Papers in Physics, vol. 8, art. 080005 (2016)

Received: 1 July 2016, Accepted: 30 August 2016
Edited by: K. Hallberg
Reviewed by: D. C. Cabra, Instituto de F́ısica La Plata, La Plata, Argentina
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.080005

www.papersinphysics.org

ISSN 1852-4249

Topological quantum phase transition in strongly correlated Kondo
insulators in 1D

Franco T. Lisandrini,1, 2 Alejandro M. Lobos,1 Ariel O. Dobry,1, 2 Claudio J. Gazza1, 2∗

We investigate, by means of a field-theory analysis combined with the density-matrix renor-
malization group (DMRG) method, a theoretical model for a strongly correlated quantum
system in one dimension realizing a topologically-ordered Haldane phase ground state.
The model consists of a spin-1/2 Heisenberg chain coupled to a tight-binding chain via
two competing Kondo exchange couplings of different type: a “s-wave” Kondo coupling
(JsK ), and a less common “p-wave” (J

p
K ) Kondo coupling. While the first coupling is the

standard Kondo interaction studied in many condensed-matter systems, the latter has
been recently introduced by Alexandrov and Coleman [Phys. Rev. B 90, 115147 (2014)]
as a possible mechanism leading to a topological Kondo-insulating ground state in one di-
mension. As a result of this competition, a topological quantum phase transition (TQPT)
occurs in the system for a critical value of the ratio JsK/J

p
K , separating a (Haldane-type)

topological phase from a topologically trivial ground state where the system can be es-
sentially described as a product of local singlets. We study and characterize the TQPT
by means of the magnetization profile, the entanglement entropy and the full entangle-
ment spectrum of the ground state. Our results might be relevant to understand how
topologically-ordered phases of fermions emerge in strongly interacting quantum systems.

I. Introduction

The study of topological quantum phases of matter
has become an area of great interest in present-day
condensed matter physics. A topological phase is
a quantum phase of matter which cannot be char-
acterized by a local order parameter, and thus falls
beyond the Landau paradigm. In particular, topo-
logical insulators (i.e., materials which are insulat-
ing in the bulk but support topologically protected
gapless states at the edges) were first proposed the-

∗E-mail: gazza@ifir-conicet.gov.ar

1 Instituto de F́ısica Rosario (CONICET), Bv 27 de
Febrero 210 bis, S2000EZP Rosario, Santa Fe, Argentina.

2 Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura,
Universidad Nacional de Rosario, Argentina.

oretically for two- and three-dimensional systems
with time-reversal symmetry [1–3], and soon after
found in experiments on HgTe quantum wells [4]
and in Bi1−xSbx [5], and Bi2Se3 [6], generating a lot
of excitement and subsequent research. The elec-
tronic structure of a topological insulator cannot be
smoothly connected to that of a trivial insulator, a
fact that is mathematically expressed in the exis-
tence of a nonzero topological invariant, an integer
number quantifying the non-local topological order
in the ground state. A complete classification based
on the dimensionality and underlying symmetries
has been achieved in the form of a “periodic table”
of topological insulators [7–9]. Nevertheless, this
classification refers only to the gapped phases of
noninteracting fermions, and leaves open the prob-
lem of characterizing and classifying strongly in-

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Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

teracting topological insulators. This is a very im-
portant open question in modern condensed-matter
physics.

Recently, Dzero et al. [10–12] proposed a
new kind of topological insulator: the topologi-
cal Kondo insulator (TKI), which combines fea-
tures of both non-interacting topological insula-
tors and the well-known Kondo insulators, a spe-
cial class of heavy-fermion system with an insu-
lating gap strongly renormalized by interactions.
Within a mean-field picture, TKIs can be under-
stood as a strongly renormalized f-electron band
lying close to the Fermi level, and hybridizing with
the conduction-electron d−bands [13–15]. At half-
filling, an insulating state appears due to the open-
ing of a low-temperature hybridization gap at the
Fermi energy induced by interactions. Due to the
opposite parities of the states being hybridized,
a topologically non-trivial ground state emerges,
characterized by an insulating gap in the bulk and
conducting Dirac states at the surface [10]. At
present, TKI materials, among which samarium
hexaboride (SmB6) is the best known example, are
under intense investigation both theoretically and
experimentally [16–19] .

From a theoretical point of view, TKIs are inter-
esting systems arising from the interplay between
strong interactions and topology. Although the
large-N meanfield approach was successful in de-
scribing qualitatively heavy-fermion systems and
TKIs in particular, it would be desirable to un-
derstand better how TKIs emerge. In order to
shed more light into this question, in a recent
work Alexandrov and Coleman proposed a one-
dimensional model for a topological Kondo insula-
tor [20], the “p-wave” Kondo-Heisenberg model (p-
KHM). Such a model consists of a chain of spin-1/2
magnetic impurities interacting with a half-filled
one-dimensional electron gas through a Kondo ex-
change (see Fig. 1). Using a standard mean-field
description [13–15], which expresses the original in-
teracting problem as an effectively non-interacting
one, the authors mentioned above found a topologi-
cally non-trivial insulating ground state (i.e., topo-
logical class D [7–9]) which hosts magnetic states at
the open ends of the chain. However, this system
was studied recently using the Abelian bosoniza-
tion approach combined with a perturbative renor-
malization group analysis, revealing an unexpected
connection to the Haldane phase at low tempera-

tures [21]. The Haldane phase is a paradigmatic
example of a strongly interacting topological sys-
tem [22–26]. The results in Ref. [21] indicate that
1D TKI systems might be much more complex and
richer than expected with the näıve mean-field ap-
proach, as they are uncapable of describing the
full complexity of the Haldane phase, and suggest
that they must be reconsidered from the more gen-
eral perspective of interacting symmetry-protected
topological (SPT) phases [25–28]. More recently,
two numerical studies using exact DMRG methods
have confirmed that 1D TKIs belong to the uni-
versality class of Haldane insulators [29, 30]. These
studies have extended the regime of validity of the
results in Ref. [21].

In this work, we theoretically investigate the ro-
bustness of the Haldane phase in one-dimensional
topological Kondo insulators, and study the effect
of local interactions that destabilize the topologi-
cal phase and drive the system to a non-topolgical
phase. Our goal is to characterize the system at,
and near to, the topological quantum phase tran-
sition (TQPT) from the perspective of symmetry-
protected topological phases, using the concepts of
entanglement entropy and entanglement spectrum
to detect the topologically-ordered ground states.
This is a novel perspective in the context of TKIs,
which might shed new light on the emergence of
topological order on strongly correlated phases of
fermions, and makes our work interesting from the
pedagogical and conceptual points of view.

II. Theoretical model

We describe the system depicted in Fig. 1 with the

Hamiltonian H = H1 +H2 +H
(s)
K +H

(p)
K , where the

conduction band is represented by a L-site tight-
binding chain

H1 = −t
L−1∑
j=1,σ

(
c
†
j,σcj+1,σ + H.c.

)
, (1)

with c
†
j,σ, the creation operator of an electron with

spin σ at site j. The Hamiltonian

H2 = JH

L−1∑
j=1

Sj ·Sj+1 (JH > 0), (2)

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Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

Figure 1: Sketch of the Kondo-Heisenberg model
under consideration. The lower leg represents a
spin-1/2 Heisenberg chain with JH > 0. The
upper leg represents a half-filled one-dimensional
tight binding chain interacting with the lower leg
through two different Kondo exchange couplings, a
“s-wave” JsK and a “p-wave” J

p
K. We also show

the fermionic and spin operators defined on each
eight-dimensional “supersite” j (see text).

corresponds to a spin-1/2 antiferromagnetic

Heisenberg chain. The terms H
(s)
K and H

(p)
K

describe two different types of Kondo exchange
couplings between H1 and H2, namely

H
(s)
K = J

s
K

L∑
j=1

Sj . sj, (3)

H
(p)
K = J

p
K

L∑
j=1

Sj . πj, (4)

with JaK > 0 (a = s,p). Eq. (3) describes the
usual antiferromagnetic Kondo exchange coupling
of a spin Sj in the Heisenberg chain to the local spin
density in the fermionic chain at site j, defined as:

sj ≡
∑
α,β

c
†
j,α

(σαβ
2

)
cj,β, (5)

where σαβ is the vector of Pauli matrices. On the
other hand, Eq. (4) describes a “p-wave” Kondo
interaction, which is unusual in that it couples the
spin Sj to the p-wave spin density in the fermionic
chain at site j, defined as: [20]

πj ≡ (6)∑
α,β

(c†j+1,α − c†j−1,α√
2

)(σαβ
2

)(cj+1,β − cj−1,β√
2

)
,

where the notation c0,σ = cL+1,σ = 0 is implied.

The case JsK > 0 and J
p
K = 0 corresponds to the

so-called “Kondo-Heisenberg model” in 1D, which
has been extensively studied in the past in con-
nection to the stripe phase of high-Tc superconduc-
tors [31–37]. These previous works indicate that, at
half-filling, that model does not support any topo-
logical phases. On the other hand, the case JsK = 0
and J

p
K > 0 is the “p-wave” Kondo-Heisenberg

model proposed recently in Ref. [20], and sub-
sequently studied in Refs. [21, 29, 30], where it
was stablished that the ground state corresponds
to the Haldane phase. In both cases, at half-
filling the system develops a Mott-insulating gap
in the fermionic chain due to “umklapp” processes
(i.e., backscattering processes with 2kF momen-
tum transfer) originated in the Kondo interactions
JsK or J

p
K, and at low temperatures (lower than

the Mott gap), the system effectively maps onto
a spin-1/2 ladder. However, the ground states of
the model in both cases cannot be smoothly con-
nected (i.e., one is topologically trivial while the
other is not), and therefore we anticipate that a
gap-closing topological quantum phase transition
(TQPT) must occur as a function of the ratio
JsK/J

p
K. Indeed, in Ref. [21], it was proposed that

while in the first case the (effective) spin-1/2 lad-
der forms singlets along the rungs, in the second
case the Kondo coupling J

p
K favors the formation

of triplets on the rungs, and therefore the system
maps onto the Haldane spin-1 chain [38–41]. From
this perspective, our toy-model Hamiltonian H al-
lows to explore a transtition from topological to
non-topological Kondo insulators, and to gain a
valuable insight of the TQPT.

III. Field-theory analysis

The purpose of this section is to provide a simple
and phenomenological understanding of the compe-
tition between the two Kondo interactions JsK and
J
p
K, which is not evident in Eqs. (3) and (4). To

that end, we introduce a field-theoretical represen-
tation of the model, valid at sufficiently low tem-
peratures. We linearize the non-interacting spec-
trum �k = −2t cos (ka) in the tight-binding chain
H1 around the Fermi energy µ = 0, and take the
continuum limit a → 0, where a is the lattice con-
stant. Then, the low-energy representation of the

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Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

fermionic operators becomes [42–46]

cj,σ√
a
∼ eikF xjR1,σ (xj) + e−ikF xjL1,σ (xj) , (7)

where R1,σ (x) and L1,σ (x) are right- and left-
moving fermionic field operators, which vary slowly
on the scale of a. Using this representation, the spin
densities become [47]:

sj
a
→
[
J1R (xj) + J1L(xj) + (−1)

j
N1 (xj)

]
, (8)

πj
a
→ 2 [J1R (xj) + J1L (xj)

−(−1)jN1 (xj)
]
, (9)

where we have defined slowly-varying spin densities
for the smooth spin configurations:

J1R (xj) =
∑
α,β

R
†
1,α (xj)

(σαβ
2

)
R1,β (xj) , (10)

J1L (xj) =
∑
α,β

L
†
1,α (xj)

(σαβ
2

)
L1,β (xj) , (11)

and for the staggered spin density

N1 (xj) =
∑
α,β

R
†
1,α (xj)

(σαβ
2

)
L1,β (xj) + H.c..

(12)

Similarly, a continuum representation for the
Heisenberg chain can be achieved, e.g., by fermion-
ization of the S=1/2 spins by means of a Jordan-
Wigner transformation. At low energies, the spin
densities become [38–41, 47]

Sj
a
→ J2R (xj) + J2L (xj) + (−1)

j
N2 (xj) . (13)

We now focus on the Kondo interaction and leave
the analysis of H1 and H2 aside, as these terms are
unimportant for the qualitative understading of the
basic mechanism leading to the TQPT. Keeping the
most relevant (in the RG sense) terms, we can write
the Kondo interaction as:

H
(s)
K + H

(p)
K → (J

s
K − 2J

p
K)

∫
dx N1 (x) .N2 (x)

(+less relevant contributions), (14)

i.e., the Kondo interaction couples the staggered
magnetization components in chains 1 and 2. In the
above expression, note that while a large JsK favors
a positive value of the effective coupling (JsK−2J

p
K),

therefore promoting the formation of local singlets
along the rungs, a large p-wave Kondo coupling
J
p
K favors an effective ferromagnetic coupling which

promotes the formation of local triplets with S = 1
(hence the connection to the S = 1 Haldane chain).
The minus sign in front of J

p
K appears as a result of

the p-wave nature of the orbitals in Eq. (6). From
this qualitative analysis we can conclude that JsK
and J

p
K will be competing interactions promoting

different ground states, and since these grounstates
cannot be adiabatically connected with each other,
a TQPT must occur.

Strictly speaking, near the critical region where
the bare coupling (JsK − 2J

p
K) vanishes, the less

relevant terms neglected in Eq. (14) should be
taken into account. However, note that opera-
tors with conformal spin 1 (i.e., operators of the
form (∂xN1) .N2 or N1. (∂xN2), see for instance
Ref. [48]) are not allowed by the inversion sym-
metry of the Hamiltonian and the p−wave symme-
try of the orbitals in Eq. (6), which demands that
cj+1,α − cj−1,α → −

(
c−(j+1),α − c−(j−1),α

)
under

the change j → −j, and therefore forbids the oc-
currence of terms proportional to ∂xN1. There-
fore, only the marginal operators J1ν.J2ν′ (with
ν = {R,L}) and terms with conformal spin big-
ger than 1 are expected in the Hamiltonian. We do
not expect these operators to change the physics
qualitatively near the critical point, and we can ig-
nore them for this simplified analysis. As we show
below, our numerical DMRG results are in accor-
dance with this qualitative picture.

IV. DMRG analysis

Before presenting the numerical results, it is worth
providing technical details on the implementation
of the DMRG method applied to the present model.
This particular Hamiltonian contains two types of
terms: (a) terms involving two local operators,
as in most condensed-matter models with nearest-
neighbor interactions, i.e., Eqs. (1)-(3), and (b)
terms involving three local operators, which result
from the expansion of Eq. (4). To make it easier
to implement, we have found useful to define first

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Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

a “supersite” representation of the system, where
each supersite combines a spin Sj and the fermionic
site along each rung (see Fig. 1), therefore span-
ning a new 8-dimensional local basis. The first
kind of terms could be easily handled with stan-
dard DMRG implementations where the system is
represented as L(j)⊗•⊗•⊗R(N−j−2), with L(j)
and R(j) the left and right blocks with j supersites,
respectively, and the two circles are the exactly-
represented middle supersites j + 1 and j + 2. This
comes at a price, however, since one is then forced
to re-express the electron creation operator c

†
j,σ,

and the spin-1/2 operator Sj in this new basis in
order to implement Eqs. (1) and (2). In this basis,

note that the Hamiltonian H
(s)
K , Eq. (3), becomes

an “on-site” term, which can be handled easily.
In the second type of contributions, the pres-

ence of three-operator terms in H
(p)
K , Eq. (4),

must be properly treated in order to avoid extra
truncation errors due to the tensor product of two
(already truncated) operators inside each left and
right blocks. Then, during each left-right DMRG
sweep iteration, we save in the previous superblock
configuration L(j − 1) ⊗ • ⊗ • ⊗ R(N − j − 3)
the exact correlation matrices between spin and
fermionic operators that involve positions j−1 (i.e.,
rightmost supersite of left block) and j (first single
site), which will become the two rightmost super-
sites of the new L(j) in the next sweep iteration
step. These correlation matrices are

[
A
z†
j−1,j,σ

]
i;i′

=

ρi;i1i2
[
Szj−1

]
i1;i1′

[
c
†
j,σ

]
i2;i2′

ρ
†
i1′i2′;i′,

and

[
A

+†
j−1,j,σ

]
i;i′

=

ρi;i1i2
[
S+j−1

]
i1;i1′

[
c
†
j,σ

]
i2;i2′

ρ
†
i1′i2′;i′,

where ρi;i1i2 is the m×8m reduced density matrix,
with m the number of states kept, and where the in-
dices i and i1 run over the truncated m-dimensional
Hilbert space while i2 runs over the 8-dimensional
supersite space. In addition, we have assumed sum-
mation over repeated internal indices. Similarly,
correlations between spin and fermionic operators

placed in the two leftmost supersites j + 3 and j + 4
of R(N −j−2) should also be kept during the cor-
responding step of the rightf-left sweep. Therefore,
we now deal with a standard “two-operator” inter-

action, for example the term L(j) ⊗• of H(p)K is

−
1

4
J
p
K

∑
σ

{(
σA

z†
j−1,j,σ + A

+†
j−1,j,σ

)
clj+1,σ + h.c.

}
.

Finally, we mention that in our implementation
we have kept up to a maximum of 800 states and
we have swept 12 times, assuring truncation errors
in the density matrix of the order 10−9 at worst.

We now turn to the results. We have used the
tight-binding parameter t as our unit of energies,
and in all of our calculations we have used the val-
ues JH/t = 1 and J

p
K/t = 2. We have studied the

evolution of the ground state upon the increase of
JsK starting from the value J

s
K = 0, where the sys-

tem is in the Haldane phase. The nature of the
topologically-ordered ground state and the precise
detection of the TQPT are determined using, re-
spectively: a) the analysis of the spin profile in the
ground state, b) the value of the entanglement en-
tropy and c) the analysis of the degeneracies in the
full entanglement spectrum. As shown recently in
seminal works [25, 26], the last two properties are
useful bona fide indicators of symmetry-protected
topological orders.

(a) Spin profile. One characteristic feature of
a S = 1 Haldane chain with open boundary con-
ditions is the presence of topologically protected
fractionalized spin-1/2 end-states, a consequence
of the broken Z2 × Z2 symmetry of the ground
state [24]. These states can be represented as:
| ↑L〉⊗| ↑R〉, | ↑L〉⊗| ↓R〉, | ↓L〉⊗| ↑R〉, | ↓L〉⊗| ↓R〉,
and correspond to the fourfold-degenerate ground
state in the thermodynamic limit L → ∞. In or-
der to detect these fractionalized spin excitations
in our system, we have defined the spin profile
as
〈
Tzj
〉

=
〈
ψM

z=1
g

∣∣Tzj ∣∣ψMz=1g 〉, where Tzj is the
z−projection of the total spin in the j-th rung
Tj = Sj + sj, and

∣∣ψMz=1g 〉 is the ground state
of the system with total spin Mz = 1 (where the
visualization of the spin states at the ends is eas-
ier). In Fig. 2 we show 〈Tzj 〉 vs j for different
values of JsK/J

p
K and for L = 80. The presence

of spin states localized at the edges can be clearly
seen for JsK/J

p
K = 0 and J

s
K/J

p
K = 0.85, where

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Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0.4

 0.5

 10  20  30  40  50  60  70  80

L=80

<
T

jz
>

Site j

JsK/J
p
K=0.00

JsK/J
p
K=0.85

JsK/J
p
K=1.60

Figure 2: Spatial profile of 〈Tzj 〉 =
〈ψM

z=1
g |Tzj |ψ

Mz=1
g 〉, i.e., the z component of

the total spin in the supersite j, computed with
the ground state of the subspace with total Mz = 1
for L = 80, and for parameters JH/t = 1 and
J
p
K/t = 2. The destruction of the topologically

protected spin-1/2 states at the ends of the chain
can be clearly seen as JsK/J

p
K is increased from

JsK/J
p
K = 0 to J

s
K/J

p
K = 1.6.

the spin density is accumulated at the ends of the
system. Note that since we are working in the sub-
space Mz = 1, this ground state corresponds to
the state | ↑L〉 ⊗ | ↑R〉. For JsK/J

p
K = 1.6, the

magnetic edge-states have already disappeared, in-
dicating that the onset of the topologically trivial
phase must occur at lower values of JsK/J

p
K. How-

ever, while this analysis is useful to understand the
nature of the topologically-ordered ground state,
it does not allow a precise determination of the
TQPT. To that end, we have studied the entan-
glement entropy and entanglement spectrum (see
below).

(b) Entanglement entropy. We have also calcu-
lated the entanglement entropy (i.e., the von Neu-
mann entropy of the reduced density matrix), de-
fined as [49]

S(L/2) = −Trρ̂L/2 ln ρ̂L/2 = −
∑
j

Λj ln Λj, (15)

where ρ̂L/2 is the reduced density matrix obtained
after tracing out half of the chain, and Λj the corre-
sponding eigenvalues of ρ̂L/2, which are the squares

of the Schmidt values. Recently, it has been clar-
ified that the entanglement of a single quantum
state is a crucial property not only from the per-
spective of quantum information, but also for con-
densed matter physics. In particular, the entan-
glement entropy has been shown to contain the
quantum dimension, a property of topologically-
ordered phases [50, 51]. Hirano and Hatsugai [52]
have computed the entanglement entropy of the
open-boundary spin-1 Haldane chain and obtained
the lower-bound value S(L/2) = ln (4) = 2 ln (2)
which, according to the edge-state picture in the
thermodynamical limit L →∞, corresponds to the
aforementioned 4 spin-1/2 edge states. In Fig. 3
we show the entanglement entropy of the system
as a function of JsK/J

p
K, for different system sizes

and in the subspace Mz = 0, where we expect to
find the ground state (i.e., the ground state of an
even-numbered antiferromagnetic chain is a global
singlet [53]). Near the critical region, the entangle-
ment entropy grows due to the contribution of the
bulk, and exactly at the TQPT the entanglement
entropy is predicted to show a logarithmic diver-
gence S(L/2) ∼ ln(L), characteristic of critical one-
dimensional systems [49]. As the size of the system
is increased, the logarithmic divergence becomes
narrower and its position shifts to larger values. Us-
ing the fitting function JsK,c (L) = J

s
K,c (∞) +a/L

2,
we have obtained the extrapolated critical point
JsK,c (∞) ≈ 1.11 J

p
K in the thermodynamic limit,

see inset (a) in Fig. 3. Note that this value is
smaller than the predicted value JsK,c = 2 J

p
K us-

ing the field-theory analysis of the previous section.
We believe this to be the effect of the neglected
marginal or irrelevant operators, which renormalize
non-universal quantities such as the critical point.

We have also confirmed the logarithmic scaling
of the entanglement entropy at the critical point,
i.e., Smax(L/2) = α ln(L) + constant, and we have
obtained a prefactor α = 0.052, see inset (b) in Fig.
3. A detailed analysis of this value and its connec-
tion to the corresponding central charge value of
the conformal field theory is beyond the scope of
the present work and is left to a subsequent publi-
cation.

In Fig. 3, note that for JsK/J
p
K < J

s
K,c/J

p
K, the

value of the entanglement entropy roughly corre-
sponds to S(L/2) ∼ 2 ln (2), consistent with the
theoretical predictions in the Haldane phase. Val-
ues of S(L/2) which are below the predicted lower-

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 0.5

 1

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 2

 2.5

 0  0.2  0.4  0.6  0.8  1  1.2  1.4

S
(L

/
2
)

JsK/J
p
K

2ln(2)

L=40

L=60

L=80

L=100

 1

 1.05

 1.1

 0  0.001

(a)

Js
K,c

(∞)=1.11Jp
K

J
s K

,c
/
J
p K

1/L2
 2.15

 2.2

 2.25

 100

(b)

α=0.052

S
m

a
x

L

Figure 3: Entanglement entropy of the reduced
density matrix, S (L/2), as function of JsK/J

p
K for

different lattice sizes. The entropy is computed
within the subspace Mz = 0 and for parameters
JH/t = 1 and J

p
K/t = 2. The maximum of S (L/2)

indicates the position of the critical point. Inset
(a): Finite-size scaling of the critical point, i.e.,
JsK,c (L) = J

s
K,c (∞) + a/L

2, from where the value
in the thermodynamic limit JsK,c (∞) ≈ 1.11 J

p
K is

obtained. Inset (b): Maximum entropy Smax(L/2)
vs L. The results reproduce the predicted logarith-
mic divergence Smax(L/2) = α ln(L) + constant,
with the fitting constant α = 0.052.

bound are presumably due to finite-size effects,
which result in a decrease of the effective quantum
dimension in small systems [52]. For JsK/J

p
K >

JsK,c/J
p
K, S(L/2) tends to zero as expected for a

topologically trivial ground state. This result can
be easily understood in the limit JsK/J

p
K → ∞,

where we expect the ground state to factorize as
a product of local singlets (we recall that we are
working in the subspace Mz = 0), for which
S(L/2) = 0.

(c) Degeneracy of the entanglement spectrum.
Finally, we focus on the full entanglement spec-
trum of the reduced density-matrix. Degeneracies
in the entanglement spectrum are intimately re-
lated to the existence of discrete symmetries which
protect the topological order. In particular, as
shown in Ref. [25], an even degeneracy constitutes
the most distinctive feature of the Haldane phase.
This fact allows to make an interesting connection

-l
n
(Λ

i)

Jsk/J
p
k

 0

 1

 2

 3

 4

 5

 6

 7

 8

 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8

Figure 4: The largest eigenvalues of the entan-
glement spectrum (extrapolated to the thermody-
namic limit L →∞), as a function of JsK. The re-
sults were obtained in the subspace Mz = 0. The
gray zone corresponds to the Haldane phase, where
the degeneracy in the spectrum of eigenvalues is
even.

between the theory of symmetry-protected topo-
logical phases and the theory of topological Kondo
insulators in one dimension, and might have impor-
tant implications in the understanding of strongly
interacting topological phases of fermions. In Fig.
4 we show the evolution of the largest eigenval-
ues extrapolated to the thermodynamic limit of
the density matrix as a function of the parame-
ter JsK/J

p
K, obtained in the subspace M

z = 0.
Note that for JsK/J

p
K < J

s
K,c/J

p
K ∼ 1.11 (i.e.,

gray-shaded area), the degeneracy of the eigenval-
ues is even, as expected for the Haldane phase. In
contrast, for values JsK/J

p
K > J

s
K,c/J

p
K, the even-

degeneracy breaks down, indicating the onset of
the trivial phase. In particular, note the evolu-
tion of the largest eigenvalue of the density matrix
Λ0 ≈ 1 (i.e., lowest “+” symbol) which becomes
non-degenerate.

V. Conclusions

Using a combination of techniques, i.e., a field-
theoretical analysis and the density-matrix renor-
malization group (DMRG), an essentially exact
method in one dimension, we have studied the tran-

080005-7



Papers in Physics, vol. 8, art. 080005 (2016) / F. T. Lisandrini et al.

sition from a topological to a non-topological phase
in a model for a one-dimensional strongly interact-
ing topological Kondo insulator. As a prototypi-
cal topological quantum phase transition with no
Landau-type local order-parameter, one must re-
sort to global quantities characterizing the ground
state. In this work, we have shown that the entan-
glement entropy and the entanglement spectrum
can be used to characterize a topological Kondo
insulator in one dimension. This system was orig-
inally understood and classified according to non-
interacting topological invariants (i.e., Chern num-
bers), employing approximate large-N mean-field
methods [20]. Here, by the means of the DMRG,
we have shown that a more appropriate way to un-
derstand this system is by using the concepts de-
veloped for symmetry-protected topological phases
[25–28]. In particular, for parameters JH/t = 1 and
J
p
K/t = 2, we have obtained the value of the crit-

ical point JsK,c/J
p
K ' 1.11 in the thermodynamic

limit L → ∞. This value is smaller than the ex-
pected from the qualitative field-theoretical estima-
tion JsK,c/J

p
K = 2, a fact that is presumably origi-

nated in the effect of marginal or irrelevant opera-
tors, which were neglected in the qualitative analy-
sis and which renormalize a non-universal quantity
such as the critical point.

Acknowledgements - F.T.L, A.O.D. and
C.J.G. acknowledge support from CONICET-PIP
11220120100389CO. A.M.L acknowledges support
from PICT-2015-0217 of ANPCyT.

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