana fermionsFidkowski and Kitaev, PRB 81, 134509 (2010); PRB 83, 075103 (2011).

Z classification reduces to Z8 one.
Definition (My ver.) The interacting Kitaev chain is said to be
in a topological phase if it can be adiabatically transformed
into a non-interacting Kitaev chain in a topological phase.
We need to check …
i) Nonvanishing energy gap
Interacting
model

Non-interacting
model

The energy gap must be
nonzero along the entire path.

ii) topo. phase of H0
I. Δ>0, |μ/2t|<1II. Δ<0, |μ/2t|<1Proof of the energy gap –outline- (1) Hamiltonian
For the frustration-free case, re-parametrization yields
is achieved bywhere we set t =1.
 Positive semi-definiteness
Each local Hamiltonian hj can be expressed as
Local supercharge?

Clearly,for s≧0. (Remember Lemma 2 & 3.)

Proof of the energy gap –outline- (2) Ground states

H(2U, θ) (interacting) and H (0, θ) (non-interacting) sharethe same ground states! H (0, θ) is like a Kohn-Sham system.
 Uniqueness of g.s. & existence of an energy gap
For s>0, we have

From the min-max theorem, (i) the g.s. of H(2U, θ) are unique if
those of H (0, θ) are unique, (ii) gap of H(2U, θ) ≧ gap of H (0, θ).
We still need to solve the non-interacting model H (0, θ) …

Spectrum of the non-interacting model: H (0, θ) Single-particle spectrum

A real matrix B, may not be diagonalizable, but can be written in
the SVD form:
whereΛ can be obtained by diagonalizing Miraculous properties of H(0, θ)Special factorization

BBT is pentadiagonal, but
C is tridiagonal & symmetric!

Exact eigenvalues of C
Zero-energy state! Topological order!!

Existence of many-body gap
 Many-body eigenstates of H(0, θ)

I. two g.s. with opposite parities.
(Unique in each parity sector.)
II. Many-body gap ()is nonzero in the infinite-L limit.

g.s.1

g.s.2

exc.1

Properties I & II also hold for H(2U, θ).
(From H(2U, θ) ≧ H(0, θ) & min-max theorem.)
H(2U, θ) is adiabatically connected to H(0, θ) which is in a
topological phase! The gap does not close along the path.
 Stability away from frustration-free line?
1. Kato’s theorem ( ||V|| < ΔE/2)
2. Cluster expansion
3. Lieb-Robinson boundMain difficulty: open boundaries …

t=Δ

Majorana edge zero modes Edge zero mode of H(0, θ) ⇔ Left or Right null vector of B

Exact zero modes even for finite L.
 Role of ΓL and ΓR in H(2U, θ)
They do not exactly commute with H(2U, θ).
Nevertheless, they map one of the g.s. to the other.
A significant overlap with many-body Majorana?

M.B. Hastings & X-G. Wen, PRB 72, 045141 (‘05).

g.s.1

g.s.2

Conclusions• Studied effect of interactions on Kitaev/Majorana chains• Solvable (frustration-free) line
• Exact ground states and proof of the gap• Exact solution of the BdG equation• Topological order and edge zero modes

Future directions• Spectral function & tunneling conductance• Many-body edge zero modes
1. Superoperator technique:
Find “zero-energy state” in the space of operators
2. Continuum limit: Sine-Gordon, boundary bound state, …
Ghoshal-Zamolodchikov (1994); Schuricht et al., PRB 83 (2011).
• Parafermionic generalizations
Frustration-free Hamiltonian? parafermionic zero-modes?