Circuit Complexity From Supersymmetric Quantum Field Theory With Morse Function
CC IRCUIT C OMPLEXITY F ROM S UPERSYMMETRIC Q UANTUM F IELD T HEORY W ITH M ORSE F UNCTION
Sayantan Choudhury , ‡ , § , Sachin Panneer Selvam , K. Shirish , National Institute of Science Education and Research, Bhubaneswar, Odisha - 752050,India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai -400085, India Department of Physics, Birla Institute of Technology and Science, Pilani, HyderabadCampus, Hyderabad - 500078, India Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, 440010, India
Abstract
Computation of circuit complexity has gained much attention in the Theoretical Physicscommunity in recent times to gain insights about the chaotic features and random fluctua-tions of fields in the quantum regime. Recent studies of circuit complexity take inspirationfrom the geometric approach of Nielsen, which itself is based on the idea of optimal quan-tum control in which a cost function is introduced for the various possible path to determinethe optimum circuit. In this paper, we study the relationship between the circuit complex-ity and Morse theory within the framework of algebraic topology using which we studycircuit complexity in supersymmetric quantum field theory describing both simple andinverted harmonic oscillators up to higher orders of quantum corrections. The expressionof circuit complexity in quantum regime would then be given by the Hessian of the Morsefunction in supersymmetric quantum field theory, and try to draw conclusion from theirgraphical behvior. We also provide a technical proof of the well known universal connectingrelation between quantum chaos and circuit complexity of the supersymmetric quantumfield theories, using the general description of Morse theory.Keywords: Circuit Complexity, Supersymmetric QFT, Morse Theory, AdS/CFT. ‡ Corresponding author, E-mail : [email protected], [email protected] § NOTE: This project is the part of the non-profit virtual international research consortium “QuantumAspects of Space-Time & Matter” (QASTM) . a r X i v : . [ h e p - t h ] J a n ontents Introduction Circuit Complexity for dummies Lie algebra formulation of the Inverted Harmonic Oscillator (IHO) Brief review of Morse Theory Definition of Morse function
Gradient flow of Morse function Circuit complexity in SUSY QFT via Morse function Effect on circuit complexity from IHO perturbation theory
Effect on circuit complexity from SHO perturbation theory Circuit complexity for φ term Circuit complexity for φ term Comparative Analysis SHO
IHO Graphical Analysis SHO
IHO Quantum Chaos from Morse function
Conclusions i Introduction
AdS/CFT correspondence has helped in providing great insights on the geometry ofthe bulk from information in the boundary CFT [1–6]. However, probing the physicsof behind the horizon is still a major challenge. It has been inspected that even if theentangled entropy of an eternal AdS black hole saturates after reaching the equilibrium,the size of the Einstein-Rosen(ER) bridge continues to grow with time, posing a problemto the dual description of boundary CFT. Considering this Susskind has introduced newobservables in bulk geometry [7–13]. One is the volume of a maximal co-dimension-onebulk surface extending to the boundary of AdS space-time and the second is the actiondefined on the Wheeler-De-Witt patch. These new observables, according to the conjectureis a dual description of complexity of the boundary field theory.One of these crucial observables is volume , which according to “
Complexity = Volume ”conjecture, states that volume V ( B ) of a maximal co-dimension-one bulk surface B thatextends to the AdS boundary and asymptotic to the time slice (cid:80) is proportional to thecomplexity of the boundary state, C V ( (cid:80) ), which is given by: C V (cid:16)(cid:88)(cid:17) = max (cid:20) V ( B ) G N l (cid:21) . (1.1) The second observable is the gravitational action estimated in the Wheeler-De-Witt(WDW) patch in the bulk region I W DW , which according to the another important con-jecture, namely the “
Complexity = Action ” conjecture; is proportional to complexity ofthe boundary field theory [14], given by: C A = I W DW π (cid:126) . (1.2) Complexity in quantum field theory and various other quantum system has been theattraction of theoretical physics community in recent times, because it not only providesinsights about what happens inside the horizon in a gravity dual, but also helps in thestudy the sensitive dependence of initial conditions of boundary fields. In other words, ithelps in understanding the out-of-time-order-correlation (OTOC) function of observablesin a quantum system which depicts the chaotic and random of quantum fields[15]. OTOCwithin the framework of supersymmetric quantum mechanics has been studied in the sim-ple harmonic oscillator, one dimensional infinite potential well and various other modelsin [16] and also the computation of complexity for non-supersymmetric quantum field the-ories has been done in earlier works [16]. The main idea was to put the free scalar fieldtheory on a lattice, which reduces it to the family of coupled oscillators, then identifying1he circuit as a path ordered exponential of Hamiltonian which forms the representation of GL (2 , R ), and then construct the euclidean metric. Minimizing the length of this metricwould finally give the expression for circuit complexity. These results were then obtainedin the framework of the inverted harmonic oscillator and interacting field theories to lookfor chaotic behavior in quantum field theories(QFT). In this paper, our main objective isto calculate circuit complexity for supersymmetric quantum field theories for simple andinverted harmonic oscillators for various higher-order quantum corrections. To do this wewill take an unusual approach and first try to bring out the connection between complexityin quantum field theories namely, supersymmetric field theories with the Morse function.In general Morse theory acts as an essential tool to study the topology of manifolds bystudying the differentiable functions through which we could identify the critical pointson that manifold namely the minima, maxima, and saddle points. To study the connec-tion between circuit complexity in supersymmetric quantum field theories with the Morsefunction, we will first identify the so-called “cost function”, an important parameter incomputing the circuit complexity which counts the number of gates operating at a partic-ular time t , to construct the optimal circuit with the Morse function on a given manifold.By computing the Hessian of the Morse function we obtain the expression for circuit com-plexity in the present context of discussion. Our motivation for doing so lies around thefact that the eigenvalues of supersymmetric Hamiltonian are closely concentrated near thecritical points of the Morse function and the number of zero eigenvalues of the groundstate is exactly equal to the Betti number of the manifold. In this paper, we will showhow the increase in the number of critical points of the manifold captures the amount ofchaos present in supersymmetric quantum field theories for the inverted harmonic oscilla-tor (IHO) and how the mathematical structure of supersymmetry (SUSY) in the regimeinverted harmonic oscillators whose potentials forms the generators of SL (2 , R ) group un-derlies chaos, and makes the number of critical points increase by a factor of exponentialwith respect to the superpotential appearing in supersymmetric quantum field theory. Forthe scalar field theories in the regime where it can be described as a simple harmonic oscil-lator (SHO), we will identify the non-dynamical auxiliary field namely the F - term in thelagrangian of SUSY theory involving scalar fields, with the gradient of the Morse function,which passes through every critical point on the manifold. In turn we will show that thecomplexity of supersymmetric quantum field theories for simple harmonic oscillator onlydepends on the absolute value of the non-dynamical auxiliary field, which also acts as anorder parameter for supersymmetry breaking. This way identifying the F - term with thegradient of the Morse function provides another way to check when the supersymmetryis broken. If the gradient flow passes through the critical points, then it could be saidthat there exists no ground states with zero energy or SUSY is spontaneously broken.Following these lines we will then compute the Hessian in the space of super-coordinatesand will derive the expression for complexity. In this connection, it is important to knowthat Edward Witten has shown that supersymmetric quantum field theory is basically a2odge Derham cohomology and has derived the Morse inequality using the formalism ofsupersymmetry, however, in this paper, we will not work along the same lines but our mainobjective would be to give quantum chaos a topological flavor in supersymmetric theories.At present, Out-of-time ordered correlation (OTOC) function has been a very importanttheoretical tool to capture the effect of chaos in any quantum system at late time scales.By following this very important notion, in this paper, we will explicitly give a technicalproof of the universal relation relating circuit complexity with the quantum chaos in termsof the previously mentioned Out-of-Time Ordered Correlation Function (OTOC) withinthe framework of supersymmetric quantum field by using the general description of Morsetheory.Organization of the paper is as follows: • In section 2, we provide a brief review of the concept of circuit complexity in thegeneral context of quantum information theory. • In section 3, we will explain the Lie algebra formulation and how the potential ofInverted Harmonic Oscillator (IHO) could be embedded in the structure of the man-ifold. • In section 4, we will give a brief review of Morse theory namely the Morse functionand the gradient flow of the Morse function for completeness. • In section 5, we will comment on the relationship between circuit complexity andMorse function over a manifold in supersymmetric quantum mechanics. • In section 6, we will explicitly compute the expression of circuit complexity for theInverted Harmonic Oscillator (IHO) upto higher order of quantum corrections. • In section 7, we will compute the complexity for supersymmetric field theories forsimple harmonic oscillators up to higher orders of quantum corrections in terms ofnon-dynamical auxiliary field. • In section 8, we will compare the results of complexity between supersymmetric andnon-supersymmetric quantum field theory for SHO and IHO in table 8.1 and 8.3. • In section 9, we will provide the detailed numerical and graphical analysis of the.obtained theoretical results for both SHO and IHO using the prescription of super-symmetric Morse quantum field theory. • In section 10, we will derive the universal relation between the circuit complexity andquantum chaos expressed in terms of OTOC function for supersymmetric quantumfield theory prescription using Morse function.3
Finally section 11, we will conclude our result and comment on how the behavior ofcomplexity for the supersymmetric model is already implemented in the structure ofa manifold. Circuit Complexity for dummies
The notion of circuit complexity was first introduced in information theory to find theminimum number of gates to get the desired state from an initial state. It involves actingthe initial state with a unitary operator or a set of quantum gates to obtain the desiredtarget state. | ψ T (cid:105) = U | ψ i (cid:105) (2.1) There exists many such set of unitary operators to get the desired final state but to finda minimum number of such operations to execute the task is what constitutes an optimalquantum circuit. Taking inspiration from this Nielsen and collaborators have developedthis idea further and have taken a geometric approach to finding the most optimal quantumcircuit to get the desired target state via unitary operations in physics [17–22], U ( t ) = P exp (cid:18) − i (cid:90) t H ( t ) dt (cid:19) where H ( t ) = (cid:88) Y I ( t ) M I , (2.2) where M I are the Pauli matrices and Y I ( t ) are referred to as the control function thatdecides the nature of gate that will act at a certain value of parameter t. This approach ofidentifying the action of quantum gates via control function could be morphed in terms offinding the extremal curves, i.e. geodesics. Hence we define a cost function F ( U ( t ) , ˙ U ( t ))which is a function of the unitary operator U and a vector at a point in the tangent spaceformed by unitaries. The idea is to minimize this cost function for various possible paths,which is described by the following expression: D ( U ( t )) = (cid:90) (cid:16) F ( U ( t ) , ˙ U ( t ) (cid:17) dt (2.3) The task now is to determine the cost function which as described above counts thenumber of gates to construct the optimal quantum circuit. A general class of cost functionsare: F p ( U, Y ) = (cid:88) p I | Y I | , (2.4) q ( U, Y ) = (cid:113)(cid:88) q I ( Y I ) , (2.5) where minimizing F would give us the expression for the length of geodesic of a Riemannsurface. This technique uses the concepts of differential geometry to construct the optimalquantum circuit and comment on the complexity of various models in theoretical physics. Lie algebra formulation of the Inverted Harmonic Oscillator (IHO)
To work out the circuit complexity in various models researchers have considered asimple, exactly solvable system known as the Inverted Harmonic Oscillator (IHO), whichis described by the Hamiltonian function: H ( p, x ) = p − ω x . (3.1) This encapsulates the sensitivity to initial conditions exhibited by the chaotic systems [23–25]. The Inverted Harmonic Oscillator (IHO) differs from the Simple Harmonic Oscillator(SHO) in a lot of ways. For instance, the energy spectrum of IHO has a continuousenergy spectrum, whereas the regular SHO has a discrete energy spectrum ( n + ) (cid:126) ω .Unlike SHO, the IHO is exactly solvable and while the SHO provides a good descriptionof the deviations from the stable equilibrium, the IHO models the decay from an unstableequilibrium. The IHO in recent times has been proved to be incredibly useful to showthe equivalence among various diverse fields. The IHO appears in the quantum hall effectand in the mechanism of Rindler Hamiltonian, whose time evolution would give rise to theHawking-Unruh effect [26–28]. The equivalence between the two phenomena can be shownin terms of the isomorphism of the underlying lie algebra [29, 30].In this paper, our main objective will not be to formally describe the lie algebra isomor-phism but instead using that to calculate the circuit complexity over a manifold via Morsefunction for supersymmetric quantum field theories. To do this, first we have to show howthe potential of IHO forms the generators of Lie algebra. We start with the settings ofthe quantum hall effect (QHE), the three quadratic potentials generate the Hamiltoniandynamics in the lowest Landau level (LLL): P LLL V P LLL = λ ( R x + R y ) , (3.2) P LLL V P LLL = λ ( R X R Y + R Y R Y ) , (3.3) P LLL V P LLL = λ ( R x − R y ) . (3.4) P LLL V P LLL = − λl B (cid:126) P LLL LP LLL , (3.5) P LLL V P LLL = 2 λl B P LLL j i P LLL , (3.6) P LLL V P LLL = − λl B (cid:126) P LLL j j P LLL . (3.7) where P LLL is the projection operator to the lowest Landau level, R X , R Y are bilinearsand j a , j b are strain generators. Therefore by projecting the potentials to the LLL bothbilinears and the strain generators lead to the quadratic Hamiltonian which is similar tothe electrostatic potential and IHO appears in quantum hall effect as might expected.We now rename the three quadratic potentials in a quantum hall effect V , V , V as K , K , K and we identify: P = R x l B , X = R y l B . (3.8) Consequently, we get the Hamiltonians in the LLL to be of the following form: K = ( P + X ) , (3.9) K = ( P X + X P ) , (3.10) K = ( P − X ) . (3.11) On the basis of LLL wave functions the potentials could be written the form of differentialoperators namely: K = 14 (cid:18) − ∂ ∂X + X (cid:19) , (3.12) K = i (cid:18) X ∂∂x + 12 (cid:19) , (3.13) K = 14 (cid:18) − ∂ ∂X − X (cid:19) . (3.14) These are precisely the generators of SL (2 , R ) Lie-algebra[31], which act as an area preserv-ing deformations of a two dimensional manifold and satisfy the following non-commutingrelations: [ K , K ] = − iK , (3.15)[ K , K ] = iK , (3.16) K , K ] = iK . (3.17) With these techniques at hand, our main objective will now be to describe a Morsefunction on a manifold formed by the Lie-algebra of SL (2 , R ) namely the potential of theIHO, and show, how the critical points of Morse function on a manifold actually play therole of the cost function, which decides the gate in action at a particular parameter t toform the optimal circuit. Brief review of Morse Theory
In this section, we are going to give a very brief review of Morse theory. The Morsefunction helps in the classification of surfaces upto homeomorphism when it passes criticalpoint of index 0 , , . The index 2 , , and 0 represents the maximum, minimum and saddlepoints of the manifold. Definition of Morse function
To define Morse function, we consider a manifold M and a function f such that: Theorem 1:
A smooth map f : X n −→ R is a Morse function if, for every critical point p ∈ X , ∃ coordinates x , ...x n , and a coordinate y around f ( p ) w.r.t. which, f ( x , ...., x n ) = − x − x − .... − x k + x k +1 + ... + x n . (4.1) such that the value of the function at the critical points vanishes. Theorem 2:
A map f −→ R is a morse function if its critical points vanishes and the Hessian of f at each critical point is non-singular .A Morse function on a compact manifold X helps to determine it’s topology, by mappingit’s critical points to an axis on a one-dimensional plane, which helps to encode a lot ofinformation about M [32]. The goal of the Morse theory is to find the invariant of themanifold by counting the critical points of chosen Morse function. Gradient flow of Morse function
The gradient flow of the Morse function creates a vector field on the surface of themanifold, which helps to define a notion of transport from one point to another. Let us7uppose we have an integral curve γ x : R −→ M such that: γ x ( t ) = φ t ( x ) , φ ( x ) = 0 . (4.2) Here φ is a smooth one-parameter group of diffeomorphism on M . Then, we get: ddt f ( γ x ( t )) = ddt ( f ◦ φ t ( x ))= df φ t ( x ) ◦ ddt φ t ( x )= df φ t ( x ) ( −∇ f ) φ t ( x )= −|| ( ∇ f ) φ t ( x ) || ≤ . (4.3)Thus the gradient flow of f is found to be decreasing down the lines of γ x . One of thereasons why we have defined the gradient flow of the Morse function will become clearin later part of this paper, where it is going to play a very crucial part in deriving thecomplexity of supersymmetric field theories for the SHO. Circuit complexity in SUSY QFT via Morse function
To compute circuit complexity within the framework of supersymmetric quantum fieldtheory using Morse function, we will consider SL (2 , R ) modular curve which encodes thepotential of IHO as generators of Lie algebra and identify it with a Riemann manifold M , and the Morse function to the cost function as described in the previous section, andthereby associate the critical points to the action of the control function which decides,which quantum gate will be active at a particular time t .The supersymmetric operators in terms of an exterior derivative and its adjoint can bedescribed as: Q = d + d ∗ , (5.1) Q = i ( d − d ∗ ) , (5.2) H = dd ∗ + d ∗ d. (5.3) The connection between supersymmetric quantum field theories and Derham operatorscould found in the ref. [33]. Then further by taking into account the following crucial fact: d = 0 = d ∗ , (5.4) we get the following subsequent supersymmetric relations in the present context, whichare given by: Q = Q = H, (5.5) Q + Q Q = 0 . (5.6) Now let us consider a Morse function f on the surface of the manifold M , and t be areal number. Then, d t = exp( − ht ) d exp( ht ) , (5.7) d ∗ t = exp( − ht ) d ∗ exp( ht ) . (5.8) Here we can show that: d t = 0 = d ∗ t , (5.9) using which we get the following expressions: Q t = d + d ∗ , (5.10) Q t = i ( d t − d ∗ t ) , (5.11) H t = d t d ∗ t + d ∗ t d t . (5.12) We will now explicitly calculate the formula for H t in terms f , to understand howcritical points come into the picture. Let, v k ( p ) and v k ∗ be an orthonormal basis oftangent vectors and the corresponding dual vectors at each point p in M . The a k and a k ∗ could be regarded as creation and annihilation operators in the present context. Wecould calculate the covariant second derivative of Morse function f in the dual basis of v k as (cid:18) D Dx i Dx j (cid:19) f , with these accords, one could then calculate the Laplacian operator H t acting on p forms on manifold M : H t = dd ∗ + d ∗ d + t ( df ) + (cid:88) t (cid:18) D Dx i Dx j (cid:19) [ v ∗ i , v j ] . (5.13) Here we define ( df ) by the following expression: ( df ) = g ij (cid:18) dfdx i (cid:19) (cid:18) dfdx j (cid:19) . (5.14) which is the square of the gradient of the Morse function f , measured with respect to theRiemannian metric g ij of M . Here the term t ( df ) plays the role of IHO potential, andalso the critical points for the Morse function f lies, where we have df = (cid:115) g ij (cid:18) dfdx i (cid:19) (cid:18) dfdx j (cid:19) = 0 . (5.15) F ( Q ( t ) , ˙ Q ( t )) −→ f. (5.16) The role of F ( Q ( t ) , ˙ Q ( t )) is to count the number of gates required to construct the optimalquantum circuit, but in the present contest it is to find the minimum of supersymmetriccharges to get: Q | E n (cid:105) bosonic = E n − ibosonic . (5.17) One could see in the above mentioned argument, that eigenfunction of H t for large t , areconcentrated near the critical points of f therefore for the identification, we could say thatthe eigenvalues of the cost function are actually the critical values of SL (2 , R ) Riemannsurface.The eigenvalues of H t acting on p forms for large t can be expressed in the followingway: λ np ( t ) = t (cid:18) A np + B nt t + C np t + ... (cid:19) . (5.18) Thus we could see that for large t the above equation, with in few orders agrees withthe computations of out-of-time-order-correlations (OTOC) from supersymmetric quantummechanics computed in [16].Now to calculate the general expression for circuit complexity, we will make use of theHessian matrix. Remember a function f is a morse function if and only if the Hessian of f at each critical point is non-singular. This can be described as: H ( f ( x )) = j ( ∇ f ( x ))( H f ) ij = ∂ f∂x i ∂x j . (5.19) we will now identify Hessian of f in the tangent space of unitary operator as: T p ( M ) × T p ( M ) −→ R. (5.20) such that we have: dU ( s ) ds = − iY Is = ( H f ) ij , (5.21) where Y Is is the control function, which counts the number of operators acting on thereference state to make the optimal circuit.Consequently, we compute: D ( Q )( s ) = (cid:112) ( Y ) + ( Y ) + ( Y ) + ( Y ) , (5.22) C ( Q ) = | H ( f ( x )) | = | j ( ∇ f ( x ))( H f ) ij | = (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) . (5.23) Effect on circuit complexity from IHO perturbation theory
To explicitly calculate circuit complexity for IHO in a supersymmetric case, we will makeuse of the concept of Witten index and show how critical point near which the eigenvaluesof H ( t ) are concentrated grows exponentially with respect to the superpotential.The excited states of supersymmetric quantum field theory always come with pair ofstates, this could be seen in the algebra of supersymmetry which has no one-dimensionalrepresentation [34]: { Q, Q ∗ } = 2 H such that Q = Q ∗ = 0 . (6.1) The Witten index Tr[( − F e − βH ], where ( − F is the well known fermion number oper-ator, carries interesting information about the ground state of supersymmetric quantumsystem especially when it is non-zero, i.e. the system has at least one ground state whenthe Witten index is non-zero, however, it doesn’t make any comment on the number ofground states of the system when it is zero [35]. We will take the Hamiltonian of thesupersymmetric quantum system in IHO regime by replacing W ( x ) by iW ( x ) and showhow the critical points of Morse function or complexity grow exponentially with respect tothe superpotential, which is given by: H ( P, W ) = P − W . (6.2) have a ground state wave function, which is defined as: ψ = exp (cid:18) − iσ (cid:90) ∞∞ W dx (cid:19) . (6.3) Now by doing supersymmetry on the two dimensional manifold the Witten index can beexpressed by the following expression [36]:
Tr[( − F exp( − βH )] = χ ( M ) . (6.4) where χ ( M ) is the Euler characteristic of the manifold, thereby using the strong Morse11nequality, we further get: χ ( M ) = Tr[( − F exp( − βH )] = (cid:88) ( − F exp( − βH ) = (cid:88) ( − γ C γ . (6.5) where C γ is the number of critical points of index γ , using which one can find out thefollowing simplified expression: C γ = (cid:12)(cid:12)(cid:12)(cid:12) ( − F − γ exp (cid:18) − β (cid:18) P − W (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (6.6) Circuit complexity for higher-order interacting term could be calculated in terms of super-field formalism Φ, in which all super-partners related by SUSY transformations could betreated as a single field. In this paper we will restrict ourselves to N = 1 supersymmetry,such that we will have only one fermionic annihilation and creation operator and therebyonly one supercharge[37–39]. Scalars and fermions related by supersymmetry correspondto different components of super-field. The most general form of super-fields in terms ofsuper-space variables can be expressed as follows: Φ( z ) = φ ( x ) + θψ + ¯ θ ¯ χ ( x ) + θσ µ ¯ θA µ ( x ) + θ F ( x ) . (6.7) Now, to write the super-field in terms of any single field component, we will apply theSUSY transformations by applying the operator i ( ξQ + ¯ ξQ ) under the projection: θ = ¯ θ = 0 , (6.8) Expressing the result in terms of other components: δ ξ φ ( x ) = i ( ξD + ¯ ξ ¯ D )Φ | θ =¯ θ =0 = − ξψ ( x ) − ¯ ξ ¯ χ ( x ) , (6.9) where D is a super-covariant derivative of the super-field which anti-commutes with thesupersymmetric charges Q and ¯ Q and under transformations maps super-field to super-field, given by: δ ξ D α Φ( z ) = D α δ ξ Φ( z ) = i ( ξQ + ¯ ξ ¯ Q ) D α Φ( z ) (6.10) They satisfy the same algebra as supersymmetric charges, thus the scalar component φ ( x ),the super-field can be recovered by exponentiating supersymmetry transformations with θ as the parameter: Φ = exp ( − δ θ ) φ ( x ) , (6.11) Thus, the super-fields can be constructed by applying the operator exp ( − δ θ ) to any com-12onent field. Hence under the previously mentioned projection θ = ¯ θ = 0 the complexityfor higher order interactions in the super-potential could be written is as follow: Complexity for φ term : C γ = ( − F − γ exp (cid:18) − β (cid:18) P − m φ − λ φ (cid:19)(cid:19) (6.12) Complexity for φ + φ term : C γ = ( − F − γ exp (cid:18) − β (cid:18) P − m φ − λ φ (cid:19)(cid:19) (6.13) Effect on circuit complexity from SHO perturbation theory
To calculate the circuit complexity for supersymmetric quantum field theory, it is con-venient to work in the super-space formalism [40, 41], i.e. we extend the 4 commutingspace-time coordinates x ν to 4 commuting and 4 ant-commuting coordinates x ν , θ α , ¯ θ α .These new coordinates satisfy the following anti-commuting relations: { θ α , θ ˙ β } = { θ α , θ β } = { θ ˙ α , θ ˙ β } = 0 . (7.1) Now, any super-multiplet in super-space coordinates could be communicated in terms ofsuper-fields and can be expressed as:
Φ = φ ( x ) − iθσ µ ¯ θ∂ µ φ ( x ) − θ ¯ θ ∂ φ ( x ) + √ θη + i √ θ ∂ µ ησ µ ¯ θ + √ θ F ( x ) . (7.2) where η is a Weyl fermion having 4 off-shell degrees of freedom and σ are the Pauli matricesand φ is a complex scalar having two degrees of freedom. The supersymmetric Lagrangianremains invariant even after the addition of the term, i.e. δ L = µ F + h.c. (7.3) The F term is an auxiliary complex bosonic field which has two off-shell degrees of freedomto match the 4 off-shell degrees of freedom of a weyl fermion, also the F term is an orderparameter for SUSY breaking, substituting F = - µ the ground state will not be invariantand supersymmetry will be spontaneously broken.13he Lagrangian in terms of components fields up to second order is: L = | ∂ µ φ | + iη † ∂ µ ¯ σ µ η + | F | + (cid:16) mF φ − m ηη + h.c (cid:17) . (7.4) We will now identify the auxiliary field F with the gradient of the Morse function whichpasses through every critical point on the surface, such that these points act as an or-der parameter for SUSY breaking. By doing the coordinate transformation, U ( u, v ) =( x ( u, v ) , θ ( u, v )) we could define: h ( u, v ) = g ◦ U ( u, v ) (7.5) such that: ∂∂x g = F. (7.6) The eigenvalues of H ( t ) are concentrated at the critical points of g , hence precisely at thecritical point where: dg = F = 0 (7.7) represents the scalar potential of the theory with no supersymmetric ground state. ThenHessian of h can be calculated as: ∂ h∂u = F x u + 2 F x u θ u , (7.8) ∂ h∂v = F x v + 2 F x v θ v , (7.9) ∂ h∂u∂v = F x u x v + F x v x u + F θ u θ v + F x uv + F x v θ u . (7.10) the above dependence on the derivative of super-space variable with respect to u and v isactually the Jacobian due to the change of the variables as mentioned above. Now solvingthe equation of motion for F , the circuit complexity can be evaluated as: C ( Q ) = F ( x u , x v , θ u , θ v , φ ) − I ( x u , x v , x uv , θ u , φ ) , (7.11) where the newly introduced function F ( x u , x v , θ u , θ v , φ ) and I ( x u , x v , x uv , θ u , φ ) are de-fined by the following expressions: F ( x u , x v , θ u , θ v , φ ) := (cid:18) mφ x u + 2 mφ x u θ u (cid:19) (cid:18) mφ x v + 2 mφ x v θ v (cid:19) , (7.12) ( x u , x v , x uv , θ u , φ ) := (cid:18) mφ x u x v + mφ x v x u + mφ x uv + mφ x v θ u (cid:19) . (7.13) Circuit complexity for φ term To calculate the complexity for higher order interacting terms we will follow the sameprocedure as above, notice that circuit complexity for SUSY field theories only dependson the value of auxiliary field which also act an order parameter for soft SUSY breaking.The Lagrangian for cubic interactions could be defined as: L = | ∂ µ φ | + iη † ∂ µ ¯ σ µ η + | F | + (cid:16) mF φ + λF φ − m ηη − λφηη + h.c (cid:17) (7.14) The kinetic term is a Kh¨ a ler potential corresponding to the θ ¯ θ term which is invariant underthe SUSY transformations. A general Kh¨ a ler potential could give rise to complicated termsin the Lagrangian, but for simplicity we will consider the most canonical kinetic terms. λ here is the coupling constant, and the corresponding F term is given by the followingexpression: F ( x ) = − φ ( m + 2 λ )2 . (7.15) As described above we will now identify gradient of the Morse function with absolutevalue of the auxiliary field F which is invariant under the SUSY transformations, and thencompute the hessian of the Morse function and thereby the complexity as described above: C ( Q ) = F ( x u , x v , θ u , θ v , φ ) − I ( x u , x v , x uv , θ u , φ ) , (7.16) where the newly introduced function F ( x u , x v , θ u , θ v , φ ) and I ( x u , x v , x uv , θ u , φ ) are de-fined by the following expressions: F ( x u , x v , θ u , θ v , φ ) := (cid:18) φ ( m + 2 λ )2 x u + 2 φ ( m + 2 λ )2 x u θ u (cid:19) (cid:18) φ ( m + 2 λ )2 x v + 2 φ ( m + 2 λ )2 x v θ v (cid:19) , (7.17) I ( x u , x v , x uv , θ u , φ ) := (cid:18) φ ( m + 2 λ )2 x u x v + φ ( m + 2 λ )2 x v x u + φ ( m + 2 λ )2 x uv + φ ( m + 2 λ )2 x v θ u (cid:19) . (7.18) Circuit complexity for φ term Super-potential allows to introduce a variety of supersymmetric interactions, here wewill study complexity for quartic interaction terms namely φ + φ . The representativeLagrangian involving quartic interaction in terms of components of super-fields is given15y: L = | ∂ µ φ | + iη † ∂ µ ¯ σ µ η + | F | + (cid:16) mF φ + λφ − m ηη − λφ ηη + h.c (cid:17) . (7.19) Here F is again representing the auxiliary field as described above, responsible for softSUSY breaking. By solving the equation of motion for F term we get: F = − mφ + λφ . (7.20) Again identifying the auxiliary field by gradient of the Morse function as described inprevious section, we could compute the circuit complexity by computing the Hessian ofthe Morse function for the corresponding perturbed term which then gives: C ( Q ) = F ( x u , x v , θ u , θ v , φ ) − I ( x u , x v , x uv , θ u , φ ) , (7.21) where the newly introduced function F ( x u , x v , θ u , θ v , φ ) and I ( x u , x v , x uv , θ u , φ ) are de-fined by the following expressions: F ( x u , x v , θ u , θ v , φ ) := (cid:18) mφ + λφ x u + 2 mφ + λφ x u θ u (cid:19) (cid:18) mφ + λφ x v + 2 mφ + λφ vθ v (cid:19) , (7.22) I ( x u , x v , x uv , θ u , φ ) := (cid:18) mφ + λφ x u x v + mφ + λφ x v x u + mφ + λφ x uv + mφ + λφ x v θ u (cid:19) . (7.23)Hence we see that circuit complexity for supersymmetric field theories only dependson the absolute value of the auxiliary field coming into light from the linear term in thesuper-potential. The above method for calculating the complexity for SHO in similar towhat we have used for IHO. The number of zero eigenvalues of supersymmetric H areprecisely equal to the Euler number of the manifold, therefore by identifying the F theterm with the gradient of the Morse function, the critical points are exactly where the F term becomes zero and determines the ground state of the system. Comparative Analysis
SHO arameter Non-SUSY QFT SUSY QFT- φ + φ φ φ + φ φ Mass Complexityfor Non-supersymmetricfield theories haspolynomial haswell as logarithmicdependence on themass parameter as log ( mδ ) Circuit complexity forNon-SUSY QFT haslogarithmic dependenceon the square of themass parameter log ( m δ ),and in the infrared (IR)region it takes the form − log k ( mk ). Complexity for Super-symmetric field theorieshas only polynomialdependence namely thequadratic exponent ofmass parameter. For large masses we observea decrease in rate andreaches a saturation value. Complexity for Su-persymmetric fieldtheories in case ofquadratic pertur-bations also haspolynomial namelythe quadratic de-pendence on massparameter.Topologicaldependence Complexityfor Non-supersymmetricfield theories hasa fractional de-pendence on thevolume of latticefor interactingterms, such as V for dimension d = 3 Complexity due to justquadratic perturbations inNon-SUSY QFT doesn’thave any topological de-pendence but depends onthe dimension of latticeused for computations Complexity for supersym-metric field theories due toquadratic interactions de-pends on the Hessian ofthe Morse function whosegradient has been iden-tified with the auxiliaryfield and also on the criti-cal points of the manifoldand hence has topologicaldependence SUSY complexitydue to quadraticterm in the su-perpotential alsodepends on topo-logical parameters(Hessian of theMorse function)defined on themanifold.Dependenceon field Complexityfor Non-supersymmetricfield theories de-pends on variousparameters ofquantized field intheories and thestrength of inter-action with eachother and normalfrequency modes Due to quadratic per-turbations complexity forNon-SUSY field theorieson a lattice depends oncomponents of the mo-mentum vectors and thenumber of oscillators inthe lattice formalism anddoesn’t depend on thecoupling parameter. Complexity for SUSYQFT only depends onthe absolute value ofnon-dynamical auxiliaryfield namely the F − term as F = mφ + λφ which isidentified as the gradientof the Morse functionwhich passes throughevery critical points onthe surface In case of quadraticperturbation com-plexity changes dueto the change of F -term which by solv-ing equation of mo-tion is given by mφ and doesn’t haveany dependence oncoupling constantGrowth ofcomplexity Complexity for per-turbating term φ for dimension d > λ → ω (cid:28) δ their is an ad-ditional logarithmic factorin the complexity and leadto divergences in the limit δ →
0. When we change coupling λ from 1 to -1 we observethat complexity first risesand then have a suddendip. As we go to more neg-ative values of lambda weobserver the rate of satu-ration to be faster. In supersymmetricfield theories wehave strangelyobserved that com-plexity first rapidlygrows and thensaturates doesn’tchange much whenwe change thecoupling λ to -1 Table 8.1 : Comparison in circuit complexity between SUSY & NON-SUSY QFT for SHO17 arametersof SHO Features of graph and LyapunovGeneralfeatures The graphs rise fast initially and then slowly saturate. The rate of saturation and complexityat saturation point depends on the order for perturbation with φ term being the slowest tosaturate. Hence the slope is significant and the Lyapunov exponent is expected to be largerfor φ theory. The φ theory saturates very quickly giving a smaller slope than the rest and asmaller Lyapunov exponent.Mass For large masses we observe a decrease in rate and saturation value of φ theory. It becomes in-differentiable to the φ term as we approach massive fields (hence a smaller Lyapunov exponent).For smaller masses the φ graph approaches φ , decreasing slightly in rate. We can expect aslight increment in the Lyapunov exponent. λ When we make λ negative we observe the saturation is slower in φ and φ perturbations, andin the case of φ perturbation we encounter a zero thereby right-shifting the point of initial risewhich will increase the value of the Lyapunov exponent. As we go to more negative values oflambda we observer the rate of saturation to be faster, and we expect the Lyapunov exponentto be smaller. Table 8.2 : Discussions on Lyapunov exponent for SHO complexity
IHO
Parameter Non-supersymmetric QFT ( φ ) Supersymmetric QFT ( φ )Mass Circuit complexity for Non-SUSY QFT in theregime of inverted harmonic oscillators hasquadratic dependence on a mass parameter inthe exponential type function namely the in-verse cosine hyperbolic function. Also it hasbeen observed that complexity starts to in-crease before the critical value λ = m . Circuit complexity for supersymmetric fieldtheories for the in inverted harmonic oscilla-tor also has quadratic dependence on the massparameter in the exponential function. Wealso observe that as mass increases the rateof change of complexity increases.Topologicaldependence Circuit complexity of Non-supersymmetricquantum field theories doesn’t have any topo-logical dependence, however it depends on thenumber of oscillators and dimension of the lat-tice Circuit complexity for supersymmetric fieldtheories depends on the critical points of themanifold, also for even values of F − γ we seethat complexity increases exponentially. λ The complexity starts to increase for λ < λ c .At the critical point the complexity sharply in-creases and beyond the critical value λ = m the model becomes unstable and we expectthe complexity to grow rapidly with decreas-ing pick up time The change in value of λ contributes to therate with increasing values resulting in fasterrates of increase and hence higher slopes. Fornegative values of λ we observe that com-plexity decreases exponentially and model be-comes irrelevant.Growth incomplexity For inverted harmonic oscillator, complexityfor non-supersymmetric field theories for ini-tial time is nearly zero, after which it exhibitlinear growth. On the contrary for the inverted harmonic os-cillator complexity doesn’t exhibit any expo-nential or linear growth which could be seenin figure (9.5). Table 8.3 : Comparison in complexity between NON-SUSY & SUSY QFT for IHO18 arameters of IHO Features of graph and LyapunovGeneral features We do not observe any saturation behavior, and the complexity values rise quicklyto very high values. Although we can not associate a Lyapunov exponent we makegeneral statements about the rate of increase (hence the slope) of the graphs. Weobserve increased slopes upon adding perturbation terms.( F − γ ) For even values the graphs are increasing exponentially whereas for odd values wesee negative complexity values and hence have not included them in our graphicalanalysis.Momentum p The constant momentum factor acts as a scale multiplying the overall complexityvalue and is hence redundant.Mass For φ perturbation, we observe as mass increases the rate also increases resulting inlarger slopes. This is also true in the case of φ theory, with the only difference beingthe symmetry along vertical-axis.Temperature The dependence on temperature can be evaluated by varying β (the inverse tempera-ture). By varying this we can observe that for high temperatures, the rate of increaseis much lesser as compared to lower temperatures in the case of IHO. This is true forboth φ and φ perturbations. One can note that this behavior is contradictory tothe upper bound that we can set for Lyapunov exponents giving us more incentive tonot associate the slope of IHO with the Lyapunov exponent. λ For negative λ value we observe a mirror inversion of the graph about vertical-axisand hence the complexity are exponentially decreasing. One can simply interpret theopposite behvior of the graphs with mass and temperature variation in the case ofnegative λ . The change in value of λ contributes to the rate with increasing valuesresulting in faster rates of increase and hence higher slopes. Table 8.4 : Discussions on Lyapunov exponent for IHO complexity Graphical Analysis
With the computed formulae for complexity for IHO and SHO in the previous sectionswe plot the graphs using different values of parameters.
SHO
For SHO, the parameters involved are the mass m and coupling coefficient λ for higherorder theories. We have chosen a simple linear co-ordinate transformation between the fieldvariables x and θ and u , v . This ensures that the Hessian is a constant and our computationsbecome much simpler. The particular coefficients of the transformation have been chosenso that complexity is positive and rising for all cases.In Fig. 9.1 we have plotted the three different complexities for a intermediate value ofmass, keeping the coupling coefficient fixed. We observe that the complexity rises andsaturates as expected.It is important to know how the complexity value behaves for different masses. Forlighter particles as we see in Fig. 9.2. We see a slight overall decrease in the complexity19 - Figure 9.1 : Complexity against φ for m = 1, λ = 1 - - - Figure 9.2 : Complexity against φ for m = 0 . λ = 1values but more importantly we observe that the graph of φ term tends very close to tothe φ term. We conclude that as mass grows smaller and smaller, the φ graph will inchcloser and be in-differentiable to the φ graph.20 Figure 9.3 : Complexity against φ for m = 100, λ = 1For larger masses we see a stark difference. Here it is the graph of the φ theory thatinches closer to the φ graph as seen in Fig. 9.3 and we can once again expect that it willbecome in-differentiable for much larger masses.The other important parameter that we need to vary is the value of λ and we startby seeing what will happen when it is made negative as seen in Fig. 9.4. We see thebehvior of the complexity remains same i.e. it grows and saturates as we go right. Aninteresting behavior occurs in the φ graph - We see an initial rise and dip, before the riseand saturation. This indicates that we have a zero in the φ theory.We observe no behavioral changes when we vary the specific value of λ apart fromthe fact that for higher values it saturates much quickly (pointing to a larger Lyapunovexponent) and saturates slower for smaller values (smaller Lyapunov exponent).21 Figure 9.4 : Complexity against φ for m = 1, λ = − IHO
For the plots of IHO we need to keep in mind that there are many more parametricvalues. We will do a graphical analysis mainly by varying mass and temperature. We willmerely state the redundancy involved in other parameters and therefore explain our choiceof fixing it. • If one looks closer to the formulae given for IHO in Sec. 6 we see the value of themomentum - p just adds an overall factor that multiplies the function thereby actingas a scaling factor. Hence we can set this to p = 1 without any issues. • Another important parameter that might affect our plots is whether the value of F − γ is Odd or Even. When it is even, we have positive complexity and when itis odd we find that we are dealing with negative complexity values which we safelyignore in the present context of discussion.Before we proceed to varying the inverse temperature ( β ) and the mass ( m ), we cantake a look at the general feature of the graphs of different perturbations in the field forboth positive and negative λ as we have done in Fig. 9.5 and Fig. 9.6.We see that for positive coupling coefficient the complexity values in all perturbationsrise, but in different rates, whereas in the case of a negative coupling coefficient we seethat they decrease rapidly after an initial period of rising (almost an inverse behavior ofthe positive λ case). We also observe symmetry for negative values of field in the φ + φ graph. It is important to note that although we can calculate the values of slope, one22 - Figure 9.5 : Complexity against φ for different perturbations of field with λ = 1 - - Figure 9.6 : Complexity against φ for different perturbations of field with λ = − Lyapunov exponent for the same because saturation does not exist.Maybe we can regard the small rise and saturation before the dip in the case of negativeLambda values, and associate a
Lyapunov exponent to it.23 - - (a) Variation of inverse temperature (b) Variation of mass Figure 9.7 : behvior of the complexity of φ + φ perturbation with varying β and m In Fig. 9.7, we have plotted the how the complexity values vary when we vary theinverse temperature and the mass. A very similar behavior is observed in the case of24 - - - (a) Variation of inverse temperature - - - (b) Variation of mass Figure 9.8 : behvior of the complexity of φ + φ perturbation with varying β and mφ + φ perturbation too, which is shown in Fig. 9.8.Since we have already established the behavior of the complexity in the case of negative λ
25e take the liberty of not explicitly plotting the above variations in the inverse temperatureand mass for that case. One simply has to imagine a decreasing (cannot associate aLyapunov exponent) mirror inversion in the negative valued domain of the field. This istrue for the cases of both perturbations that we have discussed in the present context. Quantum Chaos from Morse function
Emergence of chaos in quantum phenomena can be estimated using an out-of-time-ordercorrelation function which is firmly associated to the commutator of operator, split up intime. However, the universal relation C = − ln( OT OC ) relating complexity with OTOC[42–47] has been studied greatly in recent times to diagnose chaos in various physical mod-els. In this section we will prove this universal relation for supersymmetric case, relatingthe complexity with OTOC using the frame of Morse theory, then we will comment on theupper bound of chaos namely the Lyapunov exponent. In the above sections, by identifyingthe Morse function on a manifold with the cost function, we have calculated the complex-ity for supersymmetric field theories in various regimes in terms of the Hessian H ( f ) andalso have made use of the fact that the eigenvalues of supersymmetric Hamiltonian H t areconcentrated near the critical points of the Morse function defined on the manifold. Inthis section we will make use of the same facts and derive the universality relation. Theorem - Let φ ( t ) be an integral curve which represents the state of the particle atvarious time, then if x , ...x m be a local coordinate chart around a critical point p ∈ M such that ∂∂x , ...., ∂∂x m is an orthonormal basis for T p M with respect to the metric g , thenfor any t ∈ R the out-of-time-correlator at p is equal to the exponential of the minus thematrix of the Hessian at p , i. e. ∂∂x φ t | p = exp( − H p ( f ) t ) . (10.1) By identifying φ t as a smooth function on the surface of the manifold we have ddt φ ( t, x ) = − ( ∇ f ) (cid:18) ∂∂x φ ( t, x ) (cid:19) . (10.2) By changing the order of differentiation for any x ∈ M we get, ddt (cid:18) ∂∂x φ ( t, x ) (cid:19) = − (cid:18) ∂∂x ∇ f (cid:19) (cid:18) ∂∂x φ ( t, x ) (cid:19) . (10.3) Φ( t, x ) = ∂∂x φ ( t, x ) (10.4) is a solution of the linear system of ODE’s: ddt Φ( t, x ) = − (cid:18) ∂∂x ∇ f (cid:19) Φ( t, x ) . (10.5) Because exp (cid:18) − (cid:18) ∂∂x ∇ f (cid:19) t (cid:19) is also a solution to the linear ODE’s, we finally get: Φ( t, x ) = ∂∂x φ ( t, x ) = exp (cid:18) − (cid:18) ∂∂x ∇ f (cid:19) t (cid:19) . (10.6) Since the solution is unique, thereby, at the location of critical point p we have: ∂∂x φ t | p = exp( − H p ( f ) t ) (10.7) Hence by identifying complexity of SUSY field theories with the Hessian as describedin this paper, the above equation under good approximations could be written as: C = − ln( OT OC ) . (10.8) From the above equation, we could see that the Hessian of the Morse function which isa great tool to compute the complexity of supersymmetric theories, under good approxi-mations is perfectly consistent with universal relation and is of great interest to capturethe effect of chaos in supersymmetric field theories.we will now comment on the behavior of Lyapunov exponent in SUSY field theoriesespecially in the framework of an inverted harmonic oscillator under which it is expectedto have chaotic features, depending upon the increase in the number of critical points asdescribed in section 6. Thus we could write the expression for complexity in the IHOregime as: C i ( t ) ≈ c exp ( λ i t ) ∀ i = 1 , , ..., n. (10.9) It is to be noted that the above equation is valid only for the IHO. The index i indicatesthe higher order quantum corrections in the Lagrangian for which the complexity has beenmeasured, then mathematically Lyapunov exponent could be written as: λ i = (cid:18) d ln C i ( t ) dt (cid:19) ∀ i = 1 , , ...n. (10.10) OT OC = exp( − c exp ( λt )) (10.11) where λ is identified as Lyapunov exponent as in with reference [48–50] which captures theeffect of chaos in the quantum regime and relates different measures of complexity withOTOC through the universal relation. The above universal relation between complexitiesin different order of perturbations in the super-potential can be translated to the Lyapunovexponent through the MSS bound as: λ i ≤ λ ≤ πβ ∀ i = 1 , , ..., n. (10.12) where β is the inverse temperature. In this we have shown that the Morse function whichcould be used to classifies the topology of surfaces also captures the effect of chaos in thequantum regime of supersymmetric field theories. Conclusions
Out-of-order-correlation-function (OTOC) in the framework of supersymmetry has beenstudied before and as a result didn’t show any chaotic behavior in the regime of SHO. Ourmain focus in this paper was to bring out the relationship between circuit complexityand Morse function and comment on the complexity of supersymmetric quantum fieldtheory, in the regime of the simple and inverted harmonic oscillator (IHO), by formulatingthe potential of IHO as the generators of SL (2 , Z ) group. Circuit complexity has beencalculated for various models starting from quantum field theory (QFT) to cosmology,which involves working out the cost functions for the particular case and then minimizing it,however, in this paper, we haven’t done the same, by pointing out the relationship betweenthe cost and Morse function on a manifold we have explicitly shown how the critical pointson the surface encapsulate the action of the supersymmetric charge on the given referencestate. We have explicitly made use of the fact that the eigenvalues of the supersymmetrichamiltonian are concentrated near the critical points of the Morse function in the manifoldand then using the Witten index which comments on the symmetry breaking of the theorywe commented on the complexity of the supersymmetric field theories for the IHO, whichincreases by a factor of exponential. For computations of complexity in the regime ofsimple harmonic oscillator, we have identified the auxiliary field in the SUSY lagrangianwith the gradient of the Morse function and then computed the hessian, and found outthat circuit complexity didn’t show any dependence on initial conditions or exponentialbehavior. Next, we have the well known universal relation relating complexity and out-of-time ordered correlation function C = − ln ( OT OC ) using the general description of Morse28heory. • Remark I:
The circuit complexity for supersymmetric field theories has very deep connectionswith the Morse function defined on the surface of the manifold. We have obtainedthe expression for complexity for SUSY field theories in terms of the Hessian of theMorse function, in doing so, we have made use of the fact the eigenvalues of thesupersymmetric hamiltonian are concentrated near the critical points of the Morsefunction. • Remark II:
The behavior of complexity for supersymmetric field theories for the inverted har-monic oscillator is of prime importance, we have found that the growth of complexityin the regime of IHO is directly related to the growth of number of critical points onthe manifold, which in turn grows exponentially with respect to the superpotential,we observed similar behavior for higher order quantum corrections namely φ and φ theories. • Remark III:
We have also computed complexity for the simple harmonic oscillator, by identifyingthe auxiliary field in the SUSY lagrangian with the gradient of the Morse functionand then computed the Hessian in the superspace coordinates, and found out thatcircuit complexity didn’t show any dependence on initial conditions or exponentialbehavior. It is also worth mentioning that complexity for supersymmetric scalar fieldtheories only depends on the absolute value of the non-dynamical auxiliary field. The F - term which we have identified with gradient of the Morse function determinesweather the supersymmetry is spontaneously broken or not depending upon weatherthe gradient has passed through the critical points. On passing through the criti-cal points we get F = 0 which tells that their exists no zero enery supersymmetricground states. • Remark IV:
We have proved the well known universal relation C = − ln ( OT OC ) which relatescomplexity with the out-of-time ordered correlation function for supersymmetric fieldtheories using Morse theory. The out-of-time ordered correlation function is an ex-cellent gadget to capture the effect of chaos in the quantum regime. In this paper,we have obtained an upper bound on the Lyapunov exponent and also commentedon its various features for supersymmetric field theories purely for SHO and IHO intable ?? and 8.3 which we have obtained purely on the basis of general descriptionof Morse theory which gives quantum chaos a mathematical structure. The mainpoint of Witten’s paper on supersymmetry and Morse theory was to provide super-29ymmetry a mathematical structure, various results like Witten index which tellsif the supersymmetry is broken or not wouldn’t have been possible by the normaldescription of particle physics. • Remark V:
We have found that complexity for supersymmetric field theories differ significantlyfrom ordinary quantum field theories in the sense that for non-SUSY QFT complex-ity, slowly starts to increase at the critical point, however for SUSY field theories thegraph rises fast initially and then slowly tends to saturate, and the rate of saturationdepends on the order of quantum corrections. The φ theory saturates very rapidlyand have the smallest Lyapunov exponent among the other studied perturbations,while the complexity for theory involving φ term saturates slowly. For IHO we ob-served that complexity increases exponentially and quickly rose to very high valuesunlike ordinary QFT where it has a linear growth. • Remark VI:
We have explicitly studied the dependence of mass on the behavior of circuit com-plexity and have observed that for massive fields their is a decrease in rate of changeof complexity for φ theory, and it interesting to note that the graph becomes nearlyindistinguishable from that of free field theory. However for smaller masses the com-plexity for φ graph approaches to φ slowly with a slight decrease in the rate ofcomplexity and hence we expect an increase in the value of Lyapunov exponent. Inthe case of IHO, we observe that as mass increases the rate of change of complexityw.r.t φ also increases. • Remark VII:
We have also commented on the behavior of complexity w.r.t the coupling constant λ and have observed that for negatives values of λ in the regime of SHO the saturationis slower for φ and φ perturbations and in the case of φ theory their is a sudden dipat the initial stage to zero thereby right shifting the point of initial rise of complexity.In the case of IHO, increase in the value of λ results in increase in the rate of changecomplexity, however for negative values of λ we found that complexity decreasesexponentially for SUSY field theories.The future prospects of the present work are appended below: • Prospect I:
In this paper we have restricted ourselves to supersymmetric scalar field theories,30owever similar computations could be done for supersymmetric gauge and non-abelian gauge theories by taking in consideration the dynamical D - term[51 ? , 52],which would give further understanding about complexity and effect of chaos in su-persymmetric field theories. • Prospect II: circuit complexity for interacting quantum field theories and its re-lation with renormalization group has been studied by Arpan Bhattacharyya andcollaborator and thus it will be interesting to see what new mathematical structuredoes renormalization group flow brings out and how it is related to complexity[53–55]. • Prospect III:
In this paper, we have computed the complexity for SUSY scalar fieldtheories by making use of the properties of the Hessian matrix, however we hope thatthis is not all[56]. Use of other remarkable properties of the Morse function couldhelp in gaining much broader perspective in supersymmetric field theories and itsmatter content and interactions and the effect it has on the expansion of universe[57]. • Prospect IV:
The study of supersymmetry and its complexity in terms of Morsetheory has given it a geometrical structure, however for theories of supergravity itis still not quit clear what the right mathematical structure is[58, 59], we supposethat the work on this direction would bring interesting connections between quantumchaos and various other mathematical theories.
Acknowledgements
The research fellowship of SC is supported by the J. C. Bose National Fellowship of Sud-hakar Panda. Also SC take this opportunity to thank sincerely to Sudhakar Panda forhis constant support and providing huge inspiration. SC also would line to thank Schoolof Physical Sciences, National Institute for Science Education and Research (NISER),Bhubaneswar for providing the work friendly environment. SC also thank all the membersof our newly formed virtual international non-profit consortium “Quantum Structures ofthe Space-Time & Matter” (QASTM) for elaborative discussions. Satyaki Choudhury,Sachin Panner Selvam and K. Shirish would like to thank NISER Bhubaneswar, BITSHyderabad, VNIT Nagpur respectively for providing fellowships. Last but not the least,we would like to acknowledge our debt to the people belonging to the various part of theworld for their generous and steady support for research in natural sciences.31 eferences [1] N. Lashkari, M. B. McDermott, and M. Van Raamsdonk, “Gravitational dynamics fromentanglement ’thermodynamics’,”
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