Download Quantum spin liquids

Quantum spin liquids
SFB/TR 49 student seminar
Hamburg, October 2013
Simon Trebst
University of Cologne
[email protected]
© Simon Trebst
Motivation – a paradigm
H=
interacting
many-body system
X
z z
i j
hiji
Spontaneous symmetry breaking
© Simon Trebst
•
ground state has less symmetry than Hamiltonian
•
local order parameter
•
phase transition / Landau-Ginzburg-Wilson theory
Every rule has an exception
◆
N ✓
⇣
⌘
X
2
1
e
H=
pj
A(xj ) + eA0 (xj )
2m
c
j=1
X
+
V (|xi xj |)
electron gas
B
I
i<j
interacting
many-body system
✓
2
tr A ^ dA + A ^ A ^ A
3
◆
Sometimes, the exact opposite happens
© Simon Trebst
•
ground state has more symmetry than Hamiltonian
•
non-local order parameter
•
emergence of degenerate ground states, exotic statistics, ...
V
AlGaAs
GaAs
AlGaAs
Topological quantum matter
•
•
© Simon Trebst
Spontaneous symmetry breaking
•
ground state has less symmetry than Hamiltonian
•
Landau-Ginzburg-Wilson theory
•
local order parameter
Topological order
•
ground state has more symmetry than Hamiltonian
•
degenerate ground states
•
non-local order parameter
concepts
© Simon Trebst
Quantum spin liquids
Quantum spin liquids are exotic ground states of frustrated quantum magnets,
in which local moments are highly correlated but still fluctuate strongly
down to zero temperature.
Classification a la Xiao-Gang Wen
three spatial dimensions
spatial dimensions
energy spectrum
two spatial dimensions
gapped
gapless
Xiao-Gang Wen
MIT/Perimeter
topological
spin liquids
gapless
spin liquids
singular points
e.g. Dirac cones
© Simon Trebst
spinon surface,
Bose surface
Topological quantum matter
Energy
Xiao-Gang Wen: A ground state of a
– be fully +
+ system that
+ cannot
many-body
E
characterized
by a local order parameter.
E
Ep
Often characterized by a variety of
non-local “topological properties”.
+
–
–
© Simon Trebst
+
0
Co
ba
B
0
0
•
–
n
–
•
c
+
•
b
0
–
a
ba
Momentum
A topological
phase
can
Figure 2 | Topological
insulators
andbe
bandpositively
structures. a, The conduction and valence bands o
A
typical
3D
solid
(middle
section).
The
shaded
regions
are
the
bands
in
the bulkBof the solid, a
identified
by
its
entanglement
properties.
thick black lines are the bands at the surface. (Similar behaviour is observed in a 2D system
1D boundary.) In general, the conduction band is symmetric (red), the valence band is antisy
(blue), and spin-up and spin-down electrons (black arrows) have the same energy. En, E0 and
Knots & edge states
•
Bringing a topological and a conventional state into
spatial proximity will result in a gapless edge state –
literally a knot in the wavefunction.
•
We know this: “Counterintuitive states”
UK
© Simon Trebst
Ireland
Australia
Japan
China
Hong Kong
Knots & edge states
China
Hong Kong
Flipper bridge
© Simon Trebst
topological quantum spin liquids
– gapped quantum spin liquids –
© Simon Trebst
the toric code
the drosophila
for lattice models
of topologically ordered phases
© Simon Trebst
The toric code
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC =
A
⇥
v j vertex(v)
z
j
B
⇥
x
j
p j plaquette(p)
similar to ring exchange
introduces frustration
i
Hamiltonian has only local terms.
All terms commute → exact solution!
The vertex term
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC =
A
⇥
v j vertex(v)
z
j
B
⇥
x
j
p j plaquette(p)
•
is minimized by an even number of down-spins around a vertex.
•
Replacing down-spins by loop segments maps ground state to closed loops.
•
Open ends are (charge) excitations costing energy 2A.
z
=
A
⇥
The plaquette term
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC =
A
⇥
v j vertex(v)
z
j
B
⇥
x
j
p j plaquette(p)
•
flips all spins on a plaquette.
•
favors equal amplitude superposition of all loop configurations.
•
Sign changes upon flip (vortices) cost energy 2B.
fugacity = 1
free loop creation
isotopy
free local distortions
surgery
reconnections
Toric code – quantum loop gases
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Ground-state wavefunction is equal superposition of loop configurations.
Toric code – quantum loop gases
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Py = 1
Py =
Topological sectors defined by winding number parity Py/x =
z
i
i cx/y
1
Toric code – quantum loop gases
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Py = 1
Py =
Topological sectors defined by winding number parity Py/x =
z
i
i cx/y
1
Anyons in the toric code
A. Kitaev, Ann. Phys. 303, 2 (2003).
Two types of excitations
•
electric charges: open loop ends violate vertex constraint
•
magnetic vortices: plaquettes giving -1 when flipped
We get a minus sign taking one around the other
electric charges and magnetic vortices are mutual anyons
= (-‐‑‒1) x
quantum double models
looking beyond the drosophila
© Simon Trebst
quantum double models
Quantum double models form a larger family of lattice models
harboring non-trivial topological order, e.g. non-Abelian string nets.
loop gas configuration
(toric code)
•
Quantum double models are generally constructed
from an underlying anyon theory.
•
Key ingredient are so-called fusion rules of anyons.
M. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005);
C. Gils et al., Nature Physics 5, 834 (2009).
© Simon Trebst
string net configuration
A
Ā
}
anyon
theories
0, 1, 2, . . . k .
quantum double models
One can think of every degree of freedom as a representation
of the quantum group su(2)k or in analogy to the spin representation of SU(2), which corresponds to the limit k ! 1, as
generalized spins with even/odd integer labels corresponding
to integer/half-integer spins.
We can combine two representations
of su(2)kdouble
into one models form a larger family of lattice models
Quantum
joint representation – similar to combining two spin quantum
harboring
numbers into one joint spin quantum
number fornon-trivial
conventional topological order, e.g. non-Abelian string nets.
SU(2) spins. This process, which for the anyon theories is often called fusion, has to obey very similar rules as those for
combining two conventional SU(2) spins. In particular, they
have to obey the so-called fusion rules which also incorporate
the cut-off k in a consistent way
min[i+j,2k i j]
X
i⇥j =
(14)
l,
l=|i j|
where l increases in steps of two. Eq. (14) can be written more
compactly by introducing fusion coefficients Nijl , which are
defined via
i⇥j ⌘
k
X
l=0
Nijl l .
(15)
loop
gas configuration
Note that for the su(2)k anyon
theories
at hand these fusion
coefficients are always either 0 or 1.
The simplest example is probably the anyon theory su(2)1
with anyonic degrees of freedom 0 and 1, for which the above
Z2 anyon theory
fusion rules become
and finally the Fibonacci theory, which is the even-integer
subset of this su(2)3 anyonic
theory anyon theory
Fibonacci
0⇥0=0
0⇥1=1
1 ⇥ 1 = 0.
0⇥0=0
0⇥2=2
2 ⇥ 2 = 0 + 2.
toric code
0
1
(16)
0
0
0
A slightly less trivial example is the anyon theory su(2)2 with
anyonic degrees of freedom 0, 1 and 2, for which the fusion
rules read
© Simon Trebst
string net configuration
1
0
2
(19)
0
0
0
non-Abelian
One striking distinction between
the fusion rules for su(2)1
in (16) and for the remaining ones in (17), (18), and (19) is the
occurrence of representations, which fused with itself, gener-
2
possible labelings of ed
of allowed vertices for
ory are given (up to rota
respectively.
2
Forming
an entire gr
tices and corresponding
tude of lattice
configura
2
2
configurations it becom
translate into constraint
For the case of the su(2)
gapless quantum spin liquids
– an example –
© Simon Trebst
The Kitaev model
Jz
gapped spin liquid
Z2 topological order
x y
z
Jx + Jy + Jz = const.
z z
i j
x x
i j
HKitaev =
y y
i j
X
J
Jx
i
j
links
Rare combination of a model
of fundamental conceptual importance
and an exact analytical solution.
Jy
gapless spin liquid
w/ Majorana fermion excitations
2 Dirac cones
A. Kitaev, Ann. Phys. 321, 2 (2006)
© Simon Trebst
Solving the Kitaev model
Step 1: Majorana fermionization
x y
z
bz
bx
†
†
fermions a" a" a# a#
© Simon Trebst
bz = a# + a†"
⇣
c = i a#
y
a†#
⌘
a†"
⌘
x
z
= i by c
D=
i
x y z
}
Majorana
fermions
bx = a" + a†#
⇣
y
b = i a"
by
c
= i bx c
= i bz c
= bx b y bz c
gauge operator
[D,
]=0
physical subspace D = 1
Solving the Kitaev model
Step 2: Diagonalization of Hamiltonian
In the language of Majorana fermions, we now have
j
k
= i bj cj (i bk ck ) =
i ujk cj ck
ujk = i bj bk
which immediately allows us to write the Hamiltonian as
iX
H=
Ajk cj ck
4
hjki
Ajk = 2J ujk
bz
bx
by
c
© Simon Trebst
Hamiltonian is skew-symmetric, because
ujk =
ukj
Ajk =
Akj
Solving the Kitaev model
Step 2: Diagonalization of Hamiltonian
In the language of Majorana fermions, we now have
j
k
= i bj cj (i bk ck ) =
i ujk cj ck
ujk = i bj bk
which immediately allows us to write the Hamiltonian as
iX
H=
Ajk cj ck
4
hjki
Ajk = 2J ujk
bz
bx
by
c
Hamiltonian is skew-symmetric, because
ujk =
Ajk =
Finally, there is a gauge choice to be made
ujk
© Simon Trebst
ukj
has eigenvalues
±1
Akj
eight copies of each p
the same translationa
phase are algebraical
We now consider t
tum q is defined mod
momentum space by
the symmetric case (J
Solving the Kitaev model
Step 2: Diagonalization of Hamiltonian
The Hamiltonian has turned into a free (Majorana) fermion problem
iX
H=
Ajk cj ck
4
hjki
which can readily be diagonalized by Fourier transformation
nx
i
H(q) = A(q) =
2
ny
✓
◆
0If |Jx| if
(q)
and |Jy| decrea
⇤
i f(within
(q)
0the paralle
|Jiq·n
x| + |Jy| = |Jz|. The p
y
f (q) = Jx e
+ Jyallelogram.
e
+ Jz (Note tha
At the points ±q*
thus yielding a gapless energy spectrum of the form
iq·nx
bz
b
x
by
c
nx =
ny =
p !
1
3
,
2 2
p !
1
3
,
2 2
✏(q) = ±|f (q)|
Dirac cones
q⇤1
q⇤2
© Simon Trebst
=
=
✓
✓
2⇡ 2⇡
,p
3
3
◆
2⇡ 2⇡
,p
3
3
◆
The Kitaev model
Jz
gapped spin liquid
Z2 topological order
x y
z
Jx + Jy + Jz = const.
z z
i j
x x
i j
HKitaev =
y y
i j
X
J
Jx
i
j
links
Rare combination of a model
of fundamental conceptual importance
and an exact analytical solution.
Jy
gapless spin liquid
w/ Majorana fermion excitations
2 Dirac cones
A. Kitaev, Ann. Phys. 321, 2 (2006)
© Simon Trebst
entanglement
© Simon Trebst
Identification of topological order
Which observables can we measure numerically
to identify topological order?
•
•
© Simon Trebst
bulk properties
•
entanglement entropy & spectrum
•
ground-state degeneracy
edge properties
•
entanglement entropy
•
energy spectra
bulk properties
entanglement entropy
© Simon Trebst
Entanglement
If two quantum mechanical objects are interwoven in such a way
that their collective state cannot be described as a product state we say they are entangled.
•
"
A
B
i + sin ↵|
"
| i = cos ↵| " i|
A
i| " i
B
Calculate the reduced density matrix for one of the two parts
⇢A = | iA h |A
(traced over subsystem B)
•
A
"
A
A
ih
"
⇢A = cos2 ↵| " ih " | + sin2 ↵|
A
|
One quantitative measure of entanglement is the entanglement entropy
S(⇢A ) =
Tr( ⇢A log ⇢A )
von Neumann entropy = first Renyi entropy
Sn (⇢A ) =
1
1
Renyi entropy
© Simon Trebst
n
ln [Tr( ⇢nA
)]
S2 (⇢A ) =
ln
⇥
second Renyi entropy
Tr( ⇢2A
)
⇤
Entanglement
If two quantum mechanical objects are interwoven in such a way
that their collective state cannot be described as a product state we say they are entangled.
"
"
Tr( ⇢A log ⇢A )
p
i| " i) / 2
0.6
Renyi entropy
1
n
ln [Tr( ⇢nA )]
Renyi entropy
Sn (⇢A ) =
1
i
|
p
i| " i) / 2
S1
S2
S3
S4
S5
S6
S7
S8
0.4
0.2
|
"
second Renyi entropy
(| " i|
i| " i
| " i|
"
⇥
⇤
ln Tr( ⇢2A )
B
S = ln 2
von Neumann entropy = first Renyi entropy
S2 (⇢A ) =
A
"
S(⇢A ) =
i+|
B
i| " i
"
(| " i|
"
A
i + sin ↵|
"
| i = cos ↵| " i|
i
0
0
© Simon Trebst
π/4
π/2
α
3π/4
π
Entanglement
•
Entanglement in quantum many-body systems
•
Consider bipartition of system into two parts A and B,
and calculate the reduced density matrix for the two parts
⇢A = | iA h |A
•
(traced over subsystem B)
One quantitative measure of entanglement
is the entanglement entropy
S(⇢A ) =
Tr( ⇢A log ⇢A )
von Neumann entropy = first Renyi entropy
Sn (⇢A ) =
1
1
n
ln [Tr( ⇢nA )]
Renyi entropy
S2 (⇢A ) =
ln
⇥
second Renyi entropy
© Simon Trebst
Tr( ⇢2A
)
⇤
B
A
Boundary law
also called area law
B
A
Entanglement scales with the length L
of the boundary of the bipartition
S = ln 2
© Simon Trebst
S / aL
Corrections to the boundary law
Corrections to the boundary law can arise from
•
geometric aspects of the bipartition
B
•
S = aL + b · (# of corners)
A
(for 2D gapped state)
topological aspects of the bipartition
A
S = aL
B
© Simon Trebst
· (# of disconnected parts)
(for 2D gapped state)
Corrections to the boundary law
Corrections to the boundary law
also originate from the specific
character of the underlying quantum many-body state!
•
topological spin liquids
this talk
quantum double models
Levin-Wen / Kitaev models
S = aL
•
gapless spin liquids
• gapless modes at singular point in momentum space
S = aL + c (Lx , Ly )
•
gapless modes on surface in momentum space
“Bose metals”
S = cL ln(L)
•
(Motrunich, Sheng & Fisher)
critical points, conformal critical points, Goldstone modes, ...
SQCP = aL + c (Lx , Ly )
© Simon Trebst
Kitaev model
(honeycomb)
ScQCP = µL +
cQCP
SG = aL + b ln(L) + (Lx , Ly )
Topological entanglement entropy
Distilling the topological correction
by using a clever sequence of partitions
A1
A3
A2
A4
2
2
S = aL + b · (# of corners)
Stopo =
· (# of disconnected parts)
S A 1 + S A2 + S A 3
SA4 =
A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006);
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
© Simon Trebst
2
Topological entanglement entropy
The topological correction is universal.
v
u n
uX
= ln t
d2
i
quantum dimension
of excitation
i=1
Examples:
•
toric code (loop gas)
1
e
m
di
•
1
1
1
em
1
= ln
p
1 + 1 + 1 + 1 = ln 2 ⇡ 0.693
Fibonacci theory (string net)
1
di
1
⌧
p
1+ 5
=
2
p
= ln 1 +
A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006);
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
© Simon Trebst
2
⇡ 0.643
Let’s try this on the toric code
ARTICLES
a
toric code (loop gas)
1
e
m
di
1
1
1
5
5
hx = hz = h
h = 0.2
h = 0.3
h = 0.4
4
em
1
3
4
3
2
S(Ly)
•
S(Ly)
Examples:
N
1
1
0
= ln
p
2
0
¬ln2
1 + 1 + 1 + 1 = ln 2 ⇡ 0.693
¬1
b
0
4
2
4
6
8
Ly
0
10
12
14
16
hx = 0.3, hz = 0.0
S(Ly)
2
ARTICLES
1
0
PUBLISHED ONLINE: 11 NOVEMBER 2012 | DOI: 10.1038/NPHYS2465
¬ln2
¬1
Hong-Chen Jiang1 , Zhenghan Wang2 and Leon Balents1 *
Topological phases are unique states of matter that incorporate long-range quantum entanglement and host exotic excitations
with fractional quantum statistics. Here we report a practical method to identify topological phases in arbitrary realistic models
by accurately calculating the topological entanglement entropy using the density matrix renormalization group (DMRG). We
argue that the DMRG algorithm systematically selects a minimally entangled state from the quasi-degenerate ground states
in a topological phase. This tendency explains both the success of our method and the absence of ground-state degeneracy in
previous DMRG studies of topological phases. We demonstrate the effectiveness of our procedure by obtaining the topological
entanglement entropy for several microscopic models, with an accuracy of the order of 10 3 , when the circumference of
the cylinder is around ten times the correlation length. As an example, we definitively show that the ground state of the
quantum S = 1/2 antiferromagnet on the kagome lattice is a topological spin liquid, and strongly constrain the conditions for
identification of this phase of matter.
© Simon Trebst
Nature Physics 8, 902 (2012).
2
second-neighbou
0
Identifying topological order by entanglement
entropy
0
Figure 3 | The entan
equation (2), with L
= 0.698(8) at J2 =
lattice with Lx = 12 a
G = +¬1
G=0
3
¬1
0
2
4
6
8
10
12
14
Ly
Figure 2 | The von Neumann entropy S(Ly ) for the toric-code model in
magnetic fields. a, S(Ly ) with Ly = 4–16 at Lx = 1 for symmetric magnetic
fields at hx = hz = h = 0.2, 0.3 and 0.4. By fitting S(Ly ) = aLy
, we get
= 0.693(1), 0.691(4) and 0.001(5), respectively. b, The pure electric
case, hx = 0.3,hz = 0, and comparison of S(Ly ) in the MES obtained in the
large Lx limit (black squares) with that of the absolute ground state from
systems of dimensions Lx ⇥ Ly = 20 ⇥ 4,24 ⇥ 6,24 ⇥ 8,24 ⇥ 10 (red circles).
Extrapolation shows that the MES has the universal TEE, whereas the
absolute ground state has zero TEE.
around the entire cylinder. To take this into account, the DMRG
must fully converge the entanglement of the opposite ends of the
where Si is the s
the nearest neighb
simulation, we se
DMRG studies17
substantial spin-li
We take the ka
along a bond dire
the unit of lengt
results for the en
shown in Fig. 3 w
spacing for both s
We see that a linea
gives = 0.698(
both within one
that this phase is
quantum dimens
was proposed for
neighbour interac
of the spectrum of ! properly, but also point to a finite value
of ", and hence a topological ground state. The quantum
dimension
pffiffiffi is D ¼ 2, excluding chiral spin liquids (" ¼ 1=2
or D ¼ 2 [77]). Rigorously, DMRG only provides a lower
bound on D [80], but the bound is essentially exact as
DMRG is a method with low entanglement bias [81].
Conclusion.—Through a combination of a large number
of DMRG states, large samples with small finite size effect,
and the use of the SU(2) symmetry of the kagome model, we
have been able to corroborate earlier evidence for a QSL as
opposed to a VBC, due to energetic considerations and
complete absence of breaking of space group invariance,
although DMRG should be biased towards VBC due to its
low-entanglement nature and the use of OBC. On the basis
of the numerical evidence (spin gap, structure factor, spin,
dimer and chiral correlations, topological entanglement
entropy) numerous QSL proposals can be ruled out for the
kagome system. On the system
sizes reached, the spin gap is
ARTICLES
[85,86] albeit for a variational energy far from the groundstate energy. The concentration of weight of the structure
factor at the hexagonal Brillouin zone edge with shallow
maxima at the corners would also point to the Z2 QSL as
proposed by Sachdev [27], and a Z2 QSL is also consistent
with all other numerical findings. All this provides strong
evidence for the Z2 QSL, whereas to our knowledge, no
plausible scenario for the emergence of a double-semion
phase in the KAFM has been
discovered somodel
far, making it
Heisenberg
implausible, but of course not impossible. An analysis of the
degenerate ground-state manifold as proposed in Ref. [80],
not possible with our data, would settle the issue. Even if the
j
answer provided final evidence for a Z2 QSL,i many questions regarding the detailed microscopic structure of the
hi,ji
ground-state wave functions and the precise nature of the Z2
QSL would remain for future research.
kagomé
lattice
S. D. and U. S. thank F. on
Essler,
A. Läuchli,
C. Lhuillier,
D. Poilblanc, S. Sachdev, R. Thomale, and S. R. White for
NATURE PHYSICS
DOI: 10.1038/NPHYS2465
discussions. S. D. and U. S. acknowledge
support by
DFG.
U. S. thanks the GGI, Florence, for its hospitality. I. P. M.
5
acknowledges support
from the Australian Research
Council Centre of Excellence
for Engineered Quantum
4
Systems and the Discovery Projects funding scheme
(Project No. DP1092513).
3
Note added.—Recently, we became aware of Ref. [81],
which calculates topological
entanglement entropy from
2
von Neumann entropy for a next-nearest neighbor modification of the KAFM, 1perfectly consistent with our results
J2 = 0.10
of D ¼ 2 for the KAFM itself.
J2 = 0.15
Kagomé antiferromagnet
a
S2
S3
5
S4
h = 0.2
h = 0.3
h = 0.4
3
fit to S2
fit to S3
3
hx = hz = h
4
S5
4
5
2
fit to S4
2
1
fit to S5
1
0
0
¬ln2
0
-1
¬1
0
2
4
0
¬ln2
0
2
6
8
circumference c
b
10
4
12
6
14
8
Ly
10
4
hz = 0.0
hx = 0.3,S
FIG. 6 (color online). Renyi entropies
# of infinitely long
G = +¬1
cylinders for various # versus circumference
c, extrapolated to
G=0
3
c ¼ 0. The negative intercept is the topological entanglement
entropy ".
S(Ly)
2
S. Depenbrock, I.P. McCulloch,
and U. Schollwöck,
1
Phys. Rev. Lett. 109, 067201 (2012).
0
© Simon Trebst
~ ·S
~
S
S(Ly)
6
S(Ly)
Renyi entropy Sα
7
H=
X
12
14
¬1
0
2
4
6
Ly
8
10
12
*[email protected]
Figure 3 | The entanglement entropy S(Ly ) of the kagome J1 –J2 model in
[1] L. Balents, Nature
(London)
(2010).
equation (2),
with Ly = 464,
4–12 at199
Lx = 1.
By fitting S(Ly ) = aLy
, we get
[2] V. Elser, Phys.
Rev.
Lett.
62,
2405
(1989).
= 0.698(8) at J2 = 0.10 and = 0.694(6) at J2 = 0.15. Inset: kagome
Lx =Rev.
12 andBLy 83,
= 8. 214406 (2011).
[3] K. Matan etlattice
al., with
Phys.
067201-4
0
16
second-neighbour interactions, whose Hamiltonian is
XL. Balents,
H.-C. Jiang, X
Z. Wang, and
H = JPhysics
Si ·Sj8,
+902
J2 (2012).
Si ·Sj
1
Nature
hiji
hhijii
(2)
where Si is the spin operator on site i, and hiji (hhijii) denotes
the nearest neighbours (next-nearest neighbours). In the numerical
bulk properties
entanglement spectrum
© Simon Trebst
Entanglement spectrum
•
The entanglement entropy is one quantitative measure of entanglement
⇢A = | iA h |A
(traced over subsystem B)
B
S(⇢A ) =
Tr( ⇢A log ⇢A )
•
However, a lot of information possibly contained in
the density matrix is discarded in this simple measure.
•
The entanglement spectrum aims at unraveling some of this information
⇢A = | i A h | A =
S(⇢A ) =
X
2
↵
ln
2
↵
X
↵
↵
2
↵ | ↵ iA h ↵ |A
=e
⇠↵ /2
(Schmidt decomposition)
⇢A = e
↵
(HE =
H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
© Simon Trebst
A
HE
ln ⇢A )
“entanglement
Hamiltonian”
Entanglement spectrum
H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
The entanglement spectrum aims at unraveling
some of the information contained in the density matrix
•
⇢A = | i A h | A =
B
X
↵
↵
⇠↵ /2
=e
⇢A = e
(a)
(b)
HE
“entanglement
Hamiltonian”
week ending
(HE = 7lnMAY
⇢A )2010
PHYSICAL REVIEW LETTERS
180502 (2010)
A
2
↵ | ↵ iA h ↵ |A
8
20
20
15
15
6
4
2
10
45
0
1
0.04
0.08
0.12
N
10
1 1 23 5
53 2 11
40
5
counting related to number of modes
of the gapless edge theory (CFT)
35
5
30
25
20
0
10
6
8
10
15
12
14
20
25
30
35
0
5
10
15
20
25
30
35
R. Thomale, A. Sterdyniak, N. Regnault, and B.A. Bernevig, Phys. Rev. Lett. 104, 180502 (2010).
or online). Entanglement
spectrum for the N ¼ 11 bosons, N# ¼ 20, $ ¼ 1=2 Coulomb state on the sphere. The cut is
¼ 10 orbitals and NA ¼ 5 bosons. (a) Standard normalization on the quantum Hall sphere. The inset show the remainder
spectrum
where the entanglement levels exceed % ¼ 24. (b) CL normalization. We observe that the CL separates a set of
© Simon Trebst
Entanglement summary
•
The entanglement entropy and entanglement spectrum
are powerful bulk “observables” that allow
•
•
to unambiguously distinguish topological order from conventional order
to characterize the type of topological order to a good extent*
* in some measures, e.g. the topological entanglement entropy, some unambiguities might remain and require additional work
•
The reduced density matrix is readily available in some numerical methods such as
•
•
•
exact diagonalization
density matrix renormalization group
⇢A = | iA h |A
Other numerical techniques have caught up, in particular
•
quantum Monte Carlo ➝ replica trick ➝
Renyi entropies / entanglement entropy
You will hear about this in Peter Bröcker’s talk tomorrow morning.
•
© Simon Trebst
numerical linked cluster expansion
➝
mutual information
bulk properties
ground-state degeneracy
© Simon Trebst
Ground-state degeneracy
ST et al., Phys. Rev. Lett. 98, 070602 (2007).
topological phase
paramagnet
4
4
⇥E / exp(
energy splitting !E(L)
2
L)
2
0
0
L = 48
L = 40
L = 32
L = 24
L = 16
L = 12
L=8
-2
-4
-2
-4
-6
-6
E
-8
0.54
© Simon Trebst
0.55
0.56
0.57
0.58
0.59
magnetic field hx
L
-8
0.6
0.61
0.62
edge properties
entanglement entropy
© Simon Trebst
Edge states
“vacuum”
“vacuum”
topologically
ordered state
edge state
edge state
•
Edge states correspond to the gapless modes of a critical one-dimensional system,
which are typically described by a conformal field theory (CFT).
•
The conformal field theory can again be identified via entanglement properties
}
`
central charge
✓
✓ ◆◆
c
⇡`
S = log L sin
3
L
C. Holzhey et al., Nucl. Phys. B 424, 44 (1994);
P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004).
© Simon Trebst
` = L/2
S=
c
log L
3
Xi Xi±1 Xi = Xi ,
|xi−1 xi xi+1 "}
=
−1, 0, 0}, where the
straints arising from
xi+1 = τ , the F an are the following
•
ϕ−2 ϕ−3/2
ϕ−3/2 ϕ−1
#
$
.
&
ni + 1 + ϕ−2 ,
chain. In this exprese occupation on link
tonian H acts on the
We simulated this
finite-size excitation
f up to L = 32 sites.
vanishes linearly in
ynamical critical exal field theory. The
ated from the finiteFor two subsystems
odic (PBC) and open
ement entropy scales
c
log L .
6
(3)
up method (DMRG)
lculating these quanccording to Eqs. (3)
stimates of cPBC =
1 respectively. Since
y of these estimates
5 we can unambiguent only with central
© Simon Trebst
week ending
j j correspond
" =
e[i]
"
Edgee[i]|j
states
to|jthej jgapless
ofE VaI Ecritical
system,
P H Ymodes
SICAL R
W L E T Tone-dimensional
ERS
20 APRIL 2007
PRL 98, 160409 (2007)
'
which %are typically
described
(CFT).
" conformal field theory
! Y by a
6
6
( j~
j
&
j
i−1 i i+1
"%
j!
(2)
onian, it can be writ•
L) ∝
Edge states
i
ji+1
ji−1
&j ! i
ji
i−1
!
i i+1
0
0
0 0
i%1 i
e(
!i"j~ S:#
(6)
ji Sj ! i !j0m ;jm $e!i"j 'j :
)
i&1 i
(5)
and
= δji−1 ,ji+1
Sj0i−1 Sj0m!i
ji
i+1
The Hamiltonian of the so-defined lattice model is •obare known [12] to form atained
representation
the Temperley-Lieb
from itsoftransfer
matrix by taking, as usual [16],
algebra*
(4) for any value
k, where anisotropic
|ji − ji+1 | =
1/2 uand
theofextremely
limit,
( 1,
j!
(2j+1)(2j ! +1)
2
Sj := (k+2) sin[π
][13].
expf&a$H % c1 ' % O$a2 'g;
k+2 T #
Our model of interacting Fibonacci anyonsu’
can be cast into
1
a # xi = 1 → ji = (
this form at k = 3 by first mapping
sin!"=$k % 2'" 0, and
xi = τ → ji = 1, and then applying theP
SU (2)3 fusion rule
1
yielding
H # & sites.
is ananyunimportant constant).
3/2 × j = 3/2 − j to the
even-numbered
i (c1maps
i ’ eThis
admissible labeling |&x"Since
:= |x1the
, x2 ,operators
. . ." uniquely
j" :=
can |&be
identified with ei , this
Xi into
demonstrates that the Hamiltonian of the Fibonacci chain
is exactly that of the corresponding
1.5
1.5 k # 3 RSOS model
which is a lattice description of the tricritical Ising model at
its critical point. The latter is a well-known (supersymmet7
ric) CFT with central charge c # 10
[17,18]. Analogously
one
obtains
[19]
for
general
k
the
$k
&
1'st unitary minimal
1
1
CFT [10] of central charge c # 1 & 6=$k % 1'$k % 2'. A
periodic boundary
conditions
ferromagnetically
coupled Fibonacci chain (energetically
open boundary conditions
favoring the fusion along the # channel) is described [20]
by the critical 3-state Potts model
with c # 45 and, for
0.5
0.5
general k, by the critical Zk -parafermion CFT [15,19]
with central charge c # 2$k & 1'=$k % 2'.
Excitation spectra.—We calculated the excitation spectra
of chains up to size L # 37 0with open and periodic
0
conditions
20
30
40 boundary
80 100 120 using
200 240diagonalization, as shown
50 60
160 exact
in system
Fig. 3.size
TheLnumerical results not only confirm the CFT
predictions but also reveal some important details about the
FIG. 2: (Color online) Entropy
scaling for between
interactingcontinuous
Fibonacci
correspondence
fields and microanyons arranged along an
open (open
squares) or
scopic
observables.
In periodic
general,chain
low-energy states on a
(closed circles) versus the system size L. Logarithmic fits (solid
ring are associated with local conformal fields [21], whose
lines) give central charge estimates of cPBC = 0.701 ± 0.001 and
holomorphic
andtheantiholomorphic
cOBC = 0.70 ± 0.01 respectively, where for
open boundary con-parts belong to representations
of
the
Virasoro
algebra,
described by conformal
ditions
only the et
values
the 5Rev.
largest
systems
been
taken into
A. Feiguin
al., for
Phys.
Lett.
98, have
160409
(2007).
weights
hL and hR . The energy levels are given by
account due to large finite-size
effects.
#
$
2"v
c
E#E L%
& %h %h ;
(7)
j
e[i]ji+1
i−1
j! i
The CFT can be identified
via entanglement properties
c
S = log L
3
5
5
Even4 more information about the CFT
4
reveals
itself
in
the
energy
spectrum
L = 36
3
3
rescaled energy E(K)
.
(4)
where the ‘d-isotopy’ parameter equals the golden ratio, d =
ϕ. This representation can be seen to be identical to the standard Temperley-Lieb algebra representation associated with
SU (2)k at level k = 3. For arbitrary k, the latter contains
k + 1 anyon species labelled by j = 0, 1/2, 1, ..., k/2, satisfying the fusion rules of SU (2)k [11]. The operators ei defined
by
entropy S(L)
1
i+1
(Fxxi−1
τ τ )x! i
[Xi , Xj ] = 0 for |i − j| ≥ 2 ,
1/5 + 3
3 6/5 + 2
1/5 + 2
3/40 + 3
3/40 + 3
0+3
primary fields
descendants
1/5 + 2
0+2
7/8 + 2
3/40 + 2
⌘2
2⇡v ⇣ c
E1 = E1 L +
+ xs 1
Neveu-Schwarz
Ramond
L
12 sector
sector
0
0
2
6/5 + 1
6/5
3/40 + 2
7/8 + 1
1/5 + 1
7/8
3/40 + 1
1/5
0
3/40
0
rescaled energy E(K)
|xi−1 x! i xi+1 " (1)
i
2
4
6
8
10
12
momentum K [2π/L]
14
18
16
6
6
5
5
4
4
7/8 + 2
2
1
0+2
7/10 + 2
L = 37
primary fields
descendants
3
3/40 + 2
3/2 + 1
7/8 + 1
7/8
3/40
0
0
7/10 + 1
3
2
3/2
3/40 + 1
Ramond
sector
2
Neveu-Schwarz
sector
4
6
8
10
12
momentum K [2π/L]
14
16
1
7/10
0
18
FIG. 3 (color online). Energy spectra for periodic chains of
size L. Energies are rescaled and shifted such that the two lowest
eigenvalues match the CFT assignments. Open boxes indicate
7
CFT. Open circles give
positions of primary fields of the c # 10
positions of descendant fields as indicated. As a guide to the eye
the solid line is a cosine-fit of the dispersion.
A. Feiguin et al., Phys. Rev. Lett. 98, 160409 (2007).
00
factor exp!2"i$hL)" & hL & h"L '" matches the momenta
00
We are done!
So what did we learn?
© Simon Trebst
classification
of quantum spin liquids
© Simon Trebst
Quantum spin liquids
Quantum spin liquids are exotic ground states of frustrated quantum magnets,
in which local moments are highly correlated but still fluctuate strongly
down to zero temperature.
Classification a la Xiao-Gang Wen
three spatial dimensions
spatial dimensions
energy spectrum
two spatial dimensions
gapped
topological
spin liquids
gapless
gapless
spin liquids
S = aL
entanglement
© Simon Trebst
singular points spinon surface,
e.g. Dirac cones
Bose surface
S = aL + c (Lx , Ly ) S = cL ln(L)
Summary
•
The formation of quantum spin liquids in an interacting
quantum many-body system is one of the most fascinating
phenomena in condensed matter physics
•
The identification of topological order or gapless spin liquids
builds on concepts from
•
•
statistical physics
van Neumann & Renyi entropies
•
quantum information theory
entanglement
•
mathematical physics
boundary laws, anyon theories
The exploration of topological order is a rich and quickly
evolving research field – just at its beginning.
All slides of this presentation will become available on our group webpage at www.thp.uni-koeln.de/trebst
© Simon Trebst