A Matryoshka approach to Sine-Cosine topological models
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A Matryoshka approach to Sine-Cosine topological models
R. G. Dias1 and A. M. Marques1
1
Department of Physics & i3N, University of Aveiro, 3810-193 Aveiro, Portugal
(Dated: February 2, 2021)
We address a particular set of SSH(2n) models (2n being the number of sites in the unit cell)
that we designate by Sine-Cosine models [SC(n)], with hopping terms defined as a sequence of n
sine-cosine pairs of the form {sin(θj ), cos(θj )}, j = 1, · · · , n. These models, when squared, generate
a block-diagonal matrix representation with one of the blocks corresponding to a chain with uniform
local potentials. We further focus our study on the subset of SC(2n−1 ) chains that, when squared an
arbitrary number of times (up to n), always generate a block which is again a Sine-Cosine model, if an
arXiv:2102.00887v1 [cond-mat.mes-hall] 1 Feb 2021
energy shift is applied and if the energy unit is renormalized. We show that these n-times squarable
models [SSC(n)] and their band structure are uniquely determined by the sequence of energy unit
renormalizations and by the energy shifts associated to each step of the squaring process. Chiral
symmetry is present in all Sine-Cosine chains and edge states levels at the respective central gaps
are protected by it. The extension to higher dimensions is discussed.
The characterization
√ of squared-root topological insu- blocks corresponding to a bipartite chain (apart from an
lators ( TI) relies on the √ fact that the square of the energy shift) which is self-similar to the original chain,
matrix representation of TI Hamiltonian in the Wan- that is, it is again a sine-cosine model provided that the
nier basis is a block diagonal matrix, more precisely, it energy unit is renormalized. The sequence of energy unit
is the direct sum H 2 = HT I ⊕ H2 of two blocks HT I renormalizations associated to each step of the squaring
and H2 that have the same finite energy spectrum, after process determines the energy gaps in the spectrum of
applying a constant energy downshift, but different eigen- the original chain. The higher dimension generalizations
states (HT I being the Hamiltonian of a known of these 1D models will also have the energy gaps at the
√ topologi-
cal insulator)1–3 . This reflects the fact that TI Hamil- inversion-invariant points determined by the renormal-
tonian H is defined in a bipartite lattice [lattice with ization factors. We show that a square-root Hamiltonian
sublattices A and B, such that the Hamiltonian can be of these higher dimensional models can be also obtained
written as a sum of hopping terms (which imply finite fron the 1D counterparts introducing a π-flux per pla-
Hamiltonian matrix elements) between different sublat- quette.
tices, H = HAB + HBA ]. Sine-Cosine chains: Assume an SSH(2n) chain
As very recent examples of squared-root topological with a unit cell with 2n sites and with nearest-neighbor
insulators, one may cite the diamond chain in the pres- hopping terms ti , i = 1, · · · , 2n, for some positive integer
ence of magnetic flux3 or our work on the t1 t1 t2 t2 tight- n. The Sine-Cosine model of order n, SC(n), is defined
binding chain (where a modified Zak’s phase, a sublat- imposing that t2j−1 = sin(θj ) and t2j = cos(θj ), with
tice chiral-like symmetry, modified polarization quanti- j = 1, · · · , n (see top diagram in Fig. 1).
zation, etc., were found4 ) where HT I corresponds to the By squaring this bipartite Hamiltonian, one obtains
well-known Su-Schrieffer-Heeger (SSH) model5 . In these a block-diagonal matrix (one for each sublattice of the
cases, the topological invariants and symmetries of the bipartite chain) and one of the blocks [shown in the mid-
SSH Hamiltonian HT I map into modified topological in- dle diagram of Fig. 1(a)] corresponds to a tight-binding
2
variants of the original Hamiltonian (see Ref.4 ). The model with uniform local potentials εj = sin(θj ) +
t1 t1 t2 t2 tight-binding chain is a particular case of the 2
cos(θj ) = 1 and hopping terms tj = cos(θj ) sin(θj+1 ).
SSH(4) model6 which is a generalization of the topologi- The uniform potentials can be removed applying an en-
cal SSH chain7–9 . ergy shift of one. Note that if the hopping terms are
Recently, several methods of generating the square root globally multiplied by a hopping factor t, one can still
Hamiltonian of a given Topological insulator Hamiltonian recover the Sine-Cosine form for the Hamiltonian setting
in 1D1–3,10,11 and 2D12–15 have been proposed. These this parameter t as the unit energy so that the energy
methods do not allow its consecutive application due to shift is again one (in units of t).
the appearance of non-uniform local potentials and the In the simple case of a uniform chain with hopping
consequent loss of the bipartite property. This also re- parameter t1 , one has √ θj = π/4, for all j, and the hopping
flects the fact that the square-root lattice and the original parameter is t1 = t/ 2, so the energy shift necessary
one are not self-similar. remove the uniform potentials (which is one in units of
In this paper, we consider a particular subset of 1D t) becomes 2t21 (see Fig. 2). Obviously, the sublattice
SSH(N ) models, N being the number of sites in the unit Hamiltonian corresponds to another uniform chain with
cell, that we designate by Sine-Cosine models, such that t2 = t2 cos(π/4) sin(π/4) = t2 /2 = t21 . Note that if we
the consecutive squaring of the Hamiltonian has always consider the respective inverse operation, the square-root
a block-diagonal matrix representation with one of the of the t2 chain, the bottom zero energy level (in red) in2
(a) unit cell +
chiral
θ1
θ2
θ2
θ1
θ1
n
θn
θn
sθ
s
sin
sin
sin
s
sin
s
co
co
co
co
ε in units of t1
-2 -1 0 1 2
H2 √
θ1
θ1
θ2
θ3
energy shift 2t1
sin
sin
sin
sin
√
2
1
n
θn
H2
sθ
sθ
sθ
s
t2 = t21
co
co
co
co
H energy shift − 2t2
1 1 1 1 1 1
θ2
sin
-2 -1
H2 ε in units of t2
θ2
1
1
sθ
sθ
sin
0 1 2
co
co
+ chiral
H2
θ1
1
sθ
sin
co
1 Figure 2. The squaring process for a uniform chain (that is,
(b) θj = π/4, for all j) with PBC. The top spectrum is for a chain
with 32 sites and hopping parameter t1 and the bottom for
θ1
θ2
2
θ3
θ4
θ1
θ3
sθ
sin
sin
sin
sin
s
s a chain with 16 sites and hopping parameter t2 . The colored
co
co
co
SC(4) H(k)= lines indicate the folding energies in the following steps of the
e−ik cos θ4 squaring sequence.
Figure 1. (a) The top diagram illustrates the Sine-Cosine
chain with a unit cell of 2n sites and hopping terms t2j−1 = flecting the eik phases in the hopping terms connecting
sin(θj ) and t2j = cos(θj ), with j = 1, · · · ,n. Upon squar- one unit cell to the next), see Fig. 1(b). If the real space
ing, the Hamiltonian for one of the sublattices has the form Hamiltonian is bipartite and the unit cell has more than
shown in the bottom diagram, with uniform local potentials one site, then H(k) is also bipartite and the squaring
εj = sin(θj )2 +cos(θj )2 = 1, as illustrated in the simple case of process will generate a block diagonal matrix.
three-site chains in the bottom diagrams of (a) (b) Schematic Our Matryoshka sequence of Sine-Cosine chains is con-
representation of a SC(4) bulk Hamiltonian. The sites cor- structed starting from the last Hamiltonian in the squar-
respondent to A and B sublattices are coloured in light blue
ing process which is that of a uniform chain with a single
and light red, respectively.
site in the unit cell and applying successively a square
root operation (see Fig. 2). At each step of the square
√ root process, we obtain a Hamiltonian with a new chi-
Fig. 2 is shifted by 2t1 in the top spectrum.
ral symmetry as illustrated in Fig. 2. The first iteration
These arguments apply to an infinite chain as well as
deviates from the general expressions for the following
to chains with periodic boundary conditions. In the case
ones since the uniform chain has a single-site unit cell
of a finite chain with open boundary conditions with an
and therefore the respective bulk Hamiltonian cannot be
arbitrary number sites, impurity-like potentials may be
written in the sine-cosine form described in the beginning
generated at the edge sites of the sublattices when squar-
of this section. Therefore we describe this first iteration
ing the Hamiltonian (the local potentials are not uni-
before presenting the general expressions.
form). However, for particular values of the system size,
the same reasoning can be applied. This will be discussed
after we discuss the spectra of Sine-Cosine chains with 1. From the SSC(0) to the SSC(1) chain. The Sine-
periodic boundary conditions (PBC) in the next subsec- Cosine chain SSC(1) (corresponding to the SSH
tion. model) has hopping terms {sin θ, cos θ} and when
Application to bulk Hamiltonians: In this sub- squared, generates a set of two equal bands with en-
section, we show that, when the unit cell of PBC Sine- ergy relation ε(k) = 1 + sin(2θ) cos k that corresponds
Cosine chains has 2n sites, for certain choices of the θj , to the spectrum of the uniform tight-binding SSC(0)
the squaring process can be applied n times and, at each chain with an energy shift equal √ to one and a hop-
step, one of the Hamiltonian blocks is again a Sine-Cosine ping parameter t(0) = sin(2θ)/ 2 (that determines the
chain (and this is why we call it a Matryoshka sequence), bandwidth). So the SSC(0) chain has as band limits
√ (0)
if an energy shift is applied and if the energy unit is ± 2t(0) and the chiral level εSSC(0) (folding level un-
renormalized. We label these n-times squarable Sine- der the squaring operation) is zero. These values also
Cosine chains, SSC(n) [they are a subset of the SC(2n−1 ) determine the band pstructure
√
of the SSC(1) chain: the
chains]. Furthermore, the sequence of energy shifts and band limits are ± 1 ± 2t(0) and a new chiral sym-
energy unit renormalizations determine the energy gaps (1)
metry is present with chiral level εSSC(1) = 0. The chi-
in the respective spectrum.
A bulk Hamiltonian H(k) is the Hamiltonian of the ral level of the SSC(0) chain is present at the SSC(1)
(1)
unit cell closed onto itself with a twisted boundary (re- spectrum at the energies εSSC(0) = ±1. Note that3
band limits folding energies left edge link it generates a block corresponding to a SSC(n − 1)
chain is written as
1.5
++++
+++-
(n−1) (n) (n)
t(n−1) sin θj
++--
R L = cos θ2j−1 sin θ2j (1)
++-+
(n−1) (n−1) (n) (n)
t cos θj = cos θ2j sin θ2j+1 (2)
1.0
L R L
+---
for j = 1, · · · ,2n−2 with 2n−1 + 1 ≡ 1. This implies
+---
+-+-
R L that the global hopping factor in the SSC(n − 1) chain
is given by
+-++ 0.5
q
(n) (n) (n) (n)
t(n−1) = (cos θ2j−1 sin θ2j )2 + (cos θ2j sin θ2j+1 )2
(3)
L L L L for any value of j. These equations determine (almost
-π π
(n)
uniquely) the set {θj } of the SSC(n) if t(n−1) and
(n−1)
{θj } are known.
--++
-0.5
Similarly to what was explained in the case SSC(0) →
--+-
SSC(1), any level ε(n−1) in √the SSC(n − 1) spectrum
---- R L
becomes a pair of levels , ± 1 + t(n−1) ε(n−1) , in the
----
SSC(n − 1) spectrum. It is simple to conclude that the
-1.0 L R L band structure of the SSC(n) is characterized by the
-+-+ following sequence of energy values that give the top
-+--
-++- R L and bottom energies of each band,
-+++
-1.5
ε± ± ± · · · ± ± =
| {z }
n
(a) (b) v
u v s r
u u
√
u q
..
u
(0)
Figure 3. (a)
√ Band structure √ with t =
√ of the SSC(3) chain
t
± 1±t (n−1) 1±t (n−2) . 1±t (1) 1 ± 2t(0) ,
t
sin(0.4π)/ 2, t(1) = 0.9/ 2, and t(2) = 0.8/ 2 that gen-
erate unit cell hopping constants {sin θ1 , cos θ1 , sin θ2 , cos θ2 , (4)
sin θ3 , cos θ3 , sin θ4 , cos θ4 } ≈ {0.542, 0.840, 0.309, 0.951,
0.485, 0.875, 0.375, 0.927}. The folding levels (that inter- where all the possible combinations of signs must be
sect energy curves r at ±π/2) are shown as well as the band considered. The folding levels associated with the chi-
√
q p
limits ε±±±± = ± 1 ± t(2) 1 ± t(1) 1 ± 2t(0 (only the
ral symmetries that appear at each step of the squar-
ing process are, in the SSC(n) spectrum, given by the
signs are indicated). (b) Right (R) and left (L) edge levels of
ordered sequence of the values (all the possible com-
a SSC(3) chain with OBC and N = 2n p − 1 sites, with n = 3
and integer p > 1, for all the possible choices of the leftmost binations of signs must be considered)
hopping term that allow the squaring into SSC(j) chains.
± 1,
p
± 1 ± t(n−1) ,
the notation ε(n) means a level in the spectrum of the q p
SSC(n) chain. ± 1 ± t(n−1) 1 ± t(n−2) ,
If we introduce a global factor t(1) in the hopping con- ..
stants of the SSC(1) chain so that the hopping pa- .,
(1)
rameters become {tp sin θ, t(1) cos θ}, then the band
v s
√
u r
(1)
limits become ±t(1) 1 ± 2t(0) and εSSC(0) = ±t(1) .
u
. p
± 1 ± t(n−1) 1 ± t(n−2) . . 1 ± t(1) .
t
Note that the uniform chain band energy shift and its
bandwidth (for any choice of energy unit) determine
the hopping parameters of the SSC(1) chain and the (n)
To summarize, the set {θj } in the SSC(n) Hamiltonian
same will occur if we repeat the square root operation
[applying it to the SSC(1) chain, then to SSC(2) and is determined by the sequence of hopping factors t(j) ,
so on]. j = 1, · · · , n − 1, which are the energy units for each step
of the construction of the SSC(n) chain, starting from the
2. From the SSC(n−1) chain to the SSC(n) chain. When uniform chain. Obviously, all the bandwidths and gaps in
squaring the SSC(n) Hamiltonian, the condition that the spectrum are also determined by this sequence. Note4
that we assumed that all hopping parameters are positive 2D-SSC(n) with π flux
2D-SSC(n-1)
and this places all angles in the first quadrant. A gauge
transformation can change the sign of the hopping terms
sθ y
2
maintaining the spectrum. Even with this condition, the
co
(n)
θ2 y
y
set {θj } in the SSC(n) is not unique given the sequence
cos θ1 sin θ2
H2
sin
of hopping factors t(j) , j = 1, · · · , n−1, because there are
y
sθ y
x
still the two possible choices for the SSC(j) sublattice at cos θ1 sin θ2x
1
co
each step of the construction of the SSC(n) Hamiltonian
- +
θ1 y
(but the two choices generate the same spectrum).
sin
Note that the SSC(n) chain can be viewed as a 2n -root
θ1 x
sθ x
θ2 x
sθ x
of an uniform tight-binding chain, but a generalized one
2
1
sin
sin
co
co
due to degrees of freedom associated with global hopping
factors t(n) .
Figure 4. A square-root Hamiltonian of a 2D SSC(n − 1)
Finite systems with open boundaries: In this Hamiltonian is possible if π-flux per plaquette is introduced
section, we show how to generate edge states in any of in the 2D SSC(n) lattice. This flux implies the existence of
the gaps in the band structure of the SC(n) chain which 4 blocks (four decoupled sublattices) in the squared Hamilto-
will be protected by the chiral symmetry of a particular nian, one of them being the 2D SSC(n − 1) one (sublattice
step of the construction of the SSC(n) Hamiltonian. Note with squared sites).
that the edge states are replicated in the correspondent
gaps in the unfolding process. For example, the gaps in
Fig. 3 from top to bottom are SSC(1), SSC(2), SSC(1), possible choices of chain terminations given a system size
SSC(3), SSC(1), SSC(2), SSC(1) gaps and an edge state N = 2n p − 1. In Fig. 3(b), we show the edge state levels
associated with the SSC(1) chain will be present in all in the case of a OBC SC(3) chain with N = 2n p − 1
SSC(1) gaps. sites for all the possible choices of the leftmost hopping
Let us first explain the appearance of edge states in the term (sin θ1 , sin θ2 , sin θ3 , sin θ4 ) which agree with the
usual SSH(2) chain [equivalent to the SC(1) ≡ SSC(1) previous argument.
chain]. Edge states appear in an open boundary SSH(2) Interestingly, for these system sizes, the squaring
chain when a weak link is present at the boundaries. Our method can be extended until we reach a single level
definition of weak link is a hopping term in the unit cell and this implies that the spectrum of the OBC SSC(n)
that can be adiabatically increased from zero (with all chain is the combination of the spectrum obtained from
the other hopping terms in the unit cell finite, constant a single level with zero energy (applying successively
and larger) without closing the central gap. If one of energy shifts, energy unit renormalizations and square
(j−1)
the sublattices of the bipartite chain has one more site, √ to each level, that is, each level ε
roots generates
an edge state is always present (it changes from a right ± 1+t (j−1) ε (j−1) levels) with a spectrum that has lev-
edge state to a left edge state at the topological transi- els at the folding energies (due to the extra site in the
tion, reflecting the fact that there is always one weak link other sublattice relatively to the SSC(j) sublattice).
at one of the boundaries) and has finite support only in The presence of edge states in the central band at
this sublattice. Furthermore, its energy is exactly zero, each step of the construction can be confirmed adding
a value protected by the chiral symmetry. If both sub- the Zak’s phase of the positive bands leading as usual to
lattices have the same number of sites and we can split π in the non-trivial topological phase.
the chain in two halves, each of them with a weak link at Extension to 2D: The 2D version (for higher dimen-
the boundary, two edge sates will be present with nearly sions the reasoning is similar) of the SSC(n) chain can be
zero energy which are protected by the chiral symmetry constructed in the same way as the 2D SSH model is con-
and the band gap. structed from the SSH chain, that is, the hopping terms
In order to generate edge states in a chosen gap of the in the x direction are those of a x-SSC(n) chain and the
SSC(n) chain, one chooses the boundaries in such a way same for hopping terms in the y direction [the y-SSC(n)
that the squaring process will generate the SSC(j) chain hopping terms can be different from the x-SSC(n) ones].
that has that gap as the central one with a weak link This 2D model will have a band structure that can be
at its boundaries. Also, in order to guarantee that one characterized by the band limits at the inversion invari-
of the blocks at each squaring step is that of a bipartite ant momenta and these limits will be the sum of two
chain (apart from an energy shift), we impose that the terms of the form of Eq. 4, εx±±±···±± + εy±±±···±± .
number of sites of the SSC(n) chain is N = 2n p − 1, One may be tempted to try to construct the 2D-SSC(n)
with integer p > 1, so that the inner sublattice is the model following the method given for the SSC(n) chain so
one corresponding to the bipartite OBC SSC(j) chain that, when squaring the Hamiltonian, 2D-SSC(j) blocks
in all steps (the number of sites at each step is of the are generated. Despite the fact that the lattice is bi-
form N = 2j p − 1 ). That way, all sites of the OBC partite, one faces one difficulty: the dimension of the
SSC(j) chain will have the same local potential (equal 2D-SSC(n − 1) model is one fourth of that of the 2D-
to one). In the case of the SSC(n) chain, there are 2n−1 SSC(n) model. When squaring the Hamiltonian, the bi-5
partite property guarantees the appearance of two diago- chiral symmetry. Edge states at any gap of the original
nal blocks, each one corresponding to different sublattice chain are protected by one of these chiral symmetries.
(in Fig. 4, the two sublattices have different colors). An This sequence of chiral symmetries is lost in general if
extra factor is required in order for one of these blocks the Hamiltonian is perturbed away from the Sine-Cosine
to become a diagonal sum of two smaller blocks, reflect- form, but as long as the band gaps remain open, the edge
ing the division of the sublattice in two other sublattices states should survive.
(with pink circular sites and pink squared sites in Fig. 4, These models are determined by the sequence of en-
respectively). So we are only able to find a single square- ergy unit renormalizations in the squaring process and
root of a 2D SSC(n−1) model, and that is the 2D SSC(n) their spectrum has a very simple form in terms of these
model with π-flux per plaquette, with flux introduced by parameters. This fine tuning of their band structure
multiplying the x-hopping terms by (−1) at every other as well as the control over the presence or absence of
rung. This π flux generates destructive interference in edge states in any of the spectrum gaps makes these
the hopping terms from pink circular sites to/from pink models very appealing in the context of artificial lattices
squared sites in Fig. 4 and therefore one may interpret it such as photonic3,16–20 , optical lattices21–23 , topoelectri-
as an additional “bipartite” property. cal circuits24,25 or acoustical lattices26,27 , where the effec-
Conclusion: Square-root topological insulators have tive hopping terms can be adjusted in order to reproduce
attracted attention due to the presence of finite energy the necessary set of angles {θj }.
topological edge states in non-central gaps of the chiral Acknowledgments: This work was developed
spectrum that cannot be characterized using the usual within the scope of the Portuguese Institute for Nanos-
topological invariants. In this paper, we extend the con- tructures, Nanomodelling and Nanofabrication (i3N)
cept of SRTI by introducing a particular 1D Hamilto- projects UIDB/50025/2020 and UIDP/50025/2020.
nian [that we label n-times squarable Sine-Cosine model, RGD and AMM acknowledge funding from FCT - Por-
SSC(n)] of the family of the SSH(2n ) chains that can tuguese Foundation for Science and Technology through
be squared multiple times generating at each step a self- the project PTDC/FIS-MAC/29291/2017. AMM ac-
similar Hamiltonian (with a smaller unit cell) in what we knowledges financial support from the FCT through the
call a Matryoshka sequence, each of them with its own work contract CDL-CTTRI-147-ARH/2018.
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