Download Momentum polarization: an entanglement measure of topological spin

Xiao-Liang Qi
Stanford University
ESiCQW12, Dresden, 11/15/2012
• Topologically ordered states and topological
spin of quasi-particles
• Momentum polarization as a measure of
topological spin
• Momentum polarization from reduced density
matrix
• Analysis based on conformal field theory in
entangement spectra
• Numerical results in Kitaev model
• Summary and discussion
• Topological states of matter are gapped states that
cannot be adiabatically deformed into a trivial
reference with the same symmetry properties.
• Topologically ordered states are topological states
which has ground state degeneracy and quasi-particle
excitations with fractional charge and statistics. (Wen)
• Example: fractional quantum Hall states.
𝐵⊗
Topo.
Ordered
states
Topological
states
• Only in topologically ordered states with ground state
degeneracy, fractionalization of quantum numbers and
statistics is possible.
• A general framework to describe topologically ordered
states have been developed (for a review, see Nayak et al RMP
2008)
• A manifold with certain number and types of
topological quasiparticles define a Hilbert space.
𝑐
𝑏
𝑎
𝑐
• Particle fusion: From far away we cannot distinguish
two nearby particles from one single particle
𝑐
𝑐
Fusion rules 𝑎 × 𝑏 = 𝑐 𝑁𝑎𝑏
𝑐.
Multiple fusion channels for
Non-Abelian statistics
𝑏
𝑎
• Braiding: Winding two particles
around each other leads to a unitary
operation in the Hilbert space. From far away, 𝑎 and 𝑏
looks like a single particle 𝑐, so that
the result of braiding is not
observable from far away.
Braiding cannot change the
fusion channel 𝑐 and has to be
𝑐
𝑐
𝑖𝜃𝑎𝑏
a phase factor 𝑅𝑎𝑏 = 𝑒
• Quasi-particles obtain a Berry’s phase 𝑒 𝑖2𝜋ℎ when it’s
spinned by 2𝜋.
• Spin is required since the braiding of particles 𝑎, 𝑏
looks like spinning the fused particle 𝑐 by 𝜋.
• In general the spins ℎ𝑎,𝑏,𝑐 are related to the braiding
𝑐
𝜃𝑎𝑏
(the “pair of pants” diagram):
𝑐
𝑐
𝑎
𝑏
𝑐
2𝜃𝑎𝑏
𝑎
𝑏
= 2𝜋(ℎ𝑎 + ℎ𝑏 − ℎ𝑐 )
Examples:
1. q/𝑚 charge particle in
1/𝑚 Laughlin state:
ℎ = 𝜋𝑞 2 /𝑚
2. Three particles
(1, 𝜎, 𝜓) in the Ising
anyon theory
1 1
ℎ = (0, , )
16 2
• Topological spin of particles determines the fractional
statistics.
• Moreover, topological spin also determines one of the
Modular transformation of the theory on the torus
𝑎
𝑎
𝑎
𝑎
• Spin phase factor 𝑒 2𝜋𝑖ℎ𝑎 is the eigenvalue of the Dehn
twist operation:
• The values of topological spin can be computed
algebraically for an ideal topological state (TQFT).
• For realistic states, e.g. wavefunctions from exact
diagonalization, or variational wavefunctions, it is
generally difficult to compute the spin of particles.
• Analytic results on FQH trial wavefunctions (N. Read PRB
‘09, X. G. Wen&Z. H. Wang PRB ’08, B. A. Bernevig&V. Gurarie&S. Simon, JPA
‘09)
• Numerics on Kitaev model by calculating braiding (V.
Lahtinen & J. K. Pachos NJP ’09, A. T. Bolukbasi and J. Vala, NJP ’12)
• Numerical results on variational WF using modular Smatrix (e.g. Zhang&Vishwanath ’12)
• We propose a new and easier way to numerically
compute the topological spin for lattice models.
• Consider a lattice model on the cylinder, with lattice
translation symmetry
𝑁𝑦
𝑇𝑦 (𝑇𝑦
= 1)
• For a state with quasiparticle 𝑎 in the cylinder, rotating
the cylinder is equivalence to spinning two quasiparticles to opposite directions.
• A Berry’s phase 𝑒 𝑖2𝜋ℎ𝑎 /𝑁𝑦 is obtained at the left edge,
which is cancelled by an opposite phase at the right.
• Total momentum of the left (right) edge ±2𝜋ℎ𝑎 /
𝑁𝑦 Momentum polarization 𝑃𝑀 = 2𝜋ℎ𝑎 /𝑁𝑦
𝑎
𝑒 𝑖2𝜋ℎ𝑎 /𝑁𝑦
𝑒 −𝑖2𝜋ℎ𝑎 /𝑁𝑦
𝑎
• Viewing the cylinder as a 1D system, the translation
symmetry is an internal symmetry of 1D system, of
which the edge states carry a projective representation.
• (A generalization of the 1D results Fidkowski&Kitaev, Turner et al 10’,
Chen et al 10’)
• Ideally we want to measure
• Difficult to implement. Instead, define discrete
translation 𝑇𝑦𝐿 . Translation
of the left half cylinder by
one lattice constant
2𝜋
𝑖 𝑁 ℎ𝑎
𝑦
• Naive expectation: 𝑇𝑦𝐿 𝐺𝑎 ∼ 𝑒
𝐺𝑎 contributed by
the left edge. However the mismatch in the middle leads
to excitations and makes the result nonuniversal.
• Our key result: The non-universal contribution
is independent from topological
sector, and can be cancelled by
taking the ratio
2𝜋
𝐺𝑎 𝑇𝑦𝐿 𝐺𝑎
𝑖 𝑁 ℎ𝑎
𝑦
=
𝑒
𝐺1 𝑇𝑦𝐿 𝐺1
• Requiring knowledge about topological sectors. Even if
we don’t know which sector is trivial |𝐺1 〉, ℎ𝑎 can be
determined up to an overall constant by diagonalizing
〈𝐺𝑛 𝑇𝑦 𝐺𝑚 〉 .
• 𝑇𝑦𝐿 only acts on half of the cylinder
• The overlap 𝜆𝑎 = 𝐺𝑎 𝑇𝑦𝐿 𝐺𝑎 = tr(𝑇𝑦𝐿 𝜌𝐿𝑎 )
• 𝜌𝐿𝑎 is the reduced density matrix of the left half.
• Some properties of 𝜌𝐿𝑎 is known for generic chiral
topological states.
• Entanglement Hamiltonian 𝜌𝐿𝑎 = 𝑒 −𝐻𝐸𝑎 . (Li&Haldane ‘08) In
long wavelength limit, for chiral topological states
𝐻𝐸𝑎 ∝ 𝐻𝐶𝐹𝑇 |𝑎 + 𝑐𝑜𝑛𝑠𝑡. (Edge CFT Hamiltonian in the
same topological sector)
• Numerical observations (Li&Haldane ’08, R. Thomale et al ‘10, .etc.)
• Analytic results on free fermion systems (Turner et al ‘10,
Fidkowski ‘10), Kitaev model (Yao&Qi PRL ‘10), generic FQH ideal
wavefunctions (Chandran et al ‘11)
• A general proof of this relation between edge spectrum
and entanglement spectrum for chiral topological
states (Qi, Katsura&Ludwig 2011)
• Key point of the proof: Consider the cylinder as
obtained from gluing two cylinders
• Ground state is given by perturbed CFT 𝐻𝐿 + 𝐻𝑅 +
𝑟𝐻𝑖𝑛𝑡
B
A
“glue”
B
A
𝑟=1
𝑟𝐻𝑖𝑛𝑡
B
A
• Following the results on quantum quench of CFT
(Calabrese&Cardy 2006), a general gapped state in the
“CFT+relevant perturbation” system has the asymptotic
form in long wavelength limit
𝑡
• |𝐺𝑎 ⟩ = 𝑒 −𝜏0 𝐻𝐿 +𝐻𝑅
⋅ 𝑛=0,1,… 𝑑𝑎 (𝑛) 𝑛, 𝑑𝑎 𝑛 𝐿 𝑛, 𝑑𝑎 𝑛
• This state has an left-right
entanglement density matrix
𝜌𝐿𝑎 = 𝑍 −1 𝑒 −4𝜏0 𝐻𝐿 |𝑎 .
• Including both edges,
𝜌𝐿𝑎 = 𝑍 −1 𝑒 −(𝛽𝑙𝐻𝑙+𝛽𝑟 𝐻𝑟)
𝛽𝑙 = ∞, 𝛽𝑟 = 4𝜏0 < ∞
𝑅
𝜏0
𝐺𝑎
Maximal entangled state 𝐺0𝑎
𝑎
𝛽𝑙
𝛽𝑟
• 𝜌𝐿 describes a CFT with left movers at zero temperature and
right movers at finite temperature.
• In this approximation,
𝜆𝑎 = tr 𝑇𝑦𝐿 𝜌𝐿𝑎 = tr 𝑒 𝑖 𝐻𝑙 −𝐻𝑟 𝜌𝐿𝑎
=
𝜒𝑎 𝑒
2𝜋
𝑖−𝛽𝑙
𝑁𝑦
𝛽
−2𝜋𝑁 𝑙
𝜒𝑎 𝑒
2𝜋
−𝑖−𝛽𝑟
𝑁𝑦
𝛽
−2𝜋𝑁𝑟
𝑦 𝜒
𝑦
𝜒𝑎 𝑒
𝑎 𝑒
• 𝜒𝑎 𝑞 = tr(𝑞 𝐿0 ) is the torus partition function in sector 𝑎. In
the limit 𝛽𝑟 ≪ 𝑁𝑦 , left edge is in low T limit and right edge is
in high T limit.
• Doing a modular transformation gives the result
𝑖
2𝜋
𝑁𝑦
𝑐
24
ℎ𝑎 −
𝑐 2𝜋𝑁𝑦 𝑖
24𝛽𝑟 𝛽𝑟 −𝑖
𝜆𝑎 = 𝑒
𝑒
. 𝛽𝑟 contribution independent
from 𝑎.
• Both spin and central charge can be extracted
• Numerical verification of this formula
• Honeycomb lattice Kitaev model as
an example (Kitaev 2006)
• An exact solvable model with nonAbelian anyon
𝐻=−
𝑥 𝑥
𝐽
𝜎
𝑥
𝑥−𝑙𝑖𝑛𝑘
𝑖 𝜎𝑗 −
𝑦 𝑦
𝑦−𝑙𝑖𝑛𝑘 𝐽𝑦 𝜎𝑖 𝜎𝑗 -
𝑧 𝑧
𝐽
𝜎
𝑧
𝑧−𝑙𝑖𝑛𝑘
𝑖 𝜎𝑗
• Solution by Majorana representation
with the constraint
Physical
Hilbert
space
Enlarged
Hilbert
space
• In the enlarged Hilbert space, Hamiltonian is free
Majorana fermion
𝑎
• 𝑢𝑖𝑗
become classical 𝑍2 gauge field variables.
• Ground state in the enlarged Hilbert space
• Such a solution is suitable for any trivalent lattices
• Macroscopic ground state degeneracy in the
enlarged Hilbert space
1
1
-1
1
gauge transformation
-1
-1
• The degeneracy is removed in the physical
Hilbert space
Summation over all
possible gauge
transformations
• Due to the classical nature of the 𝑍2 gauge field, the
entanglement in the system is almost separable into that
of the fermion and gauge field sectors.
• Renyi entropy 𝑆𝑛 = 𝑆𝐹𝑛 + 𝑆𝐺𝑛 (𝑆𝐺𝑛 = 𝐿 − 1 log 2)
Density matrix
+
𝑌𝑔𝐴 𝜌𝐹 (𝑢)𝑌𝑔𝐴′
⊗ |𝑔𝐴 𝑢, 𝑤〉〈𝑔𝐴′ 𝑢, 𝑤|
𝜌𝐴 =
′ ,𝑤
𝑔𝐴 ,𝑔𝐴
(Yao&Qi ‘10)
• 𝜌𝐹 (𝑢): Fermion reduced density matrix in a certain gauge
𝑢. 𝑔𝐴 , 𝑔𝐴′ : gauge transformations. 𝑤: boundary 𝑍2
variables. 𝑌𝑔𝐴 : unitary gauge
transformation to fermions
• Translation operator average value
• 𝜆𝑎 = 𝑡𝑟 𝑇𝑦𝐿 𝜌𝐴 =
′ ,𝑤
𝑔𝐴 ,𝑔𝐴
+
𝑡𝑟𝐹 𝑇𝑦𝐿𝐹 𝑌𝑔𝐴 𝜌𝐹 𝑌𝑔𝐴
′
𝑇𝑦𝐿𝐹
⋅ 〈𝑔𝐴′ 𝑢, 𝑤|𝑇𝑦𝐿𝐺 |𝑔𝐴 𝑢, 𝑤〉
• The translation on 𝑍2 gauge fields
gives a 𝛿 function: 𝑔𝐴′ is determined
by 𝑔𝐴 , denoted by 𝑔𝐴𝑇 .
• 𝜆𝑎 =
+ 𝐿𝐹
𝑡𝑟
𝑌
𝑇 𝑌 𝜌
𝑔𝐴 ,𝑤 𝐹
𝑔𝑇 𝑦 𝑔𝐴 𝐹
𝐴
• 𝑌𝑔+𝑇 𝑇𝑦𝐿𝐹 𝑌𝑔𝐴 = 𝑇𝑦 the translation
𝐴
combined with gauge transformation.
𝑇𝑦
Gauge
transformation
• Non-Abelian phase of
Kitaev model (Kitaev 2006)
• Chern number 1 band
structure of Majorana
fermion
• 𝜋 flux in a plaquette
induces a Majorana zero
mode and is a non-Abelian
anyon.
𝐸
𝜙=𝜋
• On cylinder, 0 flux
leads to zero mode
1
+
𝛾−𝑘
𝛾𝑘+
𝜓
𝜎
𝜙=0
𝑘
𝐸
𝑘
𝑖𝑗
𝑍 −1 𝑒 −𝜂𝑖 ℎ𝐸 𝜂𝑗
• Fermion density matrix 𝜌𝐿𝐹 =
is
determined by the equal-time correlation function
〈𝜂𝑖 𝜂𝑗 〉 (Peschel ‘03)
+
• 𝑇𝑦 = exp[𝑖 𝑘,𝑛 𝑘𝛾𝑘𝑛
𝛾𝑘𝑛 ] in entanglement
+
Hamiltonian eigenstates. (𝐻𝐸 = 𝑛 𝛾𝑘𝑛
𝛾𝑘𝑛 𝜆𝑛 )
• We obtain
𝑖𝑝
ℎ
−
𝑖𝑝
𝐸
𝑒 2 cosh
2
𝜆𝜎,1 = det
ℎ𝐸
flux 0,𝜋
cosh
2
• Numerically,
𝑁𝑦
•
𝜆𝜎
ℎ𝜎 =
log
2𝜋𝑖
𝜆1
1
ℎ𝜓 = is known
2
analytically)
• Central charge 𝑐 can also
be extracted from the
comparison with CFT result
2𝜋
𝜆𝑎 = 𝑒
𝑖𝑁
𝑦
𝑐
ℎ𝑎 −24
• imag(log 𝜆1 ) =
𝑒
𝑐 2𝜋𝑁𝑦 𝑖
24𝛽𝑟 𝛽𝑟 −𝑖
𝑐 2𝜋
−
24 𝑁𝑦
+
2𝜋𝑁𝑦
1+𝛽𝑟2
• The result converges
quickly for
𝑁𝑦 >correlation length 𝜉
• Across a topological phase
𝐽𝑧
transition tuned by to
𝐽𝑥
an Abelian phase, we see
the disappearance of ℎ𝜎
• Sign of ℎ𝜎 determined by
second neighbor coupling
𝐽𝑛𝑛
• Interestingly, this method goes beyond the edge
CFT picture.
• Measurement of ℎ𝜎 and 𝑐 are independent from
edge state energy/entanglement dispersion. For a
cylinder with zig-zag edge, the dispersion is 𝐸 ∝ 𝑘 3 ,
the result still holds.
• A discrete twist of cylinder measures the topological
spin and the edge state central charge, due to the
entanglement between two parts of the cylinder
• A general approach to compute topological spin for
chiral topological states
• Numerically verified for Kitaev model. Even when the
edge state has a nonlinear dispersion in long
wavelength limit, the result still holds for Kitaev
model.
• This approach applies to many other states. Suitable
for variational Monte Carlo wavefunctions, (On going work
with Hong-Hao Tu and Yi Zhang) and MPS states (related to F.
Pollmann’s talk).