Aspects of Fermi arcs and surfaces from the perspective of Gauge -Gravity Duality

Here rH is the horizon radius of the AdS black hole (BH) where r is the radial coordinate. Curves with different colors are for different values ​​of the chemical potential (µ) at a certain fixed temperature.

Preview of the thesis

In this chapter, we discuss the Fermi arcs, the disconnected contours of the Fermi surface observed in the pseudogap phase of high-temperature superconductors. In Chapters 4 and 5, we study the nature of the Fermi surface and its evolution by tuning different types of non-minimum dipole and dipole-like couplings in the bulk.

Condensed matter background

We begin with the definition of retarded Green's function, which is given by the thermal expectation values ​​of the creation and annihilation operators, ˆ. The spectral function, as previously mentioned, is one of the most fundamental objects in the.

Figure 1.1: Filling of the ground state of a free electrons system with occupied orbitals inside a sphere of radius k F , where ϵ F = k F2 /2m is the energy of an electron having a

The holographic principle

The anti-de Sitter space

The constraint equation for the embedding space is given by: the parameter is the length scale of the geometry and can be thought of as the "radius". The AdS space is invariant under transformations of the embedded Minkowski space of the form (X′)A = ΛABXB, where the matrix Λ is of (d+ 2)×(d+ 2) dimensions and satisfies ΛTηΛ = η and Λ−1−1 > 0.

Basic Conformal Field Theory

For a Poincaré patch in the coordinator sense, the boundary of the AdS space is atr→ ∞withRdtopology. The infinitesimal generators of the conformal group can be derived by the infinitesimal transformation of the coordinates xµ → xµ +ϵµ.

The AdS/CFT Correspondence

To understand the equivalence mentioned above, we need to formulate a dictionary between all fields ϕ in the supergravity theory and the operators O in the conformal field theory side. In the strong coupling and large N limits, we do not really need to evaluate the full functional integral on the left-hand side, but we can approximate by the classical on-shell action of an AdS field theory with respect to the field boundary values ​​ϕ0 .

Figure 1.7: Cartoon of gravity theory that lives inside the bulk of AdS spacetime while the CFT is at the boundary at r = ∞

Scalar field in AdS spacetime

The coefficient A(k) of the leading term near the boundary is known in AdS/CFT terminology as the source corresponding to the dual operator O of the CFT. It is known to us that the field-theoretic path integral representation of the one-point function with a source can be written as. 1.61).

Figure 1.8: The plot of mass-squared of the gravity scalar versus the dimension ∆ of the dual field theory operator O in d = 3 dimensions

Fermion field in AdS spacetime

Set-up of background geometry and fermionic action

With this background geometry, we can write the action for spinors and solve the Dirac equation to obtain the spectral function. We can calculate the vielbeins appearing in the covariant derivative of the Dirac equations from the given metric (2.5) and find that the non-vanishing components of spin couplings are given by .

Boundary conditions

Analysis of the spectral function G R

Note that the turn-on set of the peak is approximately around ω ≈ 1.2≈k−µq, which corresponds to the divergence at ω=k in the vacuum Green's function given by [58,59]. Moreover, the spectral function is almost zero in the region where ω is between (−k−µq) and (k+µq) as in vacuum.

Figure 2.1: Plot of ImG 22 (ω) at k = 1.2 < µ q (left) and k = 3.0 > µ q (right) for the parameters m = 0 , q = 1 and µ q = √

Example 2: Computation of conductivity

Numerics and some results

After specifying the boundary condition at the horizon, we can numerically integrate equation (2.28) from the horizon to the boundary, using the NDSolveme snapshot1 method, the Mathematica function under construction, and extract the leading and subleading terms at the boundary to obtain the conductivity. σ using the formula given in equation 2.30 and we presented the results in figure 2.4. Real (left) and imaginary (right) parts of electrical conductivity obtained from experiments performed on graphene. When we compare the results in Figure 2.4 with some of the experimental results in Figure 2.5 performed on graphene [62], it is interesting to see that the conductance behavior is very close to the results obtained using the AdS/CFT duality.

Figure 2.4: Plots of the electrical conductivity (σ) versus frequency obtained by using the AdS/CFT prescription

Example 3: Holographic Fermi arcs

Numerical Results

Many interesting and emerging phenomena seem to arise from this delayed two-point function of the double fermionic operator in boundary theory. This intensity can be extremely small (almost near zero) for a given range of impulses, giving rise to the discontinuity of the Fermi surface and thereby generating an arc-like structure in impulse space. The difference can be easily understood if we assume that the mass of the scalar field is above Breitenlohner-Freedman (BF).

Figure 2.6: Fermionic spectral functions A(ω, k x , k y ), at ω = 10 −3 + iδ, δ = 10 −6

Scalar Field Solution

We investigate the AdS2 behavior of the scalar field in the limit r→ r0 and T → 0 by writing the scalar field as We have plotted the behavior of the scalar field at the horizon with temperature in the right panel of Figure 3.1. Since it is the horizon value of the scalar field that governs the fermion-meter coupling in the bulk spacetime, study the effect of.

Figure 3.1: Plot of Condensate O c (left) and horizon value of Φ (right) vs temperature (T /T c ) with O s = 0

Fermion Lagrangian and Dirac equation

Model-A

By choosing the following ansatz for the fermion field ψ(r, ⃗xi) = (−ggrr)−14e−iωt+ik.xψ˜(r, k), one can get rid of the spin connection, and finally. In equation (3.14) we see that with ky ̸= 0 the block diagonal form is lost, we now have a mixture of different spinor components. Numerically, we will integrate equation (3.18) from the horizon (r = r0) to infinity to calculate the spectral function. The boundary condition for ω ̸= 0 is still in diagonal form given by

Model-B

For the function of AdS2Green, see Appendix C.4, which includes the finite temperature-dependent scalar field and how it changes the IR CFT operators.

Numerical results and discussions

Across the phase transition: without source

We have also plotted in Figure 3.4 the evolution of Fermi surface for higher p-value which essentially changes the absolute strength of the gauge-fermion coupling. From a pole/zero duality perspective, the appearance of the gap can be realized from the pole and zero in G11 and G22 which are the diagonal components in the Green's function. Since the Fermi surface is anisotropic, we now investigate how the magnitude of the Fermi momentum kf changes as we go along the surface.

Figure 3.3: Plot of A(k f , θ) in θ − k f . Here p = 1.5 ,q = 1, m = 0, m 2 Φ = −21/10 and at very small T = 10 −3 T c

At arbitrary temperature: with source

Energy gap in the spectral function

Figure 3.6 illustrates the presence of the gap for model-B, which contains dipole-type fermion-gauge interaction with scalar field-dependent effective coupling parameter pef f = pΦ. At this point it is also important to note the symmetric nature of the energy gap for the dipole type coupling. Thus, in the holographic pseudo-gap phase, the Fermi arcs appear to be intimately connected to the partial gap of the Fermi surface.

Figure 3.6: Spectral function A(ω, k) vs (ω, k) for model-B. Below T c (≈ 0.001078), the opening of gap near ω = 0 is seen along k x direction for fixed p = 5 (lef t) and

Analytical study of Green’s function at finite temperature

The properties are usually measured by the dispersion relation near the Fermi surface, which is the pole of the Green's function. With this assumption in mind, we consider the Green's function with the horizon value of the scalar field to be temperature dependent and show the evolution of the Fermi arc with temperature. From the above expression of νkf, it is clear that it is the horizon value of the scalar field Φ(r0) that controls the properties of the fermion spectral function.

Figure 3.8: Plot of ν k f as a function of ω/T keeping ω fixed. Behaviour is in the condensed phase of the scalar field

Summary and Conclusions

Therefore, the Fermi arcs in the holographic pseudo-gap phase appear to be closely associated with the partial opening of the Fermi surface. As mentioned, we consider a complex scalar field ϕ that will be the cause of the translational invariance breakdown of the boundary field theory. The leading termχ(1) is associated with the source of the double scalar operator in limit theory, whose dimension is ∆ = 3−α− =α+.

Figure 4.1: Q-lattice profile with m 2 ϕ = −2, for the parameters T /µ=1, χ (1) /µ=0.5, k 1 /µ = 1 and k 2 /µ = 0

Fermions: Action and the spectral function

From these equations we can expand near the horizon and one can find that the leading terms of the equations of motion are given by. From this equation, we determine the incident boundary condition at the horizon for extracting the delayed Green's function at the AdS boundary, and the independent incoming boundary condition must be imposed at the horizon, i.e. 4.15) Furthermore, the asymptotic behavior of the Dirac equations near the AdS boundary (z →0) has the following form. We followed the usual prescription used to extract the Green's function using two sets of linearly independent boundary conditions given by

Results and Discussion

While in the ky direction, the spectral function indicates the existence of a Fermi surface with a high density of states. An interesting behavior of the Fermi surface also appears (see Figure 4.9) when we turn on both coupling parameters. The role of the common dipole parameters can be thought of as the strength of the disorder at the boundary.

Figure 4.3: Density plot of the spectral function A (k x , k y ) with all couplings set to zero for the parameters m ψ = 0 and q = 1

Fermions: Action and the spectral function

We recall that in Chapter 4 we studied the fermionic spectral function in the Q-lattice background with couplings of the form pψ /¯F ψ and pψ /¯F|ϕ|2ψ. Here, the scalar field ϕ is the same as considered in the lattice solution, and mψ is the fermion mass. While the second coupling, parameterized by p2, is similar to that studied in the development of the Fermi arc from the Mott insulator [5], which breaks the rotational and Lorentz symmetries of the limit theory.

Numerical Results and Discussion

The sharp peak in the spectral function essentially indicates the presence of a stable quasi-particle with a longer lifetime. The presence of these FS pairs in the spectral function indicates that the scalar field plays a crucial role in unmixing the individual spectra of p1 and p2. First, the appearance of these pairs of Fermi surfaces are mainly our interesting features outside of our numericals since these types of spectra have been observed in ARPES experiments that measure the spectral function in real condensed matter systems such as topological insulators (TIs). Dirac and Weyl semimetals [83,84,88].

Figure 5.1: Density plot of A (k x , k y ) by varying the value of p 1 . Here, we fixed p 2 = 0.3 and m ψ = 0 , q = 1

Conclusion

So far we have discussed the fermionic spectral function in the presence of different types of interaction terms as non-minimal couplings to the bulk fermions, and all our background geometry is non-magnetic. Therefore, it would be interesting and important to also study a probe fermion placed in the bulk geometry containing magnetic field. In Section 6.2 we discuss the fermionic action and the non-minimal coupling in the background geometry followed by solving the Dirac equations and the spectral function.

Figure 5.3: Here we plotted the spectral density with fermion mass m ψ = 0 and charge q = 1

Fermion action

6.4), where µ is the chemical potential of the system and h is the strength of the magnetic field, and the temperature of the system is T = 4π1 3−(µ2 +h2). The corresponding dimensionless parameter related to the magnetic field strength is H =h/µ= √3−hh 2 , which ranges from zero to infinity. which can be written in the form 6.11). These equations do not have the form of the standard Hermitian differential equation that occurs in the case of the harmonic oscillator problem.

Results and Discussion

Effects of the dipole coupling

In the above discussion, we have studied the quasiparticle decay width for different levels of fixed magnetic field. When comparing the plots for different strengths of ℘ in figure 6.5, for a fixed h there is a division in the spectrum into several peaks as a increases. This splitting is unique and can only be seen in the presence of a magnetic field with the dipole coupling term introduced in the fermionic action.

Figure 6.2: Left: Plot of Im G 22 for zero temperature, the poles near ω = 0 is clearly visible for n = 1, 2, 3..

Conclusions

In Chapter 3, which is our first technical chapter, we discussed the Fermi arcs, which are disconnected contours of the Fermi surface observed in the pseudo-gap phase of high-temperature superconductors. We mainly studied a deformation of the Fermi surface and its evolution by tuning two types of dipole coupling parameters in the bulk. Suppression in the spectral weight and distortion of Fermi surface is also seen, reminiscent of the results seen in different condensed matter experiments in real materials.

Example - 2 code

Example - 3 code

Now the spin connection has been canceled due to scaling and the Dirac equations (C.7) and (C.8) above can be packed into a single equation. Here we have multiplied by A=r2√. Define the following non-vanishing ratio G11= β1I. If we equate the elements of matrices, we proceed as follows:. If we use the above ratios, taking into account that other possible ratios are equal to zero, then we have.

AdS 2 for Fermi arcs model-A

The presence of nonzero in this case changes the dimension of the IR CFT operator. Since in equation (C.30) we have a mixture of four spinors, therefore by following the well-known methods in [66, 104] the retarded AdS2 Green's function is expressed as. To construct the AdS4 Green's function, one must match the solutions in AdS2 usually called the inner region and AdS4 called the outer region at their common boundary.

Details calculations for model-B

Furthermore, it is important to note that the dimension depends on the temperature due to the scalar field condensation, which is related to the scalar field horizon value Φ(r0).

Finite temperature AdS 2 Green’s function

Norman et. al., Destruction of the Fermi surface in underdoped high-Tc superconductors, Nature. Kotliar, Fermi arcs and hidden zeros in the Green's function in the pseudogap state, Phys. Chikamatsu et al., Gradual disappearance of the Fermi surface near the metal-insulator transition in La1−xSrxMnO3 thin films, Phys.