A Differential Geometric study of the Auxiliary Space of Interacting Topological System
Y R Kartik, Rahul S, Ranjith Kumar R, Amitava Banerjee, Sujit Sarkar
AA Differential Geometric study of the Auxiliary Space ofInteracting Topological System
Y R Kartik,
1, 2
Rahul S,
1, 2
Ranjith Kumar R,
1, 2
Amitava Banerjee, and Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar Bangalore-560 080, India. Manipal Academy of Higher Education, Madhava Nagar, Manipal, 576104, India. Department of Physics, University of Maryland College park, MD 20742, USA. (Dated: October 18, 2018)Differential geometry serves as an important tool in different branches of theoreticalphysics. Here we present a differential geometric treatment of curvature in the aux-iliary space for the non-interacting and interacting systems and also the symmetryaspect of it. We present the necessary and sufficient conditions to characterize thetopological nature of curve in the auxiliary space. We present the results both for therepulsive and attractive regimes of the parameter space. We try to picture the entireproblem from the geometric means of curvature. We characterize the curvature inthe auxiliary space, in terms of elliptical and cycloidal motion for the non-interactingand interacting system respectively. We also try to explain the topological aspectsof physics in terms of Berry connection curves. To the best of our knowledge, this isthe first application of differential geometry to the topological state of matter. Thisstudy gives a new perspective for the understanding topological state of a quantumsystem. a r X i v : . [ c ond - m a t . o t h e r] O c t Introduction
Curves and angles are the effective way of expressing the geometric properties of a physi-cal system. By this way we can define curvature, nature of wave function, periodicity andtopological properties of the system [1–3]. It is more interesting to analyze the nature ofthese properties in the presence of some interaction. The geometrical studies of the con-densed matter systems have been an interesting area of research which has rapidly picked upthe pace when the principles of topology and geometry played its foundations in quantumcondensed matter systems [4, 5]. Here in this paper, the topological behavior of the modelHamiltonian are studied from the perspective of auxiliary space, complex variable analysisand the curve theory of differential geometry.
Definition of curvature:
Curvature is a tool to measure how curved a curve is. In otherwords, curvature measures the extent to which a curve a deviates from a straight line. Fora unit speed curve γ ( t ) , where t is a parameter, curvature κ ( t ) at a point γ ( t ) is defined tobe || ¨ γ ( t ) || [6]. The main motivation is to explain the many body system in a much efficientmanner. A many body system is difficult to analyze by usual theoretical calculation. Butby simulating the system to a normal geometric structure we can explain the phenomenonslike phase transition, geometric phase, divergence in a much efficient manner. The topologi-cal configuration space of a system gives rise to the particular value of topological invariantquantity like winding number. The closed curves in the configuration space are the auxiliaryspace curves which also specifies the winding number of the system [7, 8]. Auxiliary spacecurves has a unique way of representing the topological quantum phase transition. Whenthe system is in the topological state, the auxiliary space curve encircles the origin. When itis in the non-topological state, the origin lies outside the space curve. At the point of phasetransition, the origin lies on the space curve at which case the topological invariant numbercannot be defined [9]. We will realize in this study that it is necessary but not the sufficientone. Here the auxiliary space curve is use to study the topological phase transition of thesystem both in presence and absence of the interaction.Berry’s phase is the physical entity which characterizes the topological state of the quantumsystem [7]. Berry phase exists, where there is closed interval. But Berry connection is amore fundamental quantity which tells the path of evolution of ground state wave function.Berry connection is equivalent to the vector potential in electromagnetic theory. In the sameway Berry connection is also a curve, we would like to study the similarity and differencebetween the curves in the auxiliary space curves for the different Hamiltonians.Differential geometry deals with the study of problems by means of differential calculus,integral calculus and linear algebraic techniques [10–12]. Differential geometry has a signifi-cance in general theory of relativity. The concept of manifold, curved space-time, gravity isexplained much efficiently [13–15]. Here in this manuscript we tried to analyze the role ofdifferential geometry in the topological state of matter. To the best knowledge this is thefirst study of differential geometry for the topological state of matter. Understanding the topological state state of matter
Topological insulators are the materials which conducts along the edge [16] and is an in-sulator at the bulk. The conduction along the edge is protected by two major symmetries[17–19], time reversal and particle conservation symmetry. The first experimentally real-ized 3D topological insulator state with symmetry protected surface states was discoveredin bismuth-antimony [20]. Topological Quantum Phase Transition points (TQPT) are notaccompanied with the spontaneous symmetry breaking just like Quantum Phase Transi-tions(QPT)(local order parameter). Topological invariants for topological quantum statesare protected by the non zero energy gaps, where the band gap closes at some particularpoints in the Brillouin Zone (BZ). Hence the phase transition is ill defined at this point.To understand the phase transition phenomenon, we can study winding number, which tellshow many times the Hamiltonian winds the TQPT point. Hence we can get the differencebetween the phases before and after TQPT point. The phase acquired during this evolutionis the geometric phase. For the 1D it is Zak phase [21–25]. To study the effect of interactionson the topological invariant number and topological quantum phase transitions, we applymomentum dependent interaction to our model Hamiltonian [26]. When it is applied totopological states.
Aim of the study:
To study the nature of auxiliary curves from the differential geomet-ric curvature perspective and the Berry connection for better insight into the topologicalsystem. The interaction which we propose has a physical reliability that it preserve theHermitian property of the Hamiltonians. Differential geometry is very wide and powerfulmathematical branch which has a wide application in many branches of theoretical physics[13, 27–31]. Here Our study is two folded. First one is to study the behavior of auxiliaryspace for the interactive state of system. Second is to analyze upto what extent we canuse differential geometry to study the topological properties of curves in auxiliary space ofsystem.This paper is organized in following manner. First we introduce the model Hamiltonian andthe present a detail analysis of auxiliary space curves. The characteristics and behavior ofauxiliary space curves with and without interaction is presented here. In the second part westudy Berry connection for the model Hamiltonian under different conditions. And we triedto analyze the behavior of Berry connection for topological and non-topological state of sys-tem. In the third part we give a detail analysis of differential geometric study of curvatureto the auxiliary space curves. I. Hamiltonians Under Consideration and Nature of TheirAuxiliary Space
Auxiliary space or parametric space is the set of possible combination of values of parameterscontained in the given system [9]. Here by plotting the parameters of the system with respectto each other we can get better understanding of the system [32]. Here in particular caseour system of consideration is Kitaev Hamiltonian. By this parametric plots we can get theidea about closeness of the curve, phase transition condition, behavior of the system underinteraction and other informations.
A. Basic Model Hamiltonian:
We consider the Kitaev’s chain as our model Hamiltonian [26], H = [ (cid:88) n − t ( c n † c n +1 + h.c ) − µc n † c n + | ∆ | ( c n c n +1 + h.c )] , (1)where t is the hopping matrix element, µ is the chemical potential and | ∆ | is the magnitudeof the superconducting gap. We write the Hamiltonian in the momentum space as, H = (cid:88) k> ( µ + 2 t cos k )( ψ k † ψ k + ψ − k † ψ − k ) + 2 i ∆ (cid:88) k> sin k ( ψ k † ψ − k † + ψ k ψ − k ) , (2)where ψ † ( k )( ψ ( k )) is the creation (annihilation) operator of the spin less fermion of momen-tum k .We can write the Hamiltonian in the BdG format as H BdG ( k ) = χ (1) ( k ) iχ (2) ( k ) − iχ (2) ( k ) − χ (1) ( k ) . (3)We can express the Hamiltonian by Anderson Pseudo spin approach [8, 32, 33]. One canwrite the BdG Hamiltonian in the pseudo spin basis. (cid:126)H ( k ) = χ (1) ( k ) (cid:126)y + χ (2) ( k ) (cid:126)z ⇒ H BdG ( k ) = Σ i (cid:126)χ i ( k ) .(cid:126)τ i , Where τ ’s are the Pauli spin matrices which act in the Nambu basis of H BdG .Here the set of parametric equations are, χ (1) ( H ( k )) = − t cos k − µ, χ (2) ( H ( k )) = 2∆ sin k.H BdG in the pseudo spin basis is, H ( k ) = χ (1) ( H ( k )) (cid:126)y + χ (2) ( H ( k )) (cid:126)z. (4)The energy dispersion relation, E ( k ) = (cid:112) (2 t cos k + µ ) + (2∆ sin k ) . The defined model Hamiltonian is the non-interacting Hamiltonian. Now we consider theHamiltonian in presence of interactions. We consider a very specific type of interaction forthe complete theoretical interest. The results of this study may trigger quantum simulationscientist to find this type of interaction and nature of their results.We observe that Kitaev Hamiltonian in pseudo spin basis consists of two components. Inthe present study we consider the interaction only on each components separately ( H (1) and H (2) ) and also for the presence of interaction in both components for the Hamiltonian H (3) . B. Effect of interaction on the topological state of system:
When an interaction term is added to the system, the resultant Hamiltonian can be writtenas, H = H + H I , where H is the model Hamiltonian and H I is the interaction term.Here interaction term is momentum dependent, which takes the value H I = αk , where α is the strength of the interaction. By introducing such interaction term to the componentsof the Hamiltonian, we can investigate the behavior of the system which reveals muchinformation about the system. The effect of interaction has a greater significance in thequantum simulation and this is the test bed for understand the many-body system [34].1) Kitaev model Hamiltonian in presence of the interaction, αk , which is added to the σ x component of the Hamiltonian is written in terms of Pauli basis as, H k = ( (cid:15) k − µ − αk ) σ z −
2∆ sin kσ y . (5)Presenting the Hamiltonian in matrix from as, H (1) ( k ) = − t cos( k ) − µ − αk i ∆ sin( k ) − i ∆ sin( k ) 2 t cos( k ) + µ + αk . (6)Here the set of possible parametric equations are, χ (1) ( H (1) ( k )) = − t cos k − µ − αk, χ (2) ( H (1) ( k )) = 2∆ sin k.H BdG in the pseudo spin basis is, H (1) ( k ) = χ (1) ( H (1) ( k )) (cid:126)y + χ (2) ( H (1) ( k )) (cid:126)z. (7)The energy dispersion relation can be written as, E (1) ( k ) = (cid:112) (2 t cos k + µ + αk ) + (2∆ sin k ) .2) Kitaev model Hamiltonian in presence of the interaction αk which is added to the σ y component can be written in terms of Pauli basis as, H k = ( (cid:15) k − µ ) σ z − (2∆ sin k + αk ) σ y . (8)Writing the Hamiltonian in the matrix form, H (2) ( k ) = − t cos( k ) − µ i ∆ sin( k ) + iαk − i ∆ sin( k ) − iαk t cos( k ) + µ . (9)Here the set of parametric equations are, χ (1) ( H (2) ( k )) = − t cos k − µ, χ (2) ( H (2) ( k )) = 2∆ sin k + αk.H BdG in the pseudo spin basis is, H (2) ( k ) = χ (1) ( H (2) ( k )) (cid:126)y + χ (2) ( H (2) ( k )) (cid:126)z. (10)The energy dispersion relation, E (2) ( k ) = (cid:112) (2 t cos k + µ ) + (2∆ sin k + αk ) .
3) The model Hamiltonian in presence of the interaction terms α k and α k which are addedto both the σ x and σ y components of the Hamiltonian, can be written in terms of Pauli basisas, H k = ( (cid:15) k − µ + α k ) σ z − (2∆ sin k + α k ) σ y . (11)The Hamiltonian H (3) ( k ) written in the matrix form as, H (3) ( k ) = − t cos( k ) − µ − α k i ∆ sin( k ) + iα k − i ∆ sin( k ) − iα k t cos( k ) + µ + α k . (12)Here the set of possible parametric equations are, χ (1) ( H (3) ( k )) = − t cos k − µ − α k, χ (2) ( H (3) ( k )) = 2∆ sin k + α k.H BdG in the pseudo spin basis is, H ( k ) = χ (1) ( H (3) ( k )) (cid:126)y + χ (2) ( H (3) ( k )) (cid:126)z. (13)The energy dispersion relation, E (3) ( k ) = (cid:112) (2 t cos k + µ + α k ) + (2∆ sin k + α k ) . C. A study of Hermiticity of the Hamiltonians.
Here we check the Hermiticity condition H † = ( H ∗ ) T for our Hamiltonians. H ( k ) = (cid:15) k − µ ) + 2 i ∆ sin k ( (cid:15) k − µ ) − i ∆ sin k , ( H ∗ ( k )) T = (cid:15) k − µ ) − i ∆ sin k ( (cid:15) k − µ ) + 2 i ∆ sin k T . (14) H ( k ) † = (cid:15) k − µ ) + 2 i ∆ sin k ( (cid:15) k − µ ) − i ∆ sin k = ⇒ H ( k ) = H † ( k ) . (15)For the First case, H (1) ( k ) = H (1) † ( k ) = (cid:15) k − µ − α ) + 2 i ∆ sin k ( (cid:15) k − µ − α ) − i ∆ sin k . (16) For the second case, H (2) ( k ) = H (2) † ( k ) = (cid:15) k − µ ) + 2 i ∆ sin k + iαk ( (cid:15) k − µ ) − i ∆ sin k − iαk . (17) For third case, H (3) ( k ) = H (3) † ( k ) = (cid:15) k − µ − α k ) + 2 i ∆ sin k + iα k ( (cid:15) k − µ − α k ) − i ∆ sin k − iα k . (18) We observe that all the model Hamiltonians obey the condition for Hermiticity. The additionof the interaction term does not affect the Hermitian property of the system. Basically theKitaev Hamiltonian is in the spin less fermion basis. The interaction added is momentumdependent and is also a spinless fermion term. Therefore we justify the physical relevanceof the interaction. (1). Hamiltonian H ( k ) In the fig. 1 the auxiliary space curves for the Kitaev Hamiltonian is presented using theparametric equations of the Hamiltonian. This gives the simple closed Jordan curve [35].
FIG. 1. Parametric plot for the Hamiltonian H ( k ) for different values of µ . (2). Hamiltonian H (1) ( k ) In the fig.2, the auxiliary space curve for the Hamiltonian H (1) ( k ) is presented. The auxiliaryspace curves shows an interesting behavior where, if the interaction is absent, the curveremains closed resembling the Kitaev chain’s auxiliary space curves. When the interactionis present, the auxiliary space curve no longer closed.The auxiliary space curves resembles the cycloidal pattern since the mathematical structureof the equations of cycloid and the Hamiltonian H (1) ( k ) are the same. The general expressionof the cycloid is given by [12], Cyc [ a, b ]( t ) = ( at − b sin t, a − b cos t ) . (19)The Hamiltonian H (1) ( k ) has a mathematical structure of cycloid (Please see eq.6 andeq.19). In general the cycloid is classified into two categories depending on the values ofcoefficients. Suppose in eq.19,if a < b , then the cycloid is prolate and if a > b , it is curate.From this classification, we can assign our Hamiltonian H (1) ( k ) , as prolate since the prolatecycloid is self-interacting. and also it satisfies the condition a < b . An interesting featurewhich can be observed is that, the right column is a mirror symmetric to the left column,which is due to the change in the sign of the value of α . FIG. 2. Parametric plot for the Hamiltonian H (1) ( k ) for different values of µ , α and t . (3). Hamiltonian H (2) ( k ) The auxiliary space curve of the Hamiltonian H (2) ( k ) is presented in the fig. 3.0 FIG. 3. Parametric plot for the Hamiltonian H (2) ( k ) for different values of µ , α and t . Fig. 3 represents the auxiliary space curves of the Hamiltonian H (2) ( k ) . It is observedeven here that, if the interaction is turned off, the curve becomes auxiliary space curvesi.e., simple Jordan curve as we denote for Kitaev model. In the presence of interactionterm, the auxiliary space curve no longer remains closed, it will show a standard cycloidalpattern. The left column and the right column are plotted for positive and negative valuesof α . Unlike in the Hamiltonian H (1) ( k ) case, the right column is not mirror symmetric withrespect to the left column. (4). Hamiltonian H (3) ( k ) The auxiliary space curve of the Hamiltonian H (3) ( k ) is presented in the fig. 4 and fig. 5. FIG. 4. Parametric plot for the Hamiltonian H (3) ( k ) for different values of µ , α and t . Here both α and α are opposite in sign. FIG. 5. Parametric plot for the Hamiltonian H (3) ( k ) for different values of µ , α and t . Here both α and α are opposite in sign. The curve starts to align on X-axis for small values of α and aligns on the Y-axis for thesmall values of α . The shape of the auxiliary curve depends on the strength of the α and α terms. D. An analysis from the perspective of complex variable
For a complex number z ( t ) = x ( t ) + iy ( t ) where a ≤ t ≤ b , if x ( t ) and y ( t ) are the continuousfunction of t , then the curve z ( t ) is said to be continuous. If x ( t ) and y ( t ) are differentiable,then z ( t ) is differentiable. If two curves z ( t ) (cid:54) = z ( t ) , then it is a simple curve. If t (cid:54) = t for t ∈ [ a, b ] except at z ( b ) = z ( a ) is called simple closed curve. A contour is formed by a finitenumber of connected smooth arcs. If a curve or arc C is non self intersecting is called simpleJordan arc . If it self intersects, then it is called simple closed curve or Jordan curve 1. Forthe present study the parametric equations satisfies the condition of Jordan curve. In thepresence of interaction, the curve gets open, there are some possibilities of curve being notsimple not closed type [35]. We now observe that the curve in the auxiliary space belongs tothe simple closed curve and the open curve. Therefore we conclude that the curves for thenon-interacting Kitaev chain is the simple closed curve and the curves for the interactingKitaev chain are the not simple not closed curves.2 E. Analyzing mirror symmetrical aspects of auxiliary space curves
For the Kitaev Hamiltonian, when the interaction term is added, the periodicity of the Bril-louin zone breaks. In the presence of interaction, the elliptic auxiliary space curve is nolonger closed but instead it forms a cycloidal pattern [12]. So with the addition of interac-tion term, the closed curve condition is deformed into open curve condition. The cycloidalstructure in the fig. 2, the left column curves are mirror symmetric to the right column,where in the right column the curves are plotted for negative values of α . In the presenceof interaction, the σ y component of the Hamiltonian, the orientation of the helical springstructure is along k -axis. When the sign of the interaction term α changes from positive tonegative, the rotation of the helical spring changes from clockwise to anticlockwise. Wheninteraction term is added to both the σ x and σ y components of the Hamiltonian, the orien-tation of the helical spring structure is tilted and its angle with X/Y axis depends on thestrength of the interaction terms.i.e., α and α II. Study of Berry connection for the model Hamiltonian
For a quantum mechanical eigenstate | n ( λ ) (cid:105) in a adiabatic process, which is evolving as afunction of parameter k, like | n ( λ ( k )) (cid:105) , where n ( λ ) is the non-degenerate eigenstate. Duringthis process, apart from the dynamical phase, the wave function acquires an additional geo-metric phase. This depends on the path evolved during the process [36–39]. Berry connectiondefined as ( A ( n ) µ ) ab = (cid:104) n, b | ∂ µ | n, a (cid:105) . Under the gauge transformation Berry connection givesthe relation as A ( R ) −→ g − ( R ) A ( R ) g ( R ) − ig − ( R )∆ g ( R ) [40],By using Stokes theorem, one can connect the Berry phase and Berry connection. By usingthe analytical relation, γ n = (cid:72) c A n ( R ) .dR over the closed contour C. The resulting Berryphase will be π or integral multiples of π . If the contour is not closed, Berry phase won’texists, because gauge dependency is not present. The the concept of Berry phase arisesnaturally from Berry connection [41–43]. Berry Connection is also a curve in the parametricspace that gets modified and shows its behavior for different Hamiltonians. In otherwords,Berry connection tells about the rate of change of wavefunction in the parameter space.If the wavefunction is not varying with respect to parameter, then the Berry connectionvanishes [44]. When we study auxiliary space curves, we observe that the closed curve is3transforming to a open cycloidal curve with the addition of interaction. To analyze thisnature of auxiliary space, we need to understand the behavior of wavefunction. So, here weare interested in the study of Berry connection. At the same time we are interested to knowhow far its behavior in case resembles with the curvature in the auxiliary space. The studyof Berry connection with the variation of k gives the idea about the state of the system. Forthe Kitaev Hamiltonian, at topological state condition the plot is symmetric and periodic.If the system is loosing topological properties means, the periodicity and symmetry of thesystem will be violated as shown in the following study.When the interaction is added to the model Hamiltonians, H (1) ( k ) , H (2) ( k ) and H (3) ( k ) ,the Berry connection changes remarkably. When interaction added, the periodicity of thek-space gets broken and the auxiliary space curve shows the cycloidal motion instead ofclosed curve. Here the Berry connection gives the idea of evolution of wave function in theperiodic and non-periodic k-space and the behavior of the wave function in the neighboringlattice sites. Based on the strength of α , we can study Berry connection with varying k.For the Hamiltonian H (1) ( k ) , the interaction term is added to the σ x component of theHamiltonian. The study of Berry connection with k is shown in fig. 7. Here the changingthe of sign of interaction term, we can observe the mirror symmetry in the curves. As theinteraction is added to σ x , the σ y does not shows any variation. But when the interactionis added to the σ y component fig. 8,9,10, we can observe, the breaking of mirror symmetrywhen the sign of interaction term is changed. The symmetry and periodicity of the k-spacebreaks in different manner and the behavior of the wave function is different at neighboringlattice sites. Hence the study of Berry connection with k gives better understanding of thesystem. - - - k - - - - - - A μ = = Δ = - - - k - - - A μ = = Δ = - - - k - - A μ = = Δ = FIG. 6. The variation of Berry connection for Kitaev Hamiltonian (H) with k. The upper figurerepresents the topological case( µ < t ). The middle figure represents transition state ( µ = 2 t ) andthe lower figure represents non topological state ( µ > t ). k = 0 . We have obtained the Berry connection curve for the topological quantum phasetransition as a smooth curve. Otherwise it is piecewise continuous. For the topologicaland non-topological states, it always has a finite value. Without the interaction term Berryconnection is a mirror symmetric curve. Corresponding auxiliary space curves are alsomirror symmetric. For the topological and non topological states we can find divergence atboundary values of BZ (i.e., − π and π ) but they are periodic in nature. A. Study of Berry connection for Hamiltonian H (1) ( k ) Here we consider the H (1) ( k ) model Hamiltonian. When the interaction term αk is addedto the σ x component of the Hamiltonian. The resulting Berry connection is presented in fig.7. - - - k - A μ = =- Δ = α = - - - k - A μ = =- Δ = α =- - - - k - - A μ = = Δ = α = - - - k - - A μ = = Δ = α =- - - - k - - A μ = = Δ = α = - - - k - - A μ = = Δ = α =- FIG. 7. The variation of Berry connection with k of the Hamiltonian H (1) ( k ) . The figures contain of three panels, Upper, middle and lower one. The value of µ variesBerry connection significantly. As we increase the value of chemical potential, the divergingnature transforms from negative potential area to positive potential area. The left panelshows the variation of Berry connection with k for positive values of α considered as repulsivepotential. Here the curve don’t follow any symmetry. In all three cases, the system remainsin the non-topological state. The right panel shows the variation of Berry connection fornegative values of α considered as attractive potential. Here, too, the system remains in the5non-topological state.We observe a similarity with the Berry connection curve with the auxiliary space curve thatboth of the curve touches the origin at k = 0 , one is simple closed Jordan curve and anotherone is non-simple non-closed. The system always remains in the non-topological state forthe Hamiltonian H (1) ( k ) . B. Study of Berry connection for Hamiltonian H (2) ( k ) Here we consider the H (2) ( k ) model Hamiltonian. When the interaction term αk is addedto the σ y component of the Hamiltonian. The resulting Berry connection is presented in fig.8. - - - k - - - - A μ = = Δ = α = - - - k - - - - A μ = = Δ = α =- - - - k - A μ = = Δ = α = - - - k - - A μ = = Δ = α =- - - - k - - A μ = = Δ = α = - - - k - A μ = = Δ = α =- FIG. 8. The variation of Berry connection with k of the Hamiltonian H (2) ( k ) . For the fig. 8 which represents the variation of Berry connection with respect to k for theHamiltonian H (2) ( k ) , here we can find the symmetry with k . But there is no symmetrybetween the plots for the positive and negative values of α . From this study we can under-stand that the presence of symmetry about the axis is necessary for the topological states,but not the sufficient. C. Study of Berry connection for Hamiltonian H (3) ( k ) Here we consider the Hamiltonian H (3) ( k ) . In this case the interaction term αk is added toboth σ x and σ y components of the Hamiltonian. The Berry connection study for this caseis given in fig. 9,10.6 - - - k - A μ = = Δ = α = α = - - - k - - - A μ = = Δ = α =- α =- - - - k - - - A μ = = Δ = α = α = - - - k - - - - A μ = = Δ = α =- α =- - - - k - A μ = = Δ = α = α = - - - k - A μ = = Δ = α =- α =- FIG. 9. The variation of Berry connection with k of the Hamiltonian H (3) ( k ) . In fig. 9, the left panel shows the Berry connection for the repulsive potential. Rightpanel shows the Berry connection for the attractive potential. In both the cases there is nosymmetry behavior of the curves either with k or with the negative value of α . The cycloidalmotion loses its periodicity. - - - k - A μ = = Δ = α = α =- - - - k - - - A μ = = Δ = α =- α = - - - k - - - A μ = = Δ = α = α =- - - - k - - - - A μ = = Δ = α =- α = - - - k - A μ = = Δ = α = α =- - - - k - A μ = = Δ = α =- α = FIG. 10. Berry connection plots for Kitaev Hamiltonian with interaction term α added to the σ x and σ y component. By the addition of interaction to the Kitaev Hamiltonian, the Berry connection looses itsperiodicity and becomes piecewise continuous. Berry phase is absent for these conditions.Hence we can show how Brillouin zone looses periodicity through the study of Berry con-nection.In all cases of Berry connection we can study how the addition of interaction is affecting theperiodicity of the auxiliary space. But Berry connection won’t give any idea about the phase7transition and the reason for the loss of periodicity. But Berry connection gives idea aboutthe variation of wavefunction in the parameter space, the divergence of the potential at theBrillouin Zone boundary values and about the existence of phase transition points. Here wecan not do the curvature study for Berry connection curves. Because Berry connection cannot be expressed in the form of eq. III C
D. A common comparison between the auxiliary space curves and the Berryconnection of the Hamiltonians
As we can observe in the fig. 1, the auxiliary space curve for the Kitaev chain is closed andshows topological, non-topological and transition state precisely depending on the value of µ . Similarly in the fig. 6, the upper panel represents the Berry connection of Kitaev chain inthe topological state. But here there is no closed curve as we observe in the auxiliary space.The only similarity is that at the transition point Berry connection touches the origin ofthe k-space. Similarly arguments can be put forth for the Hamiltonian H (2) ( k ) and H (3) ( k ) since also the auxiliary space curves for these Hamiltonians are not closed. Hamiltonians H (1) ( k ) , H (2) ( k ) and H (3) ( k ) does not exhibit the topological state except Kitaev chain. III. A differential geometric analysis of curves in auxiliary space
Now we discuss very briefly the basic aspects of curvature that have different geometry for thecompleteness of this study. One of our main motivation of this study is to find the topologicalstate of system in terms of differential geometry at all possible or in otherwords what are thecurvature properties of the auxiliary space curves. Curvature can be defined as the rate ofvariation of the angle that the tangent line is making at a particular point. To call a curveas a regular curve, it should have a non vanishing tangent line. Curve theory basically dealswith analyzing the basic properties of the curves. Basic properties include, the arc length,winding number with curvature and torsion of the curves. Topological invariant quantities,like winding number, Chern number depend on the topology of the auxiliary space, where fora particular topological configuration space, winding number acquires a definite value, andchange in the winding number leads to the different topological configuration of the system.The understanding of the curve concept is simplified by using the differential geometry tool8called curvature κ . The relation which relates the parameterized curve c ( t ) and the curvature κ ( t ) is given by [35], κ ( t ) = det ( ˙ c ( t ) , ¨ c ( t )) || ˙ c ( t ) || . (20)For a unit speed curve γ : I −→ R where I = [ a, b ] a closed curve interval. Then γ (cid:48) ( t ) gives the velocity vector defined by (cos θ ( t ) , sin θ ( t )) T of an integer multiple of π , as thecurve is defined in a closed interval. As the angle changes with curve, the invariant quantitywinding number is defined by θ ( b ) − θ ( a ) . If θ , θ : I −→ R satisfies the velocity equation.It results as θ = θ + 2 kπ , where k ∈ Z .The velocity term γ (cid:48) ([ a, b ]) ⊂ S R , i.e., γ (cid:48) ( t ) > for all t ∈ I and γ (cid:48) ( t ) = ( γ (cid:48) , γ (cid:48) ) T , γ (cid:48) (2) γ (cid:48) (1) = sin θ ( t )cos θ ( t ) = tan θ ( t ) . And θ ( t ) = arctan( γ (cid:48) ( t ) γ (cid:48) ( t ) ) + 2 kπ, k ∈ Z . So considering γ : R −→ R be a unit speed vector of a curve with period L and θ : R ←− R be scalar and windingnumber is given by, n γ = 12 π ( θ ( L ) − θ (0)) . (21)where ( θ ( L ) − θ (0)) is well defined irrespective of the choice if θ . Therefore it is clear fromthe above equation that to get a complete physical picture of winding number, the study ofcurve is useful. A. An analytical approach to find the closed curve condition in auxiliary space
We have, H (3) ( k ) = ( − t cos k − µ + α k ) σ x + ( α k −
2∆ sin k ) σ y . (22)We plot the parametric plot ( x ( k ) , y ( k )) , x ( k ) = − t cos k − µ + α k = r ( k ) cos θ ( k ) , y ( k ) = α k −
2∆ sin k = r ( k ) sin θ ( k ) . (23)so that, in the auxiliary plane, r ( k ) = ( − t cos k − µ + α k ) + ( α k −
2∆ sin k ) , θ ( k ) = tan − (cid:20) α k −
2∆ sin k − t cos k − µ + α k (cid:21) . (24)9To have a closed curve for k running between [ − π, π ] the curve must come back at its startingpoint, i.e, r ( k = π ) = r ( k = − π ) , θ ( k = π ) = θ ( k = − π ) mod (2 π ) (25)putting these two conditions in the expression of r ( k ) and θ ( k ) , ( − t cos( − π ) − µ + α ( − π )) +( α ( − π ) −
2∆ sin( − π )) = ( − t cos( π ) − µ + α ( π )) +( α ( π ) −
2∆ sin( π )) (26)and, tan − (cid:20) α ( − π ) −
2∆ sin( − π ) − t cos( − π ) − µ + α ( − π ) (cid:21) = tan − (cid:20) α ( π ) −
2∆ sin( π ) − t cos( π ) − µ + α ( π ) (cid:21) . (27)Thus the conditions for the curve to be closed, eq. 26 and eq. 27 can be simultaneouslysatisfied if α = α = 0 . For this condition, the curve become close and the system is in thetopological state. We prove here the necessary and sufficient condition for the topologicalstate of matter, that the origin of the auxiliary space is not only covered by the curve butalso the curve should be closed. We have already bean studied in fig 1,2 and 3, the curvesin the auxiliary space encircles the origin but the curve itself is not closed and finally as aconsequence of it, the system is in the non-topological state. B. An analysis of auxiliary space curves from perspective of Cauchy-RiemannIntegral
The winding number of a closed curve C about a point z j in the complex plane is given as[35], w z j = 12 πi (cid:73) dzz − z j (28)We also have the theorem that if f ( z ) is a mesomorphic function define inside and on asimple closed contour C, with no zeros or poles on C, then, πi (cid:73) f (cid:48) ( z ) f ( z ) dz = No of zeros of f inside C- No of poles of f inside C0We note that, if we define f ( z ) = z − z j , then the integrals in above equations are the same,so, w z j = Number of zeros ( z − z j ) inside C (cid:124) (cid:123)(cid:122) (cid:125) =1 if z j is included inside C=0 otherwise − Number of poles of ( z − z j ) inside C (cid:124) (cid:123)(cid:122) (cid:125) =0 as ( z − z j ) has no poles = 1 iff z j is included inside CThis explains why the winding number calculated without origin is 1 for a Kitaev chain ifthe curve in the auxiliary space encloses the origin. The above definition of w is valid onlyfor a closed contour C, hence cannot be applied if the curve is not closed in the auxiliaryspace.The value w ( z j ) defined as w ( z j ) = 12 πi (cid:73) dz ( z − z j ) = ( 12 πi )[ log ( z − z j )] c = ( ∆ θ j π ) (29)is called the winding number of the curve C around point z j . Here, ∆ θ j is the change in theargument of ( z − z j ) , where z traverses through the curve C and the point z j . The value w ( z j ) represents the number times that C winds around z j . In presence of interaction, thecurve is not closed, in one sense ∆ θ j is not π or integral multiple of π . Therefore theconcept of winding number, open and self intersecting curve is not valid here. C. A study of differential geometry from the perspective of curvature theory for theauxiliary space curve
Ellipse is generally defined as locus of points such that sum of distances from the foci isconstant. The standard equation of ellipse is given by, x a + y b = 1 , where a and b are semi-major and semi-minor axes. The parametric equation is given by [ a ( t ) , b ( t )] = ( a cos t, b sin t ) where ≤ t < π .1 - - - a = = - - - t κ a =
3, b = FIG. 11. The graphical representation of an ellipse (Left). The curvature of the ellipse for thecorresponding values
The curvature expression of ellipse is [11], κ = ab ( b cos t + a sin t ) . (30)The two parameters a and b in the curvature expression of ellipse are namely semi-majoraxis and semi-minor axis. From these two parameters we can analyze the curvature in threecases.First case: When a < b , the curvature is maximum on the semi-major axis ( − π and π ) andit is minimum on the semi-minor axis.Second case: When a = b , the auxiliary space curve is a circle with the constant curvature.Third case: When a > b , the curvature is minimum on the semi-major axis ( − π and π ) andit is maximum on the semi-minor axis [12]. An analysis of Cycloidal Properties of auxiliary space curves for interactingsystem
The general expression of a cycloid is given by [12], C [ a ( t ) , b ( t )] = ( at − b sin t, a − b cos t ) . (31)Depending on the location of the locus point on the circle the cycloid motion is devised intodifferent category. When a and b are equal it gives a cycloid which describes the locus ofpoint on a circle of radius a [12]. If a and b are not equal and if a < b then the cycloidis called prolate. Suppose a > b then cycloid is called curate and it is represented in the2figure.12. ( a ) a = = ( b ) a = = ( c ) a = = FIG. 12. Cycloidal motion of the locus on the circle at different conditions (a)a=b, (b)ab.
Curvature expression of Cycloid is, κ = ab cos t − b ( a − b cos t ) + b sin t ) . (32)The addition of interaction term alters the pattern of the cycloidal motion. (1). Kitaev model Here we present the results of study of differential geometry based on curve theory forthe non-interacting and interacting Hamiltonians. The matrix form of the Kitaev modelHamiltonian is, H ( k ) = − t cos( k ) − µ i ∆ sin( k ) − i ∆ sin( k ) 2 t cos( k ) + µ . (33)Considering the parametric equation of the Hamiltonian H ( k ) in the matrix form, c ( k ) = − t cos k − µ
2∆ sin k , ˙ c ( k ) = t sin k
2∆ cos k , ¨ c ( k ) = t cos k −
2∆ sin k . (34)Curvature is given by, κ = det [ ˙ c, ¨ c ] || ˙ c || = ⇒ det t sin k t cos k
2∆ cos k −
2∆ sin k ( √ t sin k + 4∆ cos k ) = − t ∆2( √ t sin k + ∆ cos k ) . (35)3Eq. 35 is an analytical expression of curvature for the Kiteav Hamiltonian. The cur-vature plot of the Kitaev Hamiltonian is presented in the fig. 13. The auxiliary spacecurve for the Kitaev chain is nothing but an ellipse (fig.1) since the parametric equationsof the Hamiltonian ( − t cos k − µ,
2∆ sin k ) resembles the mathematical form of the ellipse [ a ( t ) , b ( t )] = ( a cos t, b sin t ) where ≤ t < π . FIG. 13. The left panel represents the plots of curvature with k for the values t=2, 1, 0.7 fromtop to bottom respectively. The right panel represents corresponding auxiliary plots for the value µ = 0 For the topological state, the curvature is smooth and touches the k = 0 point on the k-axis.For the transition state, there is no curvature and the plot is parallel to the k-axis. For thenon-topological state, the curvature does not touches the k = 0 and forms a gaped state. Auxiliary space curve and curvature of Kitaev chain:
The parameterized equations of the Kitaev Hamiltonian are, ( − t cos k − µ,
2∆ sin k ) andthe analytic expression of curvature for the Kitaev chain gives the mathematical structureof the eq. 35, we conclude that the curvature plot of the standard ellipse and the curvatureof the Kitaev chain are same.For the value µ = 0 , the system remains in the topological state. We can study the curvatureof auxiliary space curve for all Hamiltonians. We can not characterize the topological andnon-topological states of the Hamiltonian from the curvature study. The reason for this is,the curvature expression does not include the term µ . From the above general discussion4on the ellipse we can characterize the auxiliary space curve of the Kitaev chain into similarthree cases. This is completely a theoretical exercise to show the nature of auxiliary spacecolumn from the perspective of differential geometry.First case: When t < ∆ , the curvature is maximum on the semi-major axis ( − π and π )and it is minimum on the semi-minor axis.Second case: When t = ∆ , the auxiliary space curve is a circle with the constant curvature.Third case: When t > ∆ , the curvature is minimum on the semi-major axis ( − π and π ) andit is maximum on the semi-minor axis. D. Results and discussions of curvature in the auxiliary space for the interactingHamiltonians: (2). Results of differential geometry studies for the Hamiltonian H (1) ( k ) The matrix form of the Hamiltonian H (1) ( k ) can be written as, H (1) ( k ) = − t cos( k ) − µ − αk i
2∆ sin( k ) − i
2∆ sin( k ) 2 t cos( k ) + µ + αk . (36)The Curvature of the Hamiltonian H ( k ) is given by, κ = det t sin k + α t cos k
2∆ cos k −
2∆ sin k ( (cid:112) (2 t sin k + α ) + 4∆ cos k ) = − t ∆ − α sin k ( (cid:112) (2 t sin k + α ) + (2∆ cos k ) ) . (37) - - - k - - - - κ ( a ) t = Δ = α = - π - π π π ( b ) t = μ = α = - - - k - - - κ ( c ) t = Δ = α = - π - π π π ( d ) t = μ = α = FIG. 14. The left panel represents the plots of curvature with k for the values t=1, α = 0 . , fromtop to bottom respectively. The right panel represents corresponding auxiliary plots for the value µ = 0 - - - k - - - - κ ( a ) t = Δ = α =- - π - π π π ( b ) t = μ = α =- - - - k - - - κ ( c ) t = Δ = α =- - π - π π π ( d ) t = μ = α =- FIG. 15. The left panel represents the plots of curvature with k for the values t=1, α = − . , − from top to bottom respectively. The right panel represents corresponding auxiliary plots for thevalue µ = 0 For the H (1) ( k ) Hamiltonian, the auxiliary space curve forms cycloidal motion. The corre-sponding curvature plots a deviation from topological states of system, these plots resemblesthe non-topological curvature patterns.The auxiliary space curves is a cycloid in the presence of interaction since the Hamiltonian H (1) ( k ) acquire mathematical structure of cycloid equation. We notice that the presenceof interaction changes the properties of differential geometry which we study the curvatureproperties in the auxiliary space. Based on the strength of the interaction term, the auxiliaryspace curve behaves as simple curve with non closed, self intersecting conditions. When theinteraction term changes its sign, the auxiliary space curves as well as curvature plots formsmirror symmetric image. (3). Results of differential geometry studies for the Hamiltonian H (2) ( k ) Hamiltonian H (2) ( k ) can be written in the matrix form as, H (2) ( k ) = − t cos( k ) − µ i ∆ sin( k ) + iαk − i ∆ sin( k ) − iαk t cos( k ) + µ . (38)Curvature is given by, κ = det t sin k t cos k
2∆ cos k + α −
2∆ sin k ( (cid:112) (2 t sin k ) + (2∆ cos k + α ) ) = − t ∆ − αt cos k ( (cid:112) (2 t sin k ) + (2∆ cos k + α ) ) . (39)Eq. 39 is the analytical expression of the curvature for the Hamiltonian H (2) ( k ) .6 FIG. 16. The left panel represents the plots of curvature with k for the values t=1, α = 0 . , . from top to bottom respectively. The right panel represents corresponding auxiliary plots for thevalue µ = 0 FIG. 17. The left panel represents the plots of curvature with k for the values t=1, α = − . , − . from top to bottom respectively. The right panel represents corresponding auxiliary plots for thevalue µ = 0 Fig 17 is not mirror symmetric. The part to be noted is that the maxima for the curvaturefor the positive values becomes minimum and visa versa. The curvature contains two minima7and one maxima for the repulsive potential. For the attractive potential, it contains oneminima and two maxima.As the previous case, the curvature expression is independent of the term µ . The increasein the strength of the interaction term results in decrease of curvature near k = 0 plane.For the Hamiltonian H (2) ( k ) , the auxiliary space curve is also a prolate cycloid because it isopen self-intersecting. The parametric equation of the Hamiltonian H (2) ( k ) is the following: H (2) ( k ) = ( − t cos( k ) − µ,
2∆ sin( k ) + iαk ) . (40)From the curvature studies for this auxiliary space curve of Hamiltonian H (2) ( k ) , it revealsthe following: The curvature at the points ( − π and π ) on the semi-major axis is maximumand the curvature on the semi-minor axis is minimum.When the interaction term changesits sign, the auxiliary space curves as well as curvature plots forms mirror symmetric image. (4). Results of differential geometry studies for the Hamiltonian H (3) ( k ) Hamiltonian H (3) ( k ) can be written in the matrix form as, H (3) ( k ) = − t cos( k ) − µ − β k i ∆ sin( k ) + iβ k − i ∆ sin( k ) − iβ k t cos( k ) + µ + β k (41)Curvature is given by, κ = det t sin k + α t cos k
2∆ cos k + α −
2∆ sin k ( (cid:112) (2 t sin k + α ) + (2∆ cos k + α ) ) = − t ∆ − α ∆ sin k + α t cos k )( (cid:112) (2 t sin k + α ) + (2∆ cos k + α ) ) (42)Eq.42 is an analytic expression of the curvature for the Hamiltonian H (3) ( k ) .8 - - - k - - κ t = Δ = α = α = - π - π π π t = μ = α = α = - - - k - - κ t = Δ = α = α = - π - π π π t = μ = α = α = FIG. 18. Left panel represents the plots of curvature with for different values of α and α . Rightpanel represents corresponding auxiliary space curves. - - - k - - κ t = Δ = α =- α =- - π - π π π t = μ = α =- α =- - - - k - - κ t = Δ = α =- α =- - π - π π π t = μ = α =- α =- FIG. 19. Left panel represents the plots of curvature with for different values of α and α . Rightpanel represents corresponding auxiliary space curves. For the negative α , variation of curvature with k is not mirror symmetric as like positive one.The point to be noted that the shape of the curve becomes reverse. There is no divergencein the auxiliary space curves but the curvature shows the divergence behavior.For H (3) ( k ) Hamiltonian, auxiliary space curves forms cycloidal pattern but in a very ar-bitrary way. We can not find any particular orientation. The corresponding curvatureshows the non-topological state. Based on the strength of α and α there arises divergencecharacters at the BZ boundary values.9 - - - k - - κ t = Δ = α = α =- - π - π π π t = μ = α = α =- - - - k - - κ t = Δ = α =- α = - π - π π π t = μ = α =- α = FIG. 20. Left panel represents the plots of curvature with for different values of α and α . Rightpanel represents corresponding auxiliary space curves. For the H (3) ( k ) Hamiltonian, the curvature study shows the diverging behavior. The cur-vature diverges at BZ boundary values. Here curvature is not continuous. The sign changeof the interaction term does not alter the nature of curvature but the divergence happensin the opposite region. The curvature plots shows the divergence at BZ boundary regionsi.e., − π and π . Conclusion:
We have presented topological aspects of the Kitaev model in the presenceand absence of interaction through differential geometric and complex variable method.The nature of curves, nature of Berry connection and the auxiliary space in presence ofinteraction have discussed. We have derived the necessary and sufficient conditions for thetopological characterization of the system through the study of the auxiliary space. Wehave also studied the Berry connection explicitly and its consequences in topological andnon-topological states.
Acknowledgments
The authors would like to acknowledge DST (EMR/2017/000898) for the funding andRRI library for the books and journals. The authors would like to acknowledge Prof. RSrikanth who has read this manuscript critically. Finally authors would like to acknowledge0ICTS Lectures/seminars/workshops/conferences/discussion meetings of different aspects ofphysics. [1] Pelham Mark Hedley Wilson.
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