Short introduction to the Keldysh formalism

For completeness, we give a short overview of the Keldysh formalism [201], originally con-structed for non-equilibrium problems. As it turned out, it is quite useful for a large variety of complicated interacting, disordered, or stochastic systems. Let us first recall how to calculate

38If needed, apply the holographic renormalization prescription to obtain a regularized finite on-shell action.

equilibrium expectation values (2.57) of observables in closed time contour as shown in Figure 3.14,

x

O

ptq y “ trp

O

ρˆptqq

tr ˆρptq “ trp

U

p´8, tq

OU

pt,´8qρˆp´8qq

tr ˆρp´8q , (3.103)

where the unitary time evolution operator

U

pt¹, tqis defined in (2.1). In the last equality we used the cyclicity of the trace and the invariance under unitary evolution. Physically, the expression (3.103) can be viewed as an evolution of an equilibrium system described by the initial density matrixρˆp´8q “ ρˆ0 to the timet where the system is subject to measurement represented by the operator

O

and finally brought back to its initial equilibrium configuration att “ ´8. The forward-backward evolution can be avoided for systems in equilibrium under the assumption that the ground state of the system evolves adiabatically, while interactions are slowly switched on. Therefore, extending the time evolution of the system to the distant future,tÑ 8amounts to adding a phase factorx0|

U

p8,´8q “ x0|e^iα. The additional phase factor can be factored out and so the backward segment of the closed time contour can be removed by subtracting the disconnected diagrams. However, for non-equilibrium systems the initial and final states are in general not identical, so the system could be relaxing into a different equilibrium state after it was driven out of the initial equilibrium state. The main idea to circumvent this problem is to keep the full closed time contour, where we start and end with a “trivial” description of our system, adiabatically switch on the “non-trivial” interactions and switch them off on our way back,i.e.

x

O

ptq y “ trp

U

p´8,8q

U

p8, tq

OU

pt,´8qρˆp´8qq

tr ˆρp´8q . (3.104)

In this case we do not need to know the state of the system attÑ 8 nor do we accumulate a phase and so there is no denominator and no need to subtract diagrams. As a consequence the partition function reads

Zr0s “ trp

U

Cρˆp´8qq

tr ˆρp´8q “1, (3.105)

since

U

C “

U

p´8,8q

U

p8,´8q “1. For interacting systems the symmetry between the for-ward (upper) and backfor-ward (lower) parts of the contour is broken, so

U

CrVs ‰1. Nonetheless, the denominator in the definition of the correlation functions (2.29) is absent.³⁹ Now redoing the steps of constructing the functional integral

ii

_to

iii

_{on page}15, we obtain for the closed time contour partition function

Z“ 1 trpρˆ0q

ż

^˜_ź2N

k“1

dϕ_k

¸ exp

˜ i

2Nÿ

i,j“1

ϕiG^´_ij¹ϕj

¸

, (3.106)

with indicesi, j, k running from one to2N in the discretization of the closed time contour as shown in Figure3.14. The propagator is given by a2Nˆ2N matrix of the form

39This drastically simplifies the treatment of disordered systems and poses an alternative to replica [202] or supersym-metric approaches [203] in removing the denominator.

t1

tN

tN`1 t2N

´8 8

t

Figure 3.14. Discretized close time contour used to determine the expectation value of a phys-ical observable. The computation must be independent of the initial conditions/-preparation of the system, so we can taket1Ñ ´8where the system is in equilib-rium described byρˆp´8q “ρˆ₀. The system is measured at a specific timetby an insertion of the respective operator and then brought back to its equilibrium state att2N. For a symmetric extension of the adiabatically increased interactions, we can enhance the time evolution domain to8. For equilibrium measurements the backward time evolution amounts to a phase factor e^iα which can be removed by subtracting the disconnected diagrams. For non-equilibrium systems the initial state att“ ´8and the final state att “ 8are in general not identical equilib-rium states, so the backward evolution cannot be omitted. The full propagator of the closed time contour consists of forward propagation fromt1totN, backward propagation fromtN`1tot2N. The pointstN andtN`1are identified as well ast1

andt2N which are controlled by the density matrixρˆ0describing the equilibrium system. Sometimes the density matrix is resolved as imaginary time evolution fromt2N tot2N´iβ which is then identified witht1.

iG^´1“

¨

˚˚

˚˚

˚˚

˚˚

˚˚

˚˚

˚˝

´1 ρˆ0

h_´ ´1 h_´ ´1

. .. ...

1 ´1 h_` . ..

. .. ´1 h_` ´1

˛

‹‹

‹‹

‹‹

‹‹

‹‹

‹‹

‹‚

, (3.107)

whereh_˘“1˘iHpϕ_i, ϕ_i´1q∆tfollows from the time evolution from stept_itot<sub>i`</sub>1“t<sub>i</sub>`∆tand the diagonal entries from the resolution of unity. The top-left block describes the forward time evolution whereas the bottom-right block describes the backward evolution and the bottom-left and top-right the correlation between forward and backward evolution and vice versa. Thus, the single entry in the forward-backward block follows from the identification oftN withtN`1 and the upper right entry in the backward-forward block from the matrix element of the density ma-trix connecting the equilibrium states att2N andt1. The last step

iv

_{on page15}in the program of constructing a true field theory poses some problems. Naïvely taking the continuum limit we lose the correlation between the different contours, in particular the statistical information encoded in the equilibrium density matrixρ0. To keep the correlations between the forward and backward parts of the closed time contour, we need to introduce fields with two independent components related to the forward and backward branch. Then the action in (3.106) is written

as

S“ ż

dtdt¹´

ϕ^` ϕ^´

¯ ptq

¨

˝G^`` G^`´

G^´` G^´´

˛

‚

´1

pt, t¹q

¨

˝ϕ^` ϕ^´

˛

‚pt¹q. (3.108)

Computing the formal⁴⁰Gaussian integral (2.45) yields the individual Green functions

@ϕ^{`</sup>ϕ<sup>`}D

“iG^``pt, t¹q “iG^Tpt, t¹q “ 1

detp´iG^´1q lim

NÑ8∆tÑ0

ph_´q,

@ϕ^`ϕ^´D

“iG^`´pt, t¹q “iG^ăpt, t¹q “ ρ0

detp´iG^´1q lim

NÑ8

∆tÑ0

ph_`h_´q,

@ϕ^´ϕ^`D

“iG^´`pt, t¹q “iG^ąpt, t¹q “ 1

detp´iG^´1q lim

NÑ8

∆tÑ0

ph_`h_´q,

@ϕ^´ϕ^´D

“iG^´´pt, t¹q “iG^Tpt, t¹q “ 1

detp´iG^´1q lim

NÑ8

∆tÑ0

ph_`q,

(3.109)

whereG^Tdenotes the anti-time ordered Green function. Using

G^Tpt, t¹q “Θpt´t¹qG^ąpt, t¹q `Θpt<sup>1</sup>´tqG<sup>ă</sup>pt, t<sup>1</sup>q, G<sup>T</sup>pt, t<sup>1</sup>q “Θpt<sup>1</sup>´tqG<sup>ą</sup>pt, t<sup>1</sup>q `Θpt´t¹qG^ăpt, t¹q,

(3.110)

one can show that not all of the above Green functions are independent, in fact

G^T`G<sup>T</sup> “G<sup>ă</sup>`G^ą. (3.111)

This identity holds for allt‰t¹, whereas fort“t¹it is violated. This follows from the relationship to the spectral functionA defined in (2.77)

Apt,x, t¹xq “i`

G^Rpt,x, t¹xq ´G^Apt,x, t¹xq˘

“i`

G^ăpt,x, t¹xq ´G^ąpt,x, t¹xq˘

, (3.112)

where fort“t¹the spectral function is independent of the state of the system and thus given by Apt,x, t¹,¹xq “δpx´x¹q. Mathematically, this forces us to take the normalization of the Heaviside distributionΘp0q “1. Thus, in order to work with the “physical” Green functions we apply the so-called Keldysh rotation, a linear transformation in the two dimensional Keldysh space

GÝÑG“U GU^´1“

¨

˝G^K G^R G^A 0

˛

‚ with U “ 1

?2

¨

˝1 1 1 ´1

˛

‚, (3.113)

and

ϕÝÑU ϕ“

¨

˝ϕ^cl ϕ^q

˛

‚“ 1

?2

¨

˝ϕ<sup>`</sup>`ϕ^´ ϕ^`´ϕ^´

˛

‚, (3.114)

40The convergence of the integral is ensured by the Keldysh component of the Keldysh rotated Green function (3.117).

with the Green function definitions G^Rpt, t¹q “G^{cl q}“ 1

2

´

G^T´G^T`G^ą´G^ă¯

“Θpt´t¹q pG^ą´G^ăq, G^Apt, t¹q “G^{q cl}“ 1

2

´

G^T´G^T´G^ą`G^ă¯

“Θpt¹´tq pG^ă´G^ąq, G^Kpt, t¹q “G^{cl cl}“ 1

2

´

G^T`G<sup>T</sup>`G^ą`G^ă

¯

“G^ă`G^ą.

(3.115)

Note that the Keldysh Green function is antihermitian, i.e.pG^Kq^: “ ´G^K. Taking the inverse of the Keldysh rotated Green function (3.113) leads to the action in Keldysh form

S” ϕ^cl, ϕ^qı

“ ż

dtdt¹´ ϕ^cl ϕ^q

¯ptq

¨

˝ 0 `

G^A˘_´1

`G^R˘_´1 “ G^´1‰K

˛

‚

´1

pt, t¹q

¨

˝ϕ^cl ϕ^q

˛

‚pt¹q. (3.116)

where the inverse of the2ˆ2matrix yields a non-trivial inverse for the Keldysh component

“G^´1‰K

“` G^R˘´1

F´F` G^A˘´1

, (3.117)

parametrized by the Hermitian matrix F to ensure the antihermiticity of the Keldysh Green function. In general, the Wigner transform of the F matrix yields the instantaneous particle distribution function at given timet. Some comments about the notation and physical interpre-tation of the action (3.116) are in order: First the superscripts ‘q’ and ‘cl’ stand for “quantum”

and “classical”, respectively. This notation is chosen because generically the classical-classical component of the action is zero,i.e.S“

ϕ^cl,0‰

“0, since we are dealing with a quantum statisti-cal system. Secondly, the antihermitian quantum-quantum component ensures the convergence of the functional integral and encodes the information about the distribution function of the statistical system. Last, but not least, the classical-quantum components encode the causal struc-ture of the physical system. In particular, the Fluctuation-Dissipation Theoremon page30can be restated using the Keldysh Green function as

G^Kpωq “2i ImG^Tpωq. (3.118) We see that the imaginary part of the response function Impωq characterizing the dissipation of the system is related to the equilibrium fluctuations encoded in G^Kpωq. More details about applications of the Keldysh formalism can be found in [51,52,204]. In the holographic context the Keldysh formalism has been used to prove (3.102) explicitly [79]. The causality structure of the Keldysh action is only satisfied if we define the retarded (advanced) Green function as

G^Rpω,kq “ ´2

G

pω,k, uqˇˇˇ

u“uB

. (3.119)

On the other hand, the Euclidean Green function can be converted into the Keldysh formalism which allows us to identify the correlation functions for thermal theories in equilibrium with the Green functions in the Keldysh formalism. Of course, this is applicable only to transport problems in equilibrium, for non-equilibrium calculations the holographic dictionary needs to be extended.⁴¹

41Approaches to render the equilibration process in holographic duals involve black hole generations by colliding gravi-tational shock waves.

With the working prescription for linear response theory, we close our survey of holography and apply it to concrete physical systems and their respective gravitational duals. In particular, all the machinery derived in this chapter and Chapter2 can be used to tackle strongly correlated systems where the intractable field theoretical treatment without quasi particles is mapped to weakly coupled gravity duals, that allow for the applicability of our beloved perturbative tools again unfolding their power to describe physical systems.

4

Universal Properties in Holographic Superconductors

Our first application of the holographic dictionary and the weak/strong duality will be a bottom-up approach geared towards condensed matter applications. As advertised in the introduction to this thesis, we are looking for universal physical quantities that do not depend on the microscop-ical details but are rather defined by universality classes in the Wilsonian sense. In particular for the classification of novel states of quantum matter arising in quantum critical regions, strongly correlated quantum liquids or high temperature superconductors, the gauge/gravity duality can provide valuable insight and might even succeed to quantify some of the dimensionless functions that are not fixed by symmetries of the effective field theory. In this chapter we will explore the holographic superconductors aiming at the understanding of universal features, such asHomes’

lawand some of its cousins as Tanner’s law to be explained in detail below. The holographic superconductors are endowed with key features reminiscent of real superconductors,e.g. there is a charged condensate which gives rise to a massive vector boson in the spirit of the Higgs mechanism, although the Meißner-Ochsenfeld effect is absent. Yet, a redistribution of spectral weight opens an energy gap in the optical conductivity and allows for an infinite DC conductivity or vanishing resistivity. The removed localUp1qgauge symmetry on the gravity side translates to a removed global Up1qgauge symmetry, so the holographic superconductor may be viewed as a strongly correlated charged superfluid. In this sense there is no true dynamical photon on the field theory side since the gauge field only sources the charged density of the system. How-ever, we may allow for local transformation of the external source which in turn lifts the global Up1qsymmetry to a background localUp1qsymmetry. As explored in [205], the additional back-ground localUp1qsymmetry permits the computation of response functions related to strongly coupled superconductors described by an effective Abelian Higgs model. The order parameter for the superconducting phase transition is identified with the vacuum expectation value of the operator dual to a bulk field with no source turned on in order to mimic a spontaneous symmetry breaking mechanism. To date there are holographic duals known for s- and p-wave supercon-ductivity [206–208] and there are some ideas how to construct a d-wave type holographic super-conductor [209–211]. Holographic supersuper-conductors have been studied extensively, numerically

117

as well as analytically [212–214]. There are also explicit top-down constructions employing a true D-brane setup where the field theory is explicitly known [215–217].

In this chapter I will present my work on holographic superconductors. Operationally, I have verified most of the numerical calculations presented in [218,219] in order to test the validity of my numerical approach for applications of the holographic dual to open questions in real world superconductors. In particular, the empirically foundHomes’ lawto be discussed in Section4.4is the central focus of my work on holographic superconductors. All results obtained to understand a holographic realization of Homes’ law presented in Section4.4are original and first published in [1]. To my knowledge, the s-wave equations of motion for arbitrary values of the charged scalar field’s mass and the system’s dimensionality have not been explicitly derived so far. In the first part of this chapter we will discuss the properties of holographic s- and p-wave supercon-ductors and determine their phase diagrams. In the second part, an empirically found universal relation, the so-calledHomes’ law, between the superfluid density at zero temperature and the conductivity at the critical temperature times the critical temperature is explained and a possible holographic realization is proposed. In a way, our approach to Homes’ law can be viewed anal-ogously to the famous ratio of shear viscosity to entropy density^η{^sof the quark-gluon plasma with an important difference: for a hydrodynamic calculation the shear viscosity is determined by the metric fluctuations about the background metric. Thus the universality of^η{^sfollows di-rectly from the equivalence principle,i.e. the metric must couple globally to all forms of energy with a single coupling constant. Adding another coupling constant characterizing the charge of the scalar field which is independent of the metric field, however, complicates this simple uni-versality in a non-trivial way. Finally, in the last part the results obtained from our proposed holographic realization of Homes’ law, are analyzed.

Im Dokument Gauge/Gravity duality (Seite 131-138)