CFT correspondence

Author’s Accepted Manuscript Propagation of few cycle optical pulses in marginal Fermi liquid and ADS/CFT correspondence N.N. Konobeeva, M.B. Belonenko

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To appear in: Physica B: Physics of Condensed Matter Received date: 29 March 2015 Revised date: 1 June 2015 Accepted date: 23 August 2015 Cite this article as: N.N. Konobeeva and M.B. Belonenko, Propagation of few cycle optical pulses in marginal Fermi liquid and ADS/CFT correspondence, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.08.041 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Propagation of few cycle optical pulses in marginal Fermi liquid and ADS/CFT correspondence N.N. Konobeeva1,* , M.B. Belonenko1,2 1

2

Volgograd State University, Volgograd, 400062, University Avenue 100, Russia

Volgograd Institute of Business, Volgograd, 400048, Uzhno-ukrainskaya str., Russia

Absract. The paper considers features of few cycle optical pulse propagation in marginal Fermi liquid. The Green functions whose poles are responsible for the dispersion law excitation states of the liquid have been derived within the framework of ADS/CFT correspondence. Marginal Fermi liquid parameters influence on the pulse shape was defined. Keywords: few cycle optical pulses; ADS/CFT correspondence; marginal Fermi liquid

1. Introduction The idea of ADS/CFT correspondence enabled significant progress in the study of quantum critical phenomena [1, 2]. The obtained achievements in recent years, associated with the use of technical methods were well developed in string theory [3, 4]. Since 1970s (in the field of critical phenomena) there was a clear understanding of the role of universality, scaling, and hence ideas conformal field theory [5-7], which determined the success of this approach. Close relationship of conformal quantum theory with quantum critical phenomena leads to the mutual development and clearer understanding of these fields. It is well known that physical theories become conformal under concrete parameters choice. The best known example is an appearance of conformal invariance close to phasetransition point vicinity. In similar way, many theories describing non-Fermi liquids properties are conformal and the Green's function for them can be derived from the AdS/CFT correspondence [7-9]. Therefore, in the last few years there has been a growing interest in

*

Corresponding author: Tel: +7-8442-464894; fax: +7-8442-546891, [email protected]

systems which are located "on the border" between the Fermi liquid and non-Fermi liquid. The electronic state of such systems is called marginal Fermi liquid. Theoretically, this state has been predicted for a long time, but it was impossible to find materials in which it would be clearly expressed. Meanwhile, a comprehensive experimental study of these materials is very important because they can contribute to the development of new theoretical approaches to their description. If we can finally understand the behavior of marginal Fermi liquid, then we can come to grips with more complex non-Fermi liquid systems. In its turn, among nonlinear optical phenomena upwards the first papers [10, 11], specific attention is attracted by few cycle optical pulses. Note that such pulses are localized in space electric field pulses, and their energy is concentrated in a bounded region of space. The propagation of few cycle pulses in optical media without substance destroying, give us the opportunity to observe and investigate the nonlinear phenomena which are rare in the slow pulses field. All of these factors have stimulated studies of the features of few cycle optical pulse propagation in marginal Fermi liquid by AdS/CFT correspondence. 2. The dispersion law excitations of AdS/CFT correspondence The two-charge black hole in AdS5 is determined by Lagrangian, which in the standard form for the gravitational field and the gauge field with U(1) symmetry [12] can be written as:

L 

2 1  1 1  R  e4 F2  12      2 8e2  4e4  2  2g  4 L ,





(1)

Where g is gravitational interaction constant, F  is stress tensor for the gauge field, L is the constant length dimension, α is the scalar field associated with the original system supercharges. The given choice of the Lagrangian after standard variational procedure is in accordance with the solution of motion equations:

2





ds 2  e 2 A hdt 2  dx 2  A  ln

e2 B 2 dr h

r 1  Q2   ln 1  2  L 3  r 

B  ln

r 2  Q2   ln 1  2  L 3  r 

 r  2Q  r h r  Q  2

2

2

2

2

2

,

(2)

1  Q2  ,   ln 1  2  6  r 

A dx   dt ,  



2Qr 2 r 2  Q2 L



here r is the radial component, Q is the black hole charge. The asymptotic behavior of Dirac spinor in AdS5 has the following form: r 

1

 0

 

 ,

  a r m    b r  m   0 1

(3)

The expectation value of the boundary spinorial operator dual to the bulk spinor ψ can be 0 r r  written as: =(0,b1,0,b2)T. In fact O  0.5( 1    )O ,     3  т which means that the  0 3 

boundary spinorial operator is left-handed, here σ3 is the Pauli matrix. By imposing the in-falling boundary condition at the horizon, we can obtain the retarded Green’s function as: 0    G1   , G  b , G    a 0   G2  

(4)

Note, if we use the alternative quantization, the Green's function has the form: Gα=-aα/bα, then the spinor operator at the border is right-handed. If m=0, G1 and G2 are related by: G2=-1/G1 [13]; herefore, the alternative quantization for G1 is the standard quantization for G2, and vice versa. Taking into account G1 and G2, the alternative quantization gives the same Fermi momenta as the standard quantization does at m=0. Nearby the Fermi level the Green's function has the form:

3

с1k1 / 2 k 1 G   c2 k1 / 2 k с1 



h1

2 h2 ei k



1 / 2 k

, с2 

1

h e  i k

1 / 2 k

2

,

(5)

 k    0.5  k   arg   0.5  k  h1  Z / vF , h2   /  2 vF where k  k , k F is Fermi momentum, k┴=k-kF, v F is Fermi velocity, Z is the function depending on the charge Q and the Fermi velocity. The main difference of the given Green's function which poles determine the dispersion law of quasiparticles lies in the fact that the quasiparticles dissipation highly depends on parameter  k 

k 2Q

. Nearby the Fermi level we

have:

( w, kF )  c( kF )w1/ 2 K .

(6)

According to the performed analysis in [14], if  k (k  k F ) >1/2 то, we have a Fermi liquid, characterized by sharp quasiparticles, herewith the case  k (k  k F ) =1 is similar to the Landau Fermi liquid. At the  k (k  k F ) <1/2 we have a non-Fermi liquid, with no sharp quasiparticles. When  k (k  k F ) =1/2 we deal with “marginal Fermi liquid with logarithmic dependence ( w, k F ) on frequency. In accordance to the mentioned above we will investigate the case at the  k (k  k F ) <1/2 and will define the dispersion law in disregard for imaginary component of frequency:

w  ck 1/ 2 k  0 .

(7)

Note that there are two parameters in Eq. (7) k (k  k F ) and с= c(k F ) (further we consider the system in which vF =1). Parameters of electronic liquid can be described by chemical potential  and temperature Т. There are many references (point out only [15-18]) where it is possible to familiarize with the relations between  k (k  k F ) , c(k F ) and  , Т. 4

3. Basic equations We know the dispersion law E(p) for quasiparticles for investigation of few cycle pulses dynamics. Next, we construct the general equation for the propagation of few cycle pulses in a marginal Fermi liquid taking into account the dispersion law E (p) has the form (7). Note that all our arguments will relate only to the area where the quasiparticles are well defined [14]. In some ways it uses semi-holographic approach [19], where the dispersion law for strongly interacting particles taken from holography, and then we apply a classical consideration. Further in construction of few cycle pulses propagation model, we will describe electromagnetic field of a pulse by virtue of the Maxwell equations in the Coulomb calibration [20] E   A ct . Vector-potential has the form: A  (0,0, Az ( x, t )) . Then to the planar wave front approximation (one-dimensional problem) we have the following expression:

 2 A 1  2 A 4   j  0, c x 2 c 2 t 2

(8)

where j is current generated by electric field pulse exposure onto quasiparticles, c is the light velocity. Here we neglect the few cycle pulses diffractive spreads out in directions perpendicular to distribution axis. As for the non-Fermi liquids, as it will be shown later, there is a region in which quasiparticle lifetime is very large than the quasiparticles assembly for the problem of few cycle pulses (about 10-14 s) dynamics can be described by the collisionless Boltzmann kinetic equation [21]: f q  A  f  0, t c t  p

(9)

where q is the charge, f  f ( p, t ) is distribution function implicitly dependent upon coordinate, moreover, the distribution function f at initial moment of time coincides with F0 (the Fermi equilibrium distribution function):

F0 

5

1 1  exp E p  k bT  ,

here kb is the Boltzmann's constant. The expression for current density j  (0,0, j z ) can be written as:

j z  q  d 3 pvz f ,

(10)

where v z  E(p) p z is the group velocity. Solving the equation (9) by characteristics method we can obtain the following: q0 q0 q0

jz  q 

 d

3

q0 q0 q0

 q  pv z  p  Az t  F0 p  ,  c 

(11)

Integration in (11) is performed with respect to the first Brillouin zone and q0 corresponds to the chosen layer near the Fermi surfaces, namely this parameter gives a contribution to observed values. It is known that dependence of basic quantities on q0 is of logarithmic type, this fact let us choose the parameter under discussion in rather arbitrary way. . 4. Dynamics of few cycle pulses Hereafter let us consider few cycle pulse with plane wave front, vector potential of which can be assigned as:

A(x , 0 )  Bexp(  x 2 /  ) dA(x , 0 ) 2 xv  Bexp(  x 2 /  ) , dt 

(12)

  ( 1  v 2 )1 / 2 where B is a pulse amplitude, v is its initial velocity. This initial condition corresponds to the fact that a few cycle pulse consisting of a single electric field vibration is fed to the sample. The equations (8, 11) were numerically solved by means of direct leap-frog scheme [21]. Evolution of electromagnetic field under its propagation is presented in the figure 1.

6

Fig.1. Dependence of electric field under vk=1/3.3 on time for different space points: a) x=10-5 m; b) x=1.5∙10-5 m; c) x=2∙10-5 m. Note that the pulse retains its form, but decreases in amplitude.

Fig.2. Dependence of electric field on time (x=10-3 m) at different values of vk: a) vk=1/3.3; b) vk=1/4.3; c) vk=1/5.3. Fig.2 shows when value  k (k  k F ) decreases, lag from the original pulse (vk=1/3.3) becomes more obvious and its amplitude increases. Thus, we can control the pulse shape using the parameter vk. It should be noted the parameter vk is associated with the critical exponent and it determines the region of the quasiparticles existence and their damping. The dependence of few-cycle pulse form on the vk is a fundamentally new. 7

The vector-potential for the initial pulse in two-dimensional case can be defined as:

A(x , 0 )  Bexp(  x 2 /  x )exp(  y 2 /  y ) dA(x , 0 )  2 xvx 2 yv y     dt y  x

 2 2  Bexp(  x /  x )exp(  y /  y ) 

Here B is a pulse amplitude, vx,y is its initial velocity towards x and y correspondingly, γx and γy determine pulse width. Thereafter, the equation (1) was supplemented with a second spatial derivative with respect to transverse coordinate y:

 2 A  2 A 1  2 A 4    j0 x 2 y 2 c 2 t 2 c Dependences of electric field on coordinate for two-dimensional pulse are presented in fig.3 and fig.4.

(a)

(b)

Fig.3. Dependence of electric field on coordinate at vk=1/3.3(t=400∙10-12 s): a) longitudinal section through the middle of the pulse y=250∙10-5 m); b) cross-section (the same time, x=210∙10-5 m). Note that the values of the parameter vk have the weak influence on the pulse shape in both transverse and longitudinal direction.

8

(a)

(b)

Fig.4. Dependence of electric field (unity corresponds to 106 V/m) at vk(k=kF)=1 on coordinates x and y (unity corresponds to 10-5 m)for different time: a) t=3∙10-12 s; b) t=10-11 s.

Given results demonstrate the possibility of two-dimensional pulse propagation, which eventually decreases in amplitude and spreads out along the transverse coordinate. This behavior can be considered as stable, because of the nature of non-linearity in Eq. (4), which prevents a pulse collapse. Note in this case the pulse does not demonstrate precise balance between the dispersion and the nonlinearity, especially between the dispersion in the lateral direction, which leads to the spreading of the pulse. This result is entirely new, since the issue of sustainable propagation of few-cycle pulses in a marginal Fermi liquid, even at a certain choice of parameters, have not been studied up to now. The behaviour of two-dimensional pulse in a non-Fermi liquid for other values of vk(k=kF) requires an additional discussion. To answer all these questions, we should make additional study with numerous calculations. In this paper we demonstrate the possibility of this method for two-dimensional pulses. Thus, it was shown that both one-dimensional and two-dimensional few cycle optical pulse propagation in the marginal Fermi liquid is a stable pattern, at that, pulse is due the process approximate balance of dispersion and nonlinearity. We see practical applications of this effect in the possibility to control the transverse width of the few cycle optical pulse. In this few cycle pulse width is determined by the length of its path in non-Fermi liquid. It should be noted that the nature of the pulse, which is in strong correlation with the value of vk(k=kF), depends essentially on the plane wave front or not. This fact makes possible experimental verification of the consequences of this approach in the study of pulse propagation in non-Fermi liquid with different wave front curvature. 9

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