PHYSICAL REVIEW B 98, 054204 (2018)
Tensor networks demonstrate the robustness of localization
and symmetry-protected topological phases
Thorsten B. Wahl
Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
(Received 8 January 2018; revised manuscript received 20 July 2018; published 14 August 2018)
We prove that all eigenstatesofmany-body localizedsymmetry-protectedtopologicalsystems with time reversal
symmetry have fourfold degenerate entanglement spectra in the thermodynamic limit. To that end, we employ
unitary quantum circuits where the number of sites the gates act on grows linearly with the system size. We find
that the corresponding matrix product operator representation has similar local symmetries as matrix product
ground states of symmetry-protected topological phases. Those local symmetries give rise to a Z
2
topological
index, which is robust against arbitrary perturbations so long as they do not break time reversal symmetry or drive
the system out of the fully many-body localized phase.
DOI: 10.1103/PhysRevB.98.054204
I. INTRODUCTION
The idea that systems out of equilibrium act as their own
heat bath was first challenged by Anderson in 1958 [1]. Later
works confirmed rigorously that in noninteracting one- and
two-dimensional systems (without broken time reversal sym-
metry or spin-orbit coupling) arbitrarily weak disorder leads to
localization of all single particle eigenstates [2]. Strikingly, in
one dimension, the resulting lack of transport survives for suffi-
ciently strong disorder if interactions are included [3–6]. Such
many-body localized (MBL) systems [7–12] retain a memory
of their initial state for arbitrarily long times, thus violating the
eigenstate thermalization hypothesis (ETH) [13–19]. Many-
body localization was observed in recent optical lattice ex-
periments on one- [20] and two-dimensional systems [21,22].
Numerical studies predict other exotic phenomena in MBL
systems, such as the logarithmic growth of entanglement
following a quantum quench [23–28] and an unconventional
transition to the thermal phase [8,29–47]. From a conceptual
point of view, MBL systems are characterized by an extensive
number of local integrals of motion (LIOM) [10,48–58] and
area-law entangled eigenstates [59–61]. Excited eigenstates
thus have similar features as the ground states of local gapped
Hamiltonians [62], which is why those eigenstates can be effi-
ciently approximated by matrix product states (MPS) [59,63–
69]. Moreover, unitary quantum circuits (a special type of ten-
sor networks [63,70–73]) encode the entire set of eigenstates
efficiently [74–77].
The absence of thermal fluctuations in MBL systems facili-
tates symmetry-breaking orders and symmetry-protected topo-
logical (SPT) orders at all energyscales, which in clean systems
can only exist at zero temperature [59,78–83]. Hence, in the
localized case all eigenstates can be SPT, which makes MBL
systems viable candidates for topological quantum memories
at arbitrary energy density [59]. Symmetry and localization
protected systems thus interface with quantum information
theory both at their theoretical description by tensor networks
and their practical potential for quantum information storage
and processing tasks [59,79].
One of the greatest accomplishments in tensor network
research so far was the classification of all gapped topological
phases in one dimension [84–86
]. This was made possible
by the insight that ground states of one-dimensional gapped
systems can be efficiently approximated by MPS [87,88]. This
suggests that tensor networks might also be used to classify
SPT MBL phases as proposed in Ref. [79].
In this article we establish tensor networks as a tool for
such a classification. Specifically, we use quantum circuits to
prove that MBL phases in one dimension protected by time
reversal symmetry fall in two different classes given by a Z
2
topological index. The only assumption we make in our proof
is that a two-layer unitary quantum circuit [77] diagonalizes the
MBL Hamiltonian exactly in the thermodynamic limit if the
length of the gates increases linearly with the system size. As
we argue, this applies to MBL systems as defined above, also
known as fully many-body localized (FMBL) systems [89],
which do not possess a mobility edge [36,90]. We find that
the global time reversal symmetry of the system gives rise
to local symmetries of the tensors—similarly to MPS with a
global symmetry [84,91]. We also prove that the topological
index determined by those local symmetries is robust against
arbitrary symmetry respecting perturbations as long as they
do not drive the system out of the FMBL phase. Finally, we
show that all eigenstates in t he SPT MBL phase have fourfold
degenerate entanglement spectra.
In the following section we give a very brief introduc-
tion into symmetry-protected topological many-body localized
phases. Section III provides a summary of the main results and
an intuitive (nontechnical) outline of the stability proof, which
follows in Sec. IV. Section V concludes the paper and gives
an outlook for future work.
Those readers interested only in the general MBL classifica-
tion idea using tensor networks and the physical implications
may skip Sec. IV.
2469-9950/2018/98(5)/054204(22) 054204-1 ©2018 American Physical Society
THORSTEN B. WAHL PHYSICAL REVIEW B 98, 054204 (2018)
II. SYMMETRY AND LOCALIZATION
PROTECTED PHASES
A. Local integrals of motion
Throughout this article we consider a disordered spins
chain in one dimension with periodic boundary conditions. For
sufficiently strong disorder, where the system is in the FMBL
phase, the Hamiltonian commutes with an extensive number of
LIOMs τ
i
z
,[H,τ
i
z
] = [τ
i
z
,τ
j
z
] = 0. τ
i
z
is related to σ
i
z
(Pauli-z
operator at site i) by a quasilocal unitary transformation U ,
i.e., τ
i
z
= Uσ
i
z
U
†
is an effective spin exponentially localized
around site i [49]. Note that U also diagonalizes the Hamilto-
nian. The eigenstates can be labeled by the eigenvaluesof the τ
i
z
operators, known as l-bits. The decay length ξ
i
of τ
i
z
depends
on the specific disorder realization. In the FMBL phase, the
likelihood of finding a decay length of order O (N )iszeroin
the limit N →∞ [49,56].
B. Symmetry and localization protected phases
In FMBL systems, all eigenstates fulfill the area law of
entanglement. This allows, in principle, for the topologi-
cal symmetry protection of the full set of eigenstates. In
one-dimensional systems, time reversal symmetry or on-site
symmetries given by an Abelian symmetry group [92]are
candidates. (Note that as opposed to ground states of clean
systems, for random disordered systems, inversion symmetry
is not an option.) In this article we will show the robustness of
time reversal symmetry-protected MBL systems and point out
what currently prevents the generalization to on-site symmetry
groups (see Sec. III).
As a paradigmatic example, consider the disordered cluster
model with random couplings [81],
H =
N
i=1
λ
i
σ
i−1
x
σ
i
z
σ
i+1
x
+ h
i
σ
i
z
+ V
i
σ
i
z
σ
i+1
z
(1)
on a chain with N sites and periodic boundary conditions.
(We define position indices modulo N.) λ
i
, h
i
, and V
i
are real
and chosen independently from a Gaussian distribution with
standard deviation σ
λ
, σ
h
, and σ
V
, respectively. In Ref. [81]
it was observed numerically that the entanglement spectra
of all eigenstates are approximately fourfold degenerate for
σ
h
,σ
V
σ
λ
and finite N. We prove the exact degeneracy in
the limit N →∞using the fact that the corresponding SPT
MBL phase is protected by time reversal symmetry, which
in this case is a combination of complex conjugation (
∗
) and
rotation by σ
z
,
H = σ
⊗N
z
H
∗
σ
⊗N
z
. (2)
In general, time reversal acts as T = Kv
⊗N
, where K denotes
complex conjugation and v an on-site unitary operation with
vv
∗
=±1. Note thatthe sign will not affect the topological clas-
sification [85], as the overall unitary v
⊗N
fulfills v
⊗N
v
∗⊗N
=
(±1)
N
1, which is 1 for even N .(IfN is odd, one can always
add a completely decoupled auxiliary spin to the chain, which
would not change the fact that the system is MBL.)
III. NONTECHNICAL SUMMARY OF RESULTS
AND INTUITIVE OUTLINE OF THE PROOF
Numerical evidence indicates that two-layer quantum
circuits with long gates approximate FMBL systems effi-
ciently [77]. For disordered systems, they are thus the full-
spectrum analogs of matrix product states (MPS) for clean
systems. This suggests that they might also be used for the
classification of symmetry-protected MBL systems—as MPS
were for clean systems. In this article we provide evidence for
this conclusion by using one-dimensional quantum circuits to
show that MBL systems protected by time reversal symmetry
fall in two different classes, where one of them is topologically
nontrivial as exemplified by a fourfold degeneracy of the
entanglement spectrum of all of its eigenstates.
The only assumption (other than being in a time reversal
symmetric MBL phase) that goes into the proof is that the
local integrals of motion can be represented efficiently by a
quantum circuit with long gates. This is basically equivalent
to not having any LIOM with a decay length of the order
of the system size, i.e., to be in the FMBL phase. We
show that, as a result, the Hamiltonian belongs to one of
two topologically inequivalent phases. We show that it is
impossible to connect the two phases adiabatically without
violating either the time reversal symmetry or the FMBL
condition. This is very reminiscent of SPT ground states of
clean Hamiltonians: As long as the symmetry is preserved,
they cannot be adiabatically connected to the t rivial phase
unless they become delocalized (having algebraically decaying
correlations), i.e., the gap of the Hamiltonian closes. This is
why MPS can be used for their classification: MPS always
have exponentially decaying correlations and represent ground
states of local gapped Hamiltonians. If t he tensors of two
(symmetric) MPS cannot be continuously connected, it is
impossible to connect the ground states they approximate
continuously without encountering a quantum phase transition,
at which correlations decay algebraically (which cannot be
captured exactly by an MPS).
In the same way, the transition between the two topolog-
ically inequivalent MBL phases must lie outside the realm
of systems that can be approximated efficiently by quantum
circuits. Hence, at the transition, at least one LIOM must
become delocalized (which does not imply the transition
resembles an MBL-to-thermal transition [80]). This corre-
spondence between MPS classifications of ground states and
quantum circuit classifications of MBL phases is summarized
in Table I.
The quantum circuits used for the proof are of the form [77]
(3)
where u
k
and v
k
are unitaries (indicated by boxes) acting on
sites. Each leg corresponds to a tensor index of dimension
2, i.e., in the above case = 4. The lower dangling legs
correspond to the approximate l-bit basis l
1
,l
2
,...,l
N
and
the upper open legs to the local physical basis. Connected
legs indicate summation over the corresponding indices.
˜
U
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TABLE I. Table showing the correspondence between MPS descriptions of ground states and quantum circuit descriptions of the entire set
of eigenstates of MBL sytems.
Property MPS Quantum circuit
Description of Ground states All eigenstates
System Translationally invariant,
a
gapped Disordered, fully many-body localized
Ansatz
Range Bond dimension D Length of unitary gates (D = 2
/2
)
Time reversal symmetry v
⊗N
|
˜
ψ
∗
=e
iθ
|
˜
ψ v
⊗N
˜
U
∗
=
˜
U
Local symmetry
Topological index ww
∗
=±1 w
j
w
∗
j
=±1 (same for all j )
Consequence fourfold degeneracy of the ground state fourfold degeneracy of the entanglement spectra
entanglement s pectrum for ww
∗
=−1 of all eigenstates for w
j
w
∗
j
=−1
a
MPS can be straightforwardly defined for nontranslationally invariant systems, but using them for a classification of phases in such a case
requires additional tools, such as the renormalization group procedure [85].
approximately diagonalizes the Hamiltonian. The error of the
optimized approximation decreases exponentially with .
For nondegenerate ground states of clean systems, time
reversal symmetry implies
T |ψ=v
⊗N
|ψ
∗
=e
iθ
|ψ. (4)
This generalizes to
v
⊗N
ψ
∗
l
1
···l
N
= e
iθ
l
1
···l
N
ψ
l
1
···l
N
(5)
for MBL systems with eigenstates |ψ
l
1
···l
N
and nondegenerate
energies (possible energy degeneracies can be removed by
adding infinitesimally small perturbations). Since quantum
circuits with long gates form an efficient approximation, the
same must be true for the approximate eigenstates |
˜
ψ
l
1
···l
N
contained in the unitary
˜
U,
v
⊗N
˜
ψ
∗
l
1
···l
N
= e
iθ
l
1
···l
N
˜
ψ
l
1
···l
N
. (6)
For
˜
U this implies
˜
U = v
⊗N
˜
U
∗
, (7)
where is the diagonal matrix with elements e
iθ
l
1
···l
N
.
1/2
can
be absorbed into the two-layer quantum circuit (see Sec. IV
for the precise reason for this), i.e.,
˜
U →
˜
U
1/2
, such that
˜
U = v
⊗N
˜
U
∗
. (8)
The absorption of such phase factors only works for time
reversal symmetry, which is what currently precludes a gener-
alization t o on-site symmetries characterized by a symmetry
group G. In graphical notation, Eq. (8) reads (we combine
groups of /2 lines into single lines with dimension 2
/2
)
(9)
with V = v
⊗/2
. Note that multiplication from left to right in
algebraic notation corresponds to top to bottom in graphical
notation. If we define u
k
= u
∗
k
and v
k
= (V ⊗ V )v
∗
k
, we discern
that Eq. (9) equates two two-layer quantum circuits,
(10)
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THORSTEN B. WAHL PHYSICAL REVIEW B 98, 054204 (2018)
If we multiply both sides from the bottom by u
†
k
for k =
1,...,nand from the top by v
†
k
,wearriveat
(11)
The left-hand side is a tensor product of u
k
u
†
k
and the right-
hand side of v
†
k
v
k
but shifted by one site with respect to each
other. Hence, it has to hold that
(12)
and
(13)
where the w
j
are unitaries. The phase factor e
iφ
k
arises because
the decomposition of Eq. (11) into a product of tensors acting
on blocks of
2
sites is unique up to overall factors (which have
to be of magnitude 1 due to unitarity). We call Eqs. (12) and(13)
a gauge transformation, as it leaves the overall quantum circuit
invariant. If we insert back the specific case of u
k
= u
∗
k
and
v
k
= (V ⊗ V )v
∗
k
, we obtain
(14)
If one takes the complex conjugate of Eqs. (14) and inserts that
back into the original Eqs. (14), one obtains
(15)
using VV
∗
=±1. The left equation implies w
2k−1
w
∗
2k−1
=
1e
iβ
k
, w
2k
w
∗
2k
= 1e
−iβ
k
and the right one w
2k
w
∗
2k
= 1e
iβ
k
,
w
2k+1
w
∗
2k+1
= 1e
−iβ
k
. We thus have a single phase β,
w
2k−1
w
∗
2k−1
= 1e
iβ
, w
2k
w
∗
2k
= 1e
−iβ
, for all k = 1,...,n.
Inserting the resulting w
2k−1
= e
iβ
w
2k−1
into itself [84] yields
e
2iβ
= 1, i.e., w
j
w
∗
j
=±1 with the same sign for all j =
1, 2,...,2n. This is the topological sign of the SPT MBL
phase: It does not depend on the site index k, i.e., it is the
same for the entire chain. One cannot adiabatically change
a unitary quantum circuit from a topological index −1toa
+1 index, as continuous variation of the unitaries {u
k
,v
k
}
corresponds according to Eqs. (14) to continuous variation of
{w
j
}, which leaves the sign of w
j
w
∗
j
=±1 invariant. This
indicates that under adiabatic perturbations of the Hamil-
tonian, it is impossible t o connect the two phases unless
the description in terms of local integrals of motion and
thus in terms of quantum circuits breaks down. At such a
transition point, at least one integral of motion must become
delocalized.
Finally, to gain an intuition as to why one of the SPT phases
has fourfold degeneracy of all eigenstates, it is illustrative to
write
˜
U as a matrix product operator (MPO),
(16)
(17)
where we use thick lines to denote the combination of two
vertical legs to one with dimension 2
. Equation (14)gives
(18)
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using w
∗
2k
w
2k
=±1, and therefore
(19)
This relation is almost i dentical to the one obtained for MPS
representing time reversal symmetric ground states [84]. The
only differences are the lower leg corresponding to the local
l-bit configuration (making it an MPO r ather than an MPS) and
the breaking of translational invariance reflected by the site-
dependent tensors A
k
and virtual symmetries w
2k−1
, w
2k+1
.
However, since w
j
w
∗
j
=±1 for all j, the same conclusions
can be drawn as in Ref. [84]: Consider the case of w
j
w
∗
j
=−1
and a specific eigenstate by fixing the l-bit configuration, i.e.,
the indices of the lower legs. The entanglement spectrum
of that eigenstate is encoded in a reduced density matrix
defined on the virtual space (horizontal legs). Due to Eq. (19),
it has to commute with w
2k−1
and w
2k+1
.Forw
j
w
∗
j
=−1
this implies that the spectrum of the reduced density matrix
has to be fourfold degenerate. Since this conclusion can
be drawn independently of the chosen l-bit configuration,
all eigenstates must have fourfold degenerate entanglement
spectra.
We thus showed that in the presence of time reversal
symmetry, MBL systems fall into one of two topologically dis-
tinct phases, which can be distinguished by the entanglement
spectra of the individual eigenstates. This is in analogy to the
classification of matrix product states with time reversal sym-
metry [84–86]. Along these lines, we expect a classification
by the second cohomology group if the system is invariant
under an on-site symmetry given by a certain symmetry
group [79]. The technical problems with this extension can
be gathered from the following section. Finally, note that the
derivations here only apply to bosonic systems; for fermionic
systems another symmetry constraint (parity) would have to be
imposed [ 93].
The rigorous demonstration of the results above is the
subject of the f ollowing section.
IV. THEOREM AND PROOF
A. Theorem
If for all sufficiently large N the following conditions are
fulfilled
(1) there exists a unitary U diagonalizing the Hamiltonian
H defining τ
z
i
= Uσ
z
i
U
†
and a two-layer quantum circuit
˜
U
with ˜τ
z
i
=
˜
Uσ
z
i
˜
U
†
such that ˜τ
z
i
− τ
z
i
op
<ce
−
ξ
i
with ξ
max
:=
max
i
ξ
i
<c
N
1−μ
for some fixed c, c
> 0 and 0 <μ<1
(efficient approximability)
(2) the Hamiltonian is invariant under time reversal opera-
tion T = Kv
⊗N
, H = T H T
†
(time reversal symmetry)
(3) conditions 1 and 2 are also fulfilled for the Hamiltonian
H + V with arbitrary infinitesimally small strictly local
perturbations, → 0 (MBL stability),
then the following holds in the thermodynamic limit (N →
∞):
(1) the Hamiltonian belongs to one of two topological
classes, where one of them has a full set of eigenstates
with fourfold degenerate entanglement spectra (topological
property)
(2) under adiabatic perturbations, the Hamiltonian cannot
leave its topological class if the above conditions are fulfilled
along the path (topological stability).
We will prove each of the two statements in turn.
B. Proof of Statement 1
We first prove the following:
Lemma 1. Condition 1 of the Theorem implies for (N ) =
αN to leading order in N that there exists a unitary U
ex-
actly diagonalizing the Hamiltonians such that U
−
˜
U
op
<
2
9/4
cN
3
e
−
αN
μ
2c
.
Proof of Lemma 1. We set U
= U , where (to be speci-
fied below) is a diagonal matrix whose nonvanishing elements
have magnitude 1. U
also diagonalizes the Hamiltonian and
has the same LIOMs τ
z
i
. Condition 1 hence implies f or U
σ
z
i
−
˜
U
†
U
σ
z
i
U
†
˜
U
op
<ce
−
ξ
i
. (20)
We write
˜
U
†
U
in blocks corresponding to degenerate sub-
spaces of σ
z
1
,
˜
U
†
U
:= (
U
11
U
12
U
21
U
22
). Then, Eq. (20) results in
U
11
U
†
11
− U
12
U
†
12
− 1 U
11
U
†
21
− U
12
U
†
22
U
21
U
†
11
− U
22
U
†
12
U
21
U
†
21
− U
22
U
†
22
− 1
op
<ce
−
ξ
i
. (21)
Since
˜
U
†
U
is unitary,
U
11
U
†
11
+ U
12
U
†
12
U
11
U
†
21
+ U
12
U
†
22
U
21
U
†
11
+ U
22
U
†
12
U
21
U
†
21
+ U
22
U
†
22
=
1 0
0 1
, (22)
we get
−2U
12
U
†
12
2U
11
U
†
21
−2U
22
U
†
12
2U
21
U
†
21
op
= 2
U
12
U
11
U
22
U
21
−U
†
12
0
0 U
†
21
op
= 2max(U
12
op
, U
21
op
) <ce
−
ξ
i
. (23)
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