A study of curvature theory for different symmetry classes ...

Pramana – J. Phys. (2021) 95:102 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-021-02134-9

A study of curvature theory for different symmetry classesof Hamiltonian

Y R KARTIK1,2, RANJITH R KUMAR1,2, S RAHUL1,2 and SUJIT SARKAR1,∗

1Theoretical Sciences Division, Poornaprajna Institute of Scientific Research, Bidalur, Bengaluru 562 164, India2Graduate Studies, Manipal Academy of Higher Education, Madhava Nagar, Manipal 576 104, India∗Corresponding author. E-mail: [email protected]

MS received 22 September 2020; revised 18 February 2021; accepted 26 February 2021

Abstract. We study and present the results of curvature for different symmetry classes (BDI, AIII and A) ofmodel Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry classto the other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian withBDI symmetry class but there is no evidence of such behaviour for Hamiltonians of AIII symmetry class. We showthe origin of torsion and its consequences on the parameter space of topological phase of the system. We find theevidence of torsion for the Hamiltonian of A symmetry class. We present Serret–Frenet equations for all modelHamiltonians in R3 space. To the best of our knowledge, this is the first application of curvature theory to the modelHamiltonian of different symmetry classes which belong to the topological state of matter.

Keywords. Curvature theory; torsion; topological state of matter.

PACS Nos 03.65.Vf; 02.40.−k; 73.22.Gk

1. Introduction

Symmetry and topology are two prominent branchesof physics that reveal many interesting features. It isbelieved that these two branches are always in agree-ment with each other [1–7]. Before the discovery oftopological phases of matter, Landau theory of sym-metry breaking was considered as the prominent toolto characterise the phases of matter. But thereafter theconcept got modified. There is no order parameter intopological states of systems. However, if a system isinvariant under some symmetry, it gives rise to invariantquantities. These invariants can be used to charac-terise the topological states of matter. Based on theseinvariants, a system can be classified into ten distinctsymmetry classes. Out of these ten non-interacting sym-metry classes, only a few exhibit topological nature in1D [4]. Recently, there were some interesting studieswhich involve the interplay and relations between dif-ferent symmetry classes [8–11].

Differential geometry deals with the study of prob-lems using differential calculus, integral calculus andlinear algebraic techniques [12–14]. Differential geom-etry is a significant mathematical structure of generaltheory of relativity using which the concept of manifold,

curved space–time, gravity can be explained much effi-ciently [15–17]. There are some notable works whichexplained PT symmetric systems through differentialgeometry [18].

Curvature study is an important step in differentialanalysis of the system and it is effectively used inthermodynamics and many-body systems to explain itsnature. Curvature is a tool to measure how curved acurve is. In other words, curvature measures the extentto which a curve deviates from a straight line. For aunit speed curve γ (t), where t is a parameter, curva-ture κ(t) at a point is defined to be ||γ (t)|| [19]. Themain motivation is to explain the many-body system ina more rigorous manner. Curves and angles are effectiveways of expressing the geometric properties of a physi-cal system [20–22]. Torsion is a natural quantity whichis associated with the curvature. It affects periodicity,spin wave dynamics and structural defects of the sys-tem [23–25]. Torsion also has a significant role in thedynamics of the adiabatic system, transport propertiesand bulk–boundary correspondence in the topologicalstate of matter [26,27].

The geometrical studies of condensed matter systemshave been an interesting area of research which hasrapidly picked up pace when the principles of topology

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and geometry were involved in the foundations of quan-tum condensed matter systems [28,29]. The physics ofgeometry of curves in R3 with spins in connection withthe dynamics of classical Heisenberg ferromagnetic sys-tem under different contexts has already been exploredin the literature (see for example, [30–32])

The main motivation of this work is to study a fewmodel Hamiltonians which belong to different symme-try classes from the perspective of curved space theoryof differential geometry [13,14]. This paper is organ-ised in the following manner. In §2 we introduce themodel Hamiltonian and present a detailed analysis ofsymmetry class Hamiltonians. In §3 we present the char-acteristics and behaviour of parameter space curves witha detailed analysis of differential geometric study of cur-vature. Here we try to analyse the origin of torsion andits consequences for the present model Hamiltonian.

2. Basic model Hamiltonian

Here we consider eight model Hamiltonians belongingto different symmetry classes [8,9]. Our model Hamil-tonian is expressed as

H = H0 + Heff , (1)

where H0 is the initial Hamiltonian and Heff is the effec-tive part of the Hamiltonian which is responsible for thetransformation from one symmetry class to the other.Here, initial Hamiltonian H0 is a 1D non-interactingtopological insulator. We can write our Hamiltonian inthe Bogoliubov–de Genne (BdG) format as

HBdG(k) = χ(1)

(0 11 0

)+ χ(2)

(0 i−i 0

)

+χ(3)

(1 00 −1

). (2)

The components can be written as χ(1) = 0, χ(2) =� sin k and χ(3) = μ + 2t cos k. The effective term(Heff ) is momentum-dependent, and is in the followingform:

Heff = δ1kσx + δ2kσy + δ3kσz = δi (ki · τ i ),

where k1 = k2 = k3 = k (a detailed study is presentedin ref. [8]). We consider a very specific type of effectiveterm which is of much theoretical interest. The resultsof this study may motivate researchers in quantum sim-ulation studies to look for this type of effective term andconsequences of their effects on the topological state ofmatter [33–35].

2.1 Hamiltonian H (1)(k) when δ1 = δ2 = δ3 = 0

Here the effective part of the Hamiltonian is zero. So,the Hamiltonian in Pauli basis can be written as

H (1)k = 2� sin kσy + (2t cos k + μ)σz. (3)

Hamiltonian in matrix form can be written as

H(1)(k)

=(

2t cos(k) + μ 2i� sin(k)−2i� sin(k) −2t cos(k) − μ

). (4)

2.2 Hamiltonian H (2)(k) when δ1 = δ3 = 0, δ2 �= 0

Here the effective term is added to the σy componentof the Hamiltonian. It can be written in terms of Paulibasis as

H (2)k = (2� sin k + δ2k)σy + (2t cos k + μ)σz. (5)

Hamiltonian in matrix form can be written as

H(2)(k)

=(

2t cos(k) + μ 2i� sin(k) + iδ2k−2i� sin(k) − iδ2k −2t cos(k) − μ

). (6)

2.3 Hamiltonian H (3)(k) when δ3 �= 0, δ1 = δ2 = 0

Here the effective term is added to the σx componentof the Hamiltonian. It can be written in terms of Paulibasis as

H (3)k = 2� sin kσy + (2t cos k + μ + δ3k)σz. (7)

Hamiltonian in matrix form can be presented as

H(3)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k)−2i� sin(k) −2t cos(k) − μ − δ3k

).

(8)2.4 Hamiltonian H (4)(k) when δ1 = 0, δ2 �= 0, δ3 �= 0

Here effective terms are added to both the σx and σycomponents of the Hamiltonian. It can be written interms of Pauli basis as

H (4)k = (2� sin k + δ2k)σy + (−2t cos k − μ + δ3k)σz.

(9)

Hamiltonian H (4)(k) in matrix form can be presentedas

H(4)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k) + iδ2k−2i� sin(k) − iδ2k −2t cos(k) − μ − δ3k

).

(10)

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2.5 Hamiltonian H (5)(k) when δ1 �= 0, δ2 = δ3 = 0

Here effective term is added to the σx component of theHamiltonian. It can be written in terms of Pauli basis as

H (5)k = (δ1k)σx + (2� sin k)σy

+(2t cos k + μ)σz . (11)

Hamiltonian H (5)(k) can be written in matrix form as

H(5)(k) =(

2t cos(k) + μ 2i� sin(k) + δ1k−2i� sin(k) + δ1k −2t cos(k) − μ

).

(12)

2.6 Hamiltonian H (6)(k) when δ1 �= 0, δ2 �= 0, δ3 = 0

Here effective terms are added to both the σx and σycomponents of the Hamiltonian. It can be written interms of Pauli basis as

H (6)k = (δ1k)σx + (2� sin k + δ2k)σy

+(2t cos k + μ)σz . (13)


H(6)(k) =(

2t cos(k) + μ 2i� sin(k) + iδ2k + δ1k−2i� sin(k) − iδ2k + δ1k −2t cos(k) − μ

). (14)

2.7 Hamiltonian H (7)(k) when δ1 �= 0, δ2 = 0, δ3 �= 0

Here effective terms are added to both theσx andσz com-ponents of the Hamiltonian. It can be written in termsof Pauli basis as

H (7)k = (δ1k)σx+(2� sin k+δ3k)σy+(2t cos k+μ)σz.

(15)


H(7)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k) + δ1k−2i� sin(k) + δ1k −2t cos(k) − μ − δ3k

).

(16)

2.8 Hamiltonian H (8)(k) when δ1 �= 0, δ2 �= 0, δ3 �= 0

Here effective terms are added to the σx , σy and σz com-ponents of the Hamiltonian. It can be written in termsof Pauli basis as

H (8)k = (δ1k)σx + (2� sin k + δ2k)σy

+(2t cos k + μ + δ3k)σz. (17)


H(8)(k) =(

2t cos(k) + μ + δ3k 2i� sin(k) + iδ2k + δ1k−2i� sin(k) −2t cos(k) − μ − δ3k − iδ2k + δ1k

). (18)

The addition of the effective term does not affect theHermitian property of the system.

Basically, the Hamiltonian is in the spinless fermionbasis. The effective term is also in spinless basis and ismomentum-dependent. Therefore, we justify the phys-ical relevance of the effective term.

The first Hamiltonian H1(k) is the Kitaev modelHamiltonian [36] which governs the topological state ofquantum matter. The other seven Hamiltonians (i.e. fromH2(k) to H8(k)) are variants of Kitaev model Hamil-tonian. We consider these additional Hamiltonians inthe spirit of theoretical studies only. Using these modelHamiltonians, we study the topological as well as geo-metric properties of quantum condensed matter systemup to some extent.

3. Curvature analysis of curves in planarparameter space

Curvature can be defined as the rate of variation of theangle that the tangent line makes at a particular point.To call a curve as a regular curve, it should have anon-vanishing tangent line. Curve theory basically dealswith analysing the basic properties of the curves. Basicproperties include, the arc length, winding number withcurvature and torsion of the curves [19]. Topologicalinvariant quantities, such as winding number and Chernnumber depend on the topology of the parameter space,where for a particular topological configuration space,winding number acquires a definite value, and change inthe winding number leads to different topological con-figurations of the system [37].

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The understanding of the curve concept is simplifiedby using the differential geometry tool called curvatureκ .

The relation which relates the parametrised curve c(k)and the curvature κ(t) is given by [38]

κ(k) = det(c(k), c(k)

||c(k)||3 , (19)

where dot represents d/dk. For a unit speed curvec : I → R

2 where I = [a, b] is a closed curveinterval. Then c(k) gives the velocity vector defined by(cos θ(k), sin θ(k))T of an integer multiple of 2π , asthe curve is defined in a closed interval. As the anglechanges along the curve, the invariant quantity windingnumber is defined by θ(b) − θ(a). If θ1, θ2 : I → R

satisfies the velocity equation, θ1 = θ2 + 2nπ , wheren ∈ Z.

The velocity term c([a, b]) ⊂ SR , i.e., c(t) > 0 forall k ∈ I and c(t) = (c1, c2)

T ,

c(2)

c(1)

= sin θ(k)

cos θ(k)= tan θ(k)

and

θ(k) = arctan

(c2(k)

c1(k)

)+ 2nπ, n ∈ Z.

So considering c:R → R2 a unit speed vector of a

curve with period L and θ :R ← R a scalar, the windingnumber is given by

wk = 1

2π(θ(L) − θ(0)), (20)

where (θ(L) − θ(0)) is well defined irrespective of thechoice of θ . Therefore, it is clear from the above equationthat to get a complete physical picture of the windingnumber, the study of the curve is useful. It is well knownthat the topological system is a closed curve which encir-cles the origin. Geometrically, the parameter space of atopological system is an ellipse and defined as the locusof points such that the sum of distances from the foci isconstant. The standard equation of ellipse is given by

x2

a2 + y2

b2 = 1,

where a and b are semi-major and semi-minor axesrespectively. The parametric equation is given by [a(k),b(k)] = (a cos k, b sin k) where 0 ≤ k < 2π .

The curvature of ellipse is given by [13]

κ(k) = ab

(b2 cos2 k + a2 sin2 k)3/2, (21)

where a and b are semi-major axis and semi-minor axisof ellipse respectively (figure 1). From these two param-eters, we can analyse the curvature in three differentcases.

First case: When a < b, the curvature is maximum onthe semi-major axis ( −π/2 and π/2) and minimum onthe semi-minor axis.

Second case: When a = b, the parameter space curve isa circle with the constant curvature.

Third case: When a > b, the curvature is minimum onthe semi-major axis (−π/2 and π/2) and maximum onthe semi-minor axis [14].

For a plane unit speed curve c: I → R2, where n(k) andκ(k) give the normal unit vector and curvature of thecurve. Then,

( ˙ν(k), ˙n(k)) = (ν(k), n(k))

(0 −κ(k)

κ(k) 0

)(22)

defines the relation ν = k · n and n = −κ · ν, whereν = ˙c(k) [13]. This is called Frenet equation whichgives the information about the curvature properties ofthe curve c(k).

For a non-vanishing curve c(k) with a non-vanishingcurvature κ(k), torsion is given by [19]

τ = (c(k) × c(k)) · ...c (k)

||c(k) × c(k)||2 . (23)

Our model Hamiltonian is written in the Pauli spinbasis. Naturally, the quantity torsion gives the curl ofthe derivatives of the curve. This results in the curveopening and helical motion on the addition of the effec-tive term αk.

To understand the kinematic properties of curve c:R → R3, we study Serret–Frenet equation for thecurve [19]. For a unit-speed curve c(k) in R3 curva-ture explains the failure of a curve to be a straight lineand torsion explains the failure of a line to be a pla-nar. Serret–Frenet formula describes the derivative oftangent (T ), normal (N ) and binormal (B) unit vectorswith respect to the arc length of the parameter of thecurve (s) [19], i.e.,

dTds

= κN

dNds

= −κT + τB

dBds

= −τN. (24)

Here B is perpendicular to T. Being perpendicular toboth T and B, B must be parallel to N. It is to benoted that, torsion (τ ) exists only for a curve with non-zero curvature. Equation (24) is known as Serret–Frenet

Pramana – J. Phys. (2021) 95:102 of 17 102

Figure 1. The graphical representation of an ellipse (left) and the corresponding curvature plots for the parameter space(right). Here we observe that the curvature is minimum at the origin but not vanishing.

Table 1. Properties of symmetry operators which are related to the present study.

Symmetry Relation Operator Nature

Time reversal [T , H ] = 0 T = K Reverses the arrow of time(T ) T HT−1 = H T 2 = 1 T : t −→ −tParticle–hole {C, H} = 0 C = σxK Transformation between electron and holes(C)

CHC−1 = −H C2 = 1 (within certain energy range)Chiral {S, H} = H S = σx Symmetric spectrum of the Hamiltonian(S)

SHS−1 = −H S2 = 1

equation and gives a better understanding of the geo-metric properties of the system. One can also write thematrix representation of the Serret–Frenet equation asfollows [19]:

d

ds(X) =

⎛⎝ 0 κ(k) 0

−κ(k) 0 τ(k)0 −τ(k) 0

⎞⎠ (X), (25)

where X = (T, N , B)T . By expressing dT /ds, dN/dsand dB/ds in terms of T, N and B one can get skew-symmetric matrix and it follows that the vectors T, Nand B are orthonormal for all values of the arc lengthparameter (s).

4. Different symmetry classes and their nature

Different symmetry classes have already been studiedand discussed extensively [8–10]. Here, in table 1 wediscuss them very briefly which are directly involvedwith the present study.

Time-reversal symmetry (TR): Time-reversal symmetryis the transformation which is anti-unitary in nature. Thetime-reversal operator just reverses the sign of momen-tum but does not affect the position. It is equivalent to

the complex conjugate operator (K).

T xT −1 = x, T kT −1 = −k, T iT −1 = −i. (26)

Time-reversal operator is the product of unitary (U )and complex conjugate operators, i.e. T = UK. Thesquare of the time-reversal operator is equal to thenegative of identity which yields Kramer’s degener-acy. According to that, one state is the time-reversal ofanother and every state is doubly degenerate. Thus, thesystem becomes time-reversal invariant [5,39,40].

Particle–hole (PH) symmetry: The particle–hole opera-tor is an anti-unitary operator and in the presence of thissymmetry, each eigenfunction with E > 0 has itsparticle–hole reversed partner, C with E < 0. The PHsymmetry is the intrinsic property of mean-field theoryof superconductivity.

Chiral symmetry: Chiral symmetry (S) or sublatticesymmetry is the product of time-reversal operator (T )

and particle–hole operator (C). Based on the behaviourof Hamiltonian with the TR, PH and chiral symmetries,it is classified into 10 symmetry classes.

In table 2, we present different symmetry classes tocharacterise the topological states of the system fordifferent dimensions (d). The first column presents dif-ferent symmetry classes, the second, third and fourth

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Table 2. Ten-fold symmetry class for a topological system. Here � is the time-reversal symmetry, � is the particle–holesymmetry, � is the charge conjugation symmetry and d is the dimensionality of the system.

Symmetry d

AZ � � � 1 2 3 4 5 6 7 8

A 0 0 0 0 Z 0 Z 0 Z 0 Z

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0AI 1 0 0 0 0 0 Z 0 Z2 Z2 Z

BDI 1 1 1 Z 0 0 0 Z 0 Z2 Z2D 0 1 0 Z2 Z 0 0 0 Z 0 Z2DIII −1 1 1 Z2 Z2 Z 0 0 0 Z 0AII −1 0 0 0 Z2 Z2 Z 0 0 0 Z

CII −1 −1 1 Z 0 Z2 Z2 Z 0 0 0C 0 −1 0 0 Z 0 Z2 Z2 Z 0 0CI 1 −1 1 0 0 Z 0 Z2 Z2 Z 0

columns present respectively the time-reversal, particle–hole and charge conjugation symmetries. The rest ofthe table is for the dimensionality (d) and the topolog-ical index system (Z and Z2). Here we mention verybriefly the topological characterisation of the system,and a detailed discussion can be found in refs [8–10].

Topological states of matter are characterised by thepresence of time-reversal, chiral and charge conjugationsymmetries. They are classified into different symme-try classes based on these symmetry operators. Theedge state in the topological systems are protected bythe time-reversal symmetry (T : t → −t) and time-reversal symmetry (commutes with the Hamiltonian,i.e., [H, T ] = 0), chiral symmetry (i.e., chiral opera-tor anticommutes with Hamiltonian, {S,H} = 0) andparticle–hole operator (anticommutes with the Hamil-tonian {S,H} = 0) decide whether the system istopological or not. Table 2 presents the condition andclassification of different symmetry classes. We observethat our model Hamiltonians belong to three different(BDI, AIII and A) symmetry classes. We present ourresults of different symmetry classes in the next section.

4.1 Results of BDI symmetry class

BDI symmetry class is characterised by the commuta-tion of time- reversal (T ) operator with the Hamiltoniananticommutation of other two operators like particle–hole (C) and chiral (S) with the Hamiltonian (eq. (2)).Here the Hamiltonians H (1)(k) and H (2)(k) belongto the BDI class [8]. The Hamiltonian H (1)(k) istopological in nature. The Hamiltonian H (2)(k) showstopologically trivial behaviour. Now we study the cur-vature properties of these Hamiltonians.

4.1.1 H (1)(k) Hamiltonian. Here we present theresults of differential geometric study based on curve

theory for the BDI Hamiltonians. The matrix form ofthe model Hamiltonian is

H(1)(k) =(

2t cos(k) + μ 2i� sin(k)−2i� sin(k) −2t cos(k) − μ

). (27)

Here the set of possible parametric equations are

χ(1)(H (1)(k)) = 0

χ(2)(H (1)(k)) = 2� sin k,

χ(3)(H (1)(k)) = 2t cos k + μ, (28)

HBdG Hamiltonian in the pseudospin basis is

H (1)(k) = χ(2)(H (1)(k))σy + χ(3)(H (1)(k))σz. (29)

In terms of vectors, one can write the above equation asHBdG = χ (k) · τ , where τ are the Pauli spin matricesacting in the particle–hole (Nambu) basis of HBdG [41].The energy dispersion relation

E (1)(k) =√

(2t cos k + μ)2 + (2� sin k)2.

Considering the parametric equation of the Hamilto-nian H (1)(k) in the matrix form

c(k) =[

2� sin k2t cos k + μ

], c(k) =

[2� cos k−2t sin k

],

c(k) =[−2� sin k

−2t cos k

]. (30)

Curvature is given by

κ = det[c, c]||c||3 =

det

(2� cos k - 2� sin k−2t sin k −2t cos k

)(√

4t2 sin2 k + 4�2 cos2 k)3

= −2t�(√t2 sin2 k + �2 cos2 k

)3 . (31)

Figure 2 represents the curvature plot for HamiltonianH (1)(k). The parameter space curve for the Hamiltonian

Pramana – J. Phys. (2021) 95:102 of 17 102

a

b

Figure 2. The left figure represents the plots of curvature with k for γ = 2, 1, 0.7 for red, blue and green respectively. Theright figure represents the corresponding parameter plots for μ = 0.

H (1)(k) is nothing but an ellipse (figure 1) due to themathematical structure of the parametric equation.

When μ = 0, the system remains in the topologicalstate. We can study the curvature of parameter spacecurve for all the Hamiltonians. We cannot characterisethe topological and non-topological states of the Hamil-tonian from the curvature study because the curvatureexpression does not include the term μ. From the abovegeneral discussion on the ellipse we can characterisethe parameter space curve of the H (1)(k) Hamilto-nian into similar three cases which is described below.This is completely a theoretical study to understand thebehaviour of the parameter space curve of the modelHamiltonians from the perspective of differential geom-etry.

First case: When t < �, the curvature is maximum onthe semi-major axis (−π/2 and π/2) and minimum onthe semi-minor axis.

Second case: When t = �, the parameter space curveis a circle with constant curvature.

Third case: When t > �, the curvature is minimum onthe semi-major axis (−π/2 and π/2) and maximum onthe semi-minor axis.

4.1.2 H (2)(k) Hamiltonian. Hamiltonian H (2)(k) canbe written in the matrix form as

H(2)(k)

=(

2t cos(k) + μ 2i� sin(k) + iδ2k−2i� sin(k) − iδ2k −2t cos(k) − μ

). (32)

Here the set of parametric equations are

χ(1)(H (2)(k)) = 0

χ(2)(H (2)(k)) = 2� sin k + δ2k,

χ(3)(H (2)(k)) = 2t cos k + μ. (33)

HBdG Hamiltonian in the pseudospin basis is [41]

H (2)(k) = χ(2)(H (2)(k))σy + χ(3)(H (2)(k))σz. (34)

The energy dispersion relation

E (2)(k) =√

(2t cos k + μ)2 + (2� sin k + δ2k)2.

Hence, the curvature for H (2)(k) is

κ =det

[2� cos k + δ2 −2� sin k

−2t sin k −2t cos k

](√

(2t sin k)2 + (2� cos k + δ2)2)3

= −4t� − 2δ2t cos k(√(2t sin k)2 + (2� cos k + δ2)2

)3 . (35)

Equation (35) is the analytical expression of the curva-ture for the Hamiltonian H (2)(k).

Figure 3 consists of two panels for two different valuesof δ2. The upper and lower panels represent the param-eter space δ2 = 0.5 and δ2 = 1 respectively. Each panelconsists of two figures, the left one is for curvature andthe right one is for the corresponding parameter spacecurves. We observe that as value of δ2 increases, thecurvature also increases. Here we observe an interest-ing feature that the curvature as well as parameter plotsare mirror symmetric about κ-axis. This is true for bothHamiltonians of BDI symmetry class. The parameterspace curve splits into two as we increase the value of δ2.

The parameter space curves of Hamiltonian H (2)(k)resembles the cycloidal pattern due to its mathematicalstructure. The general expression of the cycloid is givenby [14]

Cyc[a, b](t) = (at − b sin t, a − b cos t). (36)

In general, the cycloid is classified into two categoriesdepending on the values of coefficients. In eq. (36), ifa < b, then the cycloid is prolate and if a > b, it

102 of 17 Pramana – J. Phys. (2021) 95:102

a b

c d

Figure 3. (a) Plots of curvature (κ) with k for δ2 = 0.5, (b) the corresponding parameter plots for the curvature plot (a), (c)plots of curvature (κ) with k for δ2 = 1 and (d) the corresponding parameter plots for the curvature plot (c). In all the plots,the red, blue and green curves represent t = 2, 1, 0.7 respectively.

is curate. From this classification, we can assign ourHamiltonian H (2)(k), as prolate as the prolate cycloid isself-interacting and also it satisfies the condition a < b.

With the introduction of effective term, the propertiesof Hamiltonian changes, which are reflected in the cur-vature of parameter space. Based on the strength of theeffective term, the parameter space curve behaves as asimple curve with non-closed, self-intersecting condi-tions.

For this BDI symmetry class, we have presented thecurvature study of two different Hamiltonians. Hamilto-nian H (1)(k) is the model Hamiltonian without effectiveterm. In Hamiltonian H (2)(k), the effective term is addedto the σy component. In both cases, the curvature is mir-ror symmetric about the κ-axis.

4.2 Results of AIII symmetry class

AIII symmetry is characterised by the absence of time-reversal and particle–hole symmetry. But it obeys chi-ral symmetry condition (figure 2). AIII symmetry classcontains two Hamiltonians H (3)(k) and H (4)(k). BothHamiltonians are topologically trivial in one dimensionand satisfies all the symmetry properties.

4.2.1 H (3)(k) Hamiltonian. The matrix form of theHamiltonian H (3)(k) is

H(3)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k)−2i� sin(k) −2t cos(k) − μ − δ3k

).

(37)


χ(1)(H (3)(k)) = 0

χ(2)(H (3)(k)) = 2� sin k

χ(3)(H (3)(k)) = 2t cos k + μ + δ3k. (38)


H (3)(k) = χ(2)(H (3)(k))σy

+χ(3)(H (3)(k))σz. (39)


E (3)(k) =√

(2t cos k + μ + δ2k)2 + (2� sin k)2.

The curvature of the Hamiltonian H (3)(k) is

κ(k) =det

(2� cos k −2� sin k

−2t sin k + δ3 −2t cos k

)(√

(−2t sin k + δ3)2 + 4�2 cos2 k)3

= −4t� + 2δ3� sin k(√(2t sin k + δ3)2 + (2α cos k)2

)3 . (40)

Figure 4 consists of two panels for two different valuesof δ3. The upper and lower panels represent the param-eter space δ3 = 0.5 and δ3 = 1 respectively. Each panel

Pramana – J. Phys. (2021) 95:102 of 17 102

a b

c d

Figure 4. (a) Plots of curvature (κ) with k for δ2 = 0.5, (b) the corresponding parameter plots for the curvature plot (a), (c)plots of curvature (κ) with k for δ2 = 1and (d) the corresponding parameter plots for the curvature plot (c). In all the plots,the red, blue and green curves represent t = 2, 1, 0.7 respectively.

consists of two plots, the left one is for curvature andthe right one is for the corresponding parameter spacecurves. We observe that as the value of δ3 increases, thecurvature also increases.

It reveals in this study that the AIII symmetry classlacks the mirror symmetry about κ axis. As the value ofδ1 increases, the peaks become steep but their position isunaltered. As in the previous case, the curvature expres-sion is independent of the term μ. The increase in thestrength of the effective term results in a decrease of thecurvature near k = 0. For the Hamiltonian H (3)(k), theparameter space curve is also a prolate cycloid becauseit is open self-intersecting.

From the curvature studies for this parameter spacecurve of Hamiltonian H (3)(k), it reveals that the curva-ture at the points (−π and π ) on the semi-major axis ismaximum and the curvature on the semi-minor axis isminimum. When the effective term changes its sign, theparameter space curves as well as curvature plots formmirror symmetric images [8].


H(4)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k) + iδ2k−2i� sin(k) − iδ2k −2t cos(k) − μ − δ3k

).

(41)


χ(1)(H (4)(k)) = 0

χ(2)(H (4)(k)) = 2� sin k + δ2k,

χ(3)(H (4)(k)) = 2t cos k + μ + δ3k. (42)


H(k)(4) = χ(2)(H (4)(k))σy + χ(3)(H (4)(k))σz. (43)


E (4)(k) =√

(2� sin k + δ2k)2 + (2t cos k + μ + δ3k)2.

The curvature is given by

κ(k) =Det

[2� cos k + δ2 −2� sin k−2t sin k + δ3 −2t cos k

](√

(−2t sin k + δ3)2 + (2� cos k + δ2)2)3

= −4t� − 2(δ3� sin k + δ2t cos k)(√(−2t sin k + δ3)2 + (2� cos k + δ2)2

)3 .

(44)

Equation (44) is an analytic expression of the curvaturefor the Hamiltonian H (4)(k).

Figure 5 consists of two panels for two different valuesof δ2 and δ3. The upper and lower panels represent theparameter space δ2 = 1, δ3 = 0.5 and δ2 = 0.5, δ3 = 1respectively. Each panel consists of two figures, the leftone is for curvature and the right one is for the corre-sponding parameter space curves. We observe that with

102 of 17 Pramana – J. Phys. (2021) 95:102

ab

cd

Figure 5. (a) Plots of curvature (κ) with k for δ1 = 0.5, (b) the corresponding parameter plots for the curvature plot (a), (c)plots of curvature (κ) with k for δ2 = 0.5 and (d) the corresponding parameter plots for the curvature plot (c). In all the plots,the red, blue and green curves represent t = 2, 1, 0.7 respectively.

increasing value of δ2, the curvature also increases. Itclearly shows the evidence of divergence in the curva-ture plots. H (4)(k) shows that the asymmetry nature isthe same as H (3)(k) Hamiltonian. For H (4)(k) Hamil-tonian, parameter space curves form cycloidal patternbut in a very arbitrary way. There is no specific wayof orientation. The corresponding curvature shows thenon-topological state. Based on the strength of δ2 and δ3,divergence characters arise at the BZ boundary values.The curvature plots show the divergence at BZ boundaryregions, i.e.,−π and π .

In the AIII symmetry class, we have presented twomodel Hamiltonians. Hamiltonian H (3)(k) contains theeffective term in the σz part and Hamiltonian H (4)(k)contains effective term both in the σy and σz compo-nents. Both of these Hamiltonians show distorted curveswhere curvature lacks mirror symmetry about the κ-axis.

Both BDI as well as AIII symmetry classes have dis-tinct geometric properties. Through the curvature study,we can analyse the nature of parameter space, cycloidalmotion of the parameter space with and without the addi-tion of effective term. When the effective term is addedto the σy or σz component of the Hamiltonian, the systemremains in the R2 space and we observe only curvature.But the cycloidal motion of the R2 parameter space isnothing other than the helical motion in the R3 space.Hence, we consider the R3 space to investigate the tor-sional effect of effective term on the model Hamiltonian.

4.3 Results of A symmetry class

Symmetry class A is characterised by the absence oftime-reversal (T ), particle–hole (C) and chiral (S) withthe Hamiltonian 2. Here, the Hamiltonians H (5)(k),H (6)(k), H (7)(k) and H (8)(k) belong to the A class[8]. These Hamiltonians show topologically trivialbehaviour for a one-dimensional system.


H(5)(k) =(

2t cos(k) + μ 2i� sin(k) + δ1k−2i� sin(k) + δ1k 2t cos(k) + μ

).

(45)


χ(1)(H (5)(k)) = δ1k,

χ(2)(H (5)(k)) = 2� sin k,

χ(3)(H (5)(k)) = 2t cos k + μ. (46)


H(k)(5) = χ(1)(H (5)(k))σx + χ(2)(H (5)(k))σy

+χ(3)(H (5)(k))σz. (47)


E (5)(k) =√

(δ1k)2 + (2� sin k)2 + (−2t cos k − μ)2.

Pramana – J. Phys. (2021) 95:102 of 17 102

The parameter space of H (5)(k) belongs to R3 spaceand forms the circular helix as

helix [a, b] (k) = (a cos(k), a sin(k), bk), (48)

where a is the radius and b is the slope of the helix (herefor all the cases we take � = t to achieve unit speedcurve properties). The projection of R3 onto R2 mapsthe helix onto a circle.

Here the curve is

c(k) =⎡⎣ δ1k

2� sin k2t cos k + μ

⎤⎦ , c(k) =

⎡⎣ δ1

2� cos k−2t sin k

⎤⎦ ,

c(k) =⎡⎣ 0

−2� sin k−2t cos k

⎤⎦ (49)

and thus the curvature κ = ||c(k)|| = 2 which repre-sents the non-vanishing curvature. Hence, it is possibleto find normal vector for all values of k. Thus,

n(k) = c(k)

κ(k)= 1

2

⎡⎣ 0


⎤⎦ . (50)

Bi-normal vector is given by

b(k) = c × n(k) =⎡⎣ δ1


⎤⎦ × 1

2

⎡⎣ 0


⎤⎦

= 1

2

⎡⎣ −4t�

−2�δ1 sin k−2tδ1 cos k

⎤⎦ . (51)

The torsion is given by

〈n(k), b(k)〉 =⟨

1

2

⎡⎣ 0

−2� cos k2t sin k

⎤⎦ ,

1

2

⎡⎣ −4t�


⎤⎦

⟩

= t�δ1. (52)

Thus, the curvature as well as the torsion give constantvalues for H (5)(k).

Serret–Frenet equation (eq. (24)) for H (5)(k) Hamil-tonian is

T (k) =⎡⎣ 0


⎤⎦ ,

N (k) = − 2

⎡⎣ δ1


⎤⎦ + t�δ1

2

⎡⎣ −4t�


⎤⎦ ,

B(k) = − t�δ1

2

⎡⎣ 0


⎤⎦ . (53)

Serret–Frenet equation (eq. (24)) for the HamiltonianH (5)(k) gives the understanding about the dynamics ofH (5)(k) Hamiltonian. When the H (5)(k) Hamiltonian isprojected from R3 → R2, one can obtain the H (1)(k)Hamiltonian.

Figure 6 shows the study of curvature as well tor-sion to H (5)(k) Hamiltonian. The left panel indicatesthe parameter space and the right panel indicates thecorresponding curvature and torsion for different valuesof δ1. From the plot it is clear that, by increasing thevalues of δ1, the amplitude of curvature and torsion alsoincrease. Hence, the curvature and torsion are directlyproportional to δ1.


H(6)(k) =(

2t cos(k) + μ 2i� sin(k) + iδ2k + δ1k−2i� sin(k) − iδ2k + δ1k 2t cos(k) + μ

). (54)


χ(1)(H (6)(k)) = δ1k,

χ(2)(H (6)(k)) = 2� sin k + δ2k,

χ(3)(H (6)(k)) = 2t cos k + μ. (55)


H(k)(6) = χ(1)(H (6)(k))σx + χ(2)(H (6)(k))σy

+χ(3)(H (6)(k))σz. (56)


E (6)(k)

=√

(δ1k)2 + (2� sin k + δ2k)2 + (−2t cos k − μ)2.

Here the curve is

c(k) =⎡⎣ δ1k

2� sin k + δ2k2t cos k + μ

⎤⎦ , c(k) =

⎡⎣ δ1

2� cos k + δ2−2t sin k

⎤⎦ ,

c(k) =⎡⎣ 0


⎤⎦ (57)

and thus the curvature κ = ||c(k)|| = 2 which repre-sents the non-vanishing curvature. Hence, it is possible

102 of 17 Pramana – J. Phys. (2021) 95:102

ad

b e

c f

Figure 6. (Left) Parameter plots for the Hamiltonian H (5)(k). (Right) Plots of curvature (κ) and torsion (τ ) with k fort = � = 1 and α = 0, 0.5 and 1. The red and blue lines in the right panel represent the corresponding normal curvatures andtorsion respectively.

to find normal vector for all values of k. Thus,

n(k) = c(k)

κ(k)= 1

2

⎡⎣ 0


⎤⎦ . (58)


b(k) = c × n(k) =⎡⎣ δ1


⎤⎦

×1

2

⎡⎣ 0


⎤⎦

= 1

2

⎡⎣−4t� − 2tδ2 cos k


⎤⎦ . (59)


〈n(k), b(k)〉

=⟨

1

2

⎡⎣ 0


⎤⎦ ,

1

2

⎡⎣−4t� − 2tδ2 cos k


⎤⎦

⟩

= t�δ1. (60)

Thus, the curvature as well as the torsion give constantvalues for H (6)(k). Serret–Frenet equation (eq. (24)) forthe Hamiltonian H (6)(k) can be written as

T (k) =⎡⎣ 0


⎤⎦ ,

N (k) = − 2

⎡⎣ δ1


⎤⎦

+ t�δ1

2

⎡⎣−4t� − 2tδ2 cos k


⎤⎦ ,

Pramana – J. Phys. (2021) 95:102 of 17 102

B(k) = − t�δ1

2

⎡⎣ 0


⎤⎦ . (61)

Thus, Serret–Frenet equation for the Hamiltonian H (6)(k)gives an understanding about the dynamics of H (6)(k)Hamiltonian. When the H (6)(k) Hamiltonian is pro-jected from R3 → R2, one can obtain the H (2)(k)Hamiltonian.


H(7)(k)

=(

2t cos(k) + μ + δ3k 2i� sin(k) + δ1k−2i� sin(k) + δ1k −2t cos(k) − μ − δ3k

).

(62)


χ(1)(H (7)(k)) = δ1k,

χ(2)(H (7)(k)) = 2� sin k,

χ(3)(H (7)(k)) = 2t cos k + μ + δ3k. (63)


H(k)(7) = χ(1)(H (7)(k))σx + χ(2)(H (7)(k))σy

+χ(3)(H (7)(k))σz. (64)


E (7)(k)

=√

(δ1k)2 + (2� sin k)2 + (2t cos k + μ + δ3k)2.

Here, the curve is

c(k) =⎡⎣ δ1k

2� sin k2t cos k + μ + δ3k

⎤⎦ ,

c(k) =⎡⎣ δ1

2� cos k−2t sin k + δ3

⎤⎦ ,

c(k) =⎡⎣ 0


⎤⎦ (65)


n(k) = c(k)

κ(k)= 1

2

⎡⎣ 0


⎤⎦ . (66)


b(k) = c × n(k) =⎡⎣ δ1


⎤⎦

×1

2

⎡⎣ 0


⎤⎦

= 1

2

⎡⎣−4t� − 2�δ3 sin k


⎤⎦ . (67)


〈n(k), b(k)〉

=⟨

1

2

⎡⎣ 0


⎤⎦ ,

1

2

⎡⎣−4t� − 2�δ3 sin k


⎤⎦

⟩

= t�δ1. (68)

Thus, the curvature as well as the torsion give constantvalues for H (7)(k). Serret–Frenet equation (eq. (24)) forthe Hamiltonian H (7)(k) can be written as

T (k) =⎡⎣ 0


⎤⎦ ,

N (k) = − 2

⎡⎣ δ1


⎤⎦

+ t�δ1

2

⎡⎣−4t� − 2�δ3 sin k


⎤⎦ ,

B(k) = − t�δ1

2

⎡⎣ 0


⎤⎦ . (69)

Thu, Serret–Frenet equation for the Hamiltonian H (7)(k)gives an understanding about the dynamics of H (7)(k)Hamiltonian. When the H (7)(k) Hamiltonian is pro-jected from R3 → R2, one can obtain the H (3)(k)Hamiltonian.

4.3.4 H (8)(k)Hamiltonian. Hamiltonian H (8)(k) canbe written in the matrix form as

H(8)(k)

=⎛⎜⎝

2t cos(k) + μ + δ3k 2i� sin(k) + iδ2k+δ1k

−2i� sin(k) + δ1k 2t cos(k) + μ + δ3k−iδ2k

⎞⎟⎠ .

(70)

102 of 17 Pramana – J. Phys. (2021) 95:102


χ(1)(H (8)(k)) = δ1k,

χ(2)(H (8)(k)) = 2� sin k + δ2k,

χ(3)(H (8)(k)) = 2t cos k + μ + δ3k. (71)


H(k)(8) = χ(1)(H (8)(k))σx + χ(2)(H (8)(k))σy

+χ(3)(H (8)(k))σz. (72)


E (8)(k) =√

(δ1k)2 + (2� sin k + δ2k)2 + (2t cos k + μ + δ3k)2.

Here the curve is

c(k) =⎡⎣ δ1k

2� sin k + δ2k2t cos k + μ + δ3k

⎤⎦ ,

c(k) =⎡⎣ δ1

2� cos k + δ2−2t sin k + δ3

⎤⎦ ,

c(k) =⎡⎣ 0


⎤⎦ (73)


n(k) = c(k)

κ(k)= 1

2

⎡⎣ 0


⎤⎦ . (74)


b(k) = c × n(k) =⎡⎣ δ1


⎤⎦

×1

2

⎡⎣ 0


⎤⎦

= 1

2

⎡⎣−4t� − 2tδ2 cos k − 2�δ3 sin k


⎤⎦ . (75)


〈n(k), b(k)〉 =⟨

1

2

⎡⎣ 0


⎤⎦ ,

1

2

⎡⎣−4t� − 2tδ2 cos k − 2�δ3 sin k


⎤⎦

⟩

= t�δ1. (76)

Thus, the curvature as well as the torsion give constantvalues for H (8)(k). By using eq. (24), Serret–Frenetequation for the H (8)(k) Hamiltonian can be written as

T (k) =⎡⎣ 0


⎤⎦ ,

N (k) = − 2

⎡⎣ δ1


⎤⎦

+ t�δ1

2

⎡⎣−4t� − 2tδ2 cos k − 2�δ3 sin k


⎤⎦ ,

B(k) = − t�δ1

2

⎡⎣ 0


⎤⎦ . (77)

Thus, Serret–Frenet equation for the H (8)(k) Hamil-tonian gives an understanding about the dynamics of theH (8)(k) Hamiltonian. When the H (8)(k) Hamiltonian isprojected from R3 → R2 , one can obtain the H (4)(k)Hamiltonian.

Thus, it is very clear that the projection of R3 → R2

space (χ2 − χ3 parameter space) signals the changes inthe geometrical properties of the model Hamiltonian. Inthe R3 space, the Hamiltonian belongs to the symmetryclass A, but when it is projected to R2 space, it belongsto either BDI or AIII symmetry class. It is a very impor-tant point that, under the given conditions, Hamiltoniansbelonging to the symmetry class A show the same cur-vature and torsion. But the Hamiltonians belonging toBDI and AIII symmetry classes have different curvatureexpressions.

4.4 Geodesic properties of the curve for H (5)(k)Hamiltonian

Geodesics are the shortest path between two points in asurface. Geodesics always have a constant speed. Some-times geodesics can be expressed as geodesic curvature(kg). Hence, as a part of curvature study, we consider

Pramana – J. Phys. (2021) 95:102 of 17 102

a unit-speed curve on a circular cylinder which actuallyforms a helix on the surface. It is interesting that at theintersection of a cylinder, the plane perpendicular to itsrulings is always a geodesic. Here we consider H (5)(k)Hamiltonian and calculate the geodesic by geometricaloperations.

Local isometry is the quantity which can give a clearunderstanding about this. For a unit cylinder W with theconditions x2 + y2 = 1, there always exists geodesicwith the circles obtained by intersecting W with planesparallel to the x–y plane. Because of the local isometricproperty, one can connect the points (u, v, 0) of the x–y plane to the points (cos u, sin u, v) of the W plane.This makes a geodesic from x–y plane to the geodesicon W . The line which is not parallel to the y-axis in thex–y plane gives the equation y = mx + c, where m andc are constants. Parametrising the line by x = k andy = mk + c we get c(k) = (cos(k), sin(k),mk + c)which is nothing other than the similar helix consideredin H (5)(k). Here we clearly give the geodesic curve forH (5)(k). Let there be a circular cylinder,

W = {X = (χ1, χ2, χ3) ∈ R3|χ22 + χ2

3 = 1,

χ1 = k, k ∈ R}. (78)

Here, we consider H (5)(k) Hamiltonian with the condi-tion μ = 0, t = � = 1/2 and δ1 = 1. The minimumcondition for a curve c : I → W on W to be a geodesicis that the curve c(k) should be inclined on W . Let thecurve c(k) be a geodesic on the circular cylinder W .

Now

c(k) = dc

dk= V1.

If the angle between V1 and d/dχ3 is φ(k), then forevery k [42],⟨V1,

d

dχ1= cos(φ(k))

⟩. (79)

By taking covariant derivatives with respect to V1⟨DV1V1,

d

dχ1

⟩+

⟨V1DV1,

d

dχ1

⟩= − sin(φ(k))

dφ

dk(80)

or in other words⟨k1V2,

d

dχ1

⟩= − sin(φ(k))

dφ

dk, (81)

where

V2 = c(k)

||c(k)|| , ||c(k)|| = k1.

Then⟨c,

d

dχ1

⟩= − sin(φ(k))

dφ

dk.

Here the curve c(k) is a unit speed curve (undergiven conditions) and a geodesic on the circular heli-cal. Hence, we get c(k) = λN . For a N vector areadefined by Np = (p1, p2, p3, . . . , pn−1 = 0) forp = (p1, p2, . . . , pn) ∈ W is the unit normal vectorarea of W . So,⟨N ,

∂n

∂kn

⟩= sin(φ(k))

dφ

dk= 0. (82)

Now

sin(φ(k)) = 0 ordφ(k)

dk= 0.

So

dφ(k)

dk= 0 �⇒ φ(k) = 0 or φ(k) = constant.

This shows that the curve is an inclined curve with d/dχ1as the axis on the circular cylinder W .

In other words,⟨V1,

dV1

dχ1

⟩= cos(φ(k)), φ(k) �= π/2 (φ = constant).

(83)

Hence,⟨k1V2,

d

dχ1

⟩= 0.

The covariant derivative with respect to V1 is⟨dV1

dk,

d

dχ1

⟩= 0 �⇒

⟨N ,

d

dχ1

⟩= 0. (84)

It shows 〈c(k), c(k)〉 = 0 and N , c(k) = 0, where

N = λd

dχ1∧ c(k)

c = βd

dχ1∧ c(k). (85)

Then c(k) = βN , which clearly shows that the inclinedcurve is a geodesic under a given parameter space.

Here, we consider just H (5)(k) Hamiltonian undersome particular parameter space to calculate thegeodesics. We choose the parameter space in such away that the curve c(k) remains unit-speed. In otherHamiltonians, it is not possible to achieve unit-speedcurve. And we consider unit cylinder with the condi-tion x2 + y2 = 1. This case is only possible in H (5)(k)Hamiltonian. When the effective term is added to eitherσy or σz , the curve fails to be a unit-speed curve. As thiscondition is not possible in other Hamiltonians, we onlycalculate geodesic curvature for H (5)(k) Hamiltonian.

102 of 17 Pramana – J. Phys. (2021) 95:102

4.5 Consequences of effective term and its physicalinterpretation

The differential geometric analysis of the parameterspace makes us understand the nature of Hamiltoniansof different symmetry classes. This effort successfullyexplains the curvature study of the parameter space withthe addition of effective term αk and the transition ofsystem from topological to topologically trivial phase.Curvature and torsion are integral parts of a geometri-cal system and one can understand the physical systemin a better way by this study. In the study of space–time geometry, mass is responsible for curvature andspin is responsible for torsion [25]. In the same way, forour present model, the dependence of momentum vec-tor k in terms ‘sine’ and ‘cosine’ is responsible for thecurvature and effective term αk is responsible for thetorsional effects. The cycloidal motion in a R2 space isa cycloid when it is projected to a R3 space and a unit-speed cycloid in a R3 space is a unit-speed circle whenit is projected to a R2 space. This helps us to under-stand the relation between the geometry and physics ofa quantum condensed mater system.

When the same analogy is applied to a lattice model,the initial Hamiltonian H0(k) represents a tight-bindingmodel and the effective term δi k represents the externalinteraction term which is linear momentum (in somecases it is similar to magnetic field). Because of thenature of the effective term, it gives rise to torsion inthe lattice system. So it results in the curve opening ofparameter space and cycloidal motion.

For the tight-binding models, this type of torsionresults in dislocations and disclinations [25]. It is sim-ilar to the disorder and defect in the crystal lattices. Inour Hamiltonians, the periodicity of the Bloch spacebreaks and the system transforms from topological tonon-topological phase. This transformation is the resultof torsion. Even though the system transforms fromtopological to trivial phase, the model remains in therespective symmetry classes (BDI, AIII and A).

5. Conclusion

We have presented entirely new and insightful results ofcurvature analysis for different symmetry classes, eachsystem class containing different Hamiltonians with dif-ferent topological properties. We have shown explicitlythe merits and limitations of curvature study in the pres-ence of effective term. We have analysed the behaviourof the system from topological to non-topological statewith the addition of effective term to the model Hamil-tonian. We have shown explicitly the presence of mirrorsymmetry for the curvature study of BDI symmetry

class but that symmetries are absent for the AIII andA symmetry classes. We have introduced the concept oftorsion in topological state of matter, thereby explainedthe transformation of the system from topological tonon-topological state and we observed a transformationof symmetry classes, when there is a projection from R3

space to R2 space. We have given the geodesic proper-ties of certain Hamiltonian under given conditions. Thiswork provides a new perspective on the curvature anal-ysis for the topological state of matter.

Acknowledgements

SS would like to acknowledge DST (EMR/2017/

000898) for the funding and RRI library for the booksand journals. YRK would like to thank Admar MuttEducation Foundation for the scholarship. The authorswould like to acknowledge Dr R Srikanth, Dr B SRamachandra and Prof. C Sivaram for reading thismanuscript critically and for giving useful suggestions.This research was supported in part by the InternationalCentre for Theoretical Sciences (ICTS) during a visit forparticipating in the program – Geometry and Topologyfor Lecturers (Code: ICTS/gtl2018/06).

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A study of curvature theory for different symmetry classes ...

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