The dynamics

4.2.1 Quench setup

The system is initially prepared (at timest <0) in such a way that its initial configura-tion (at timet= 0) is neither correlated with disorder (J_ij’s) nor with the reservoirs. This can be realized, for instance, by coupling the system to an equilibrium bath at temperature T₀ ≫J,Γso that any correlation in the system is suppressed. At timet= 0the quench is performed by suddenly coupling the system to theLandRreservoirs. These are supposed to be “good reservoirs” in the sense that their properties are not affected by the state of the system.

This setup generates non-equilibrium dynamics at times t > 0 for multiple reasons.

First of all, the rapid quenching procedure puts the system in a non-equilibrium initial con-dition with respect to its new environment. Moreover, the latter is not an equilibrium bath but a bias drive the role of which is to constantly destabilize the system. Finally, as a conse-quence of its disordered interactions, the system of rotors experiences intrinsic difficulties to reach equilibrium. Indeed, even if it were embedded within an equilibrium environment it would show a glassy phase [258–260] in some parts of the phase diagram.

Since system and reservoirs are decoupled at timest < 0, the initial density matrix of the whole system is given by

̺(t= 0)_tot=̺(t= 0) ⊗^N

i=1

̺_Li ^N⊗

i=1

̺_Ri. (4.9)

̺_Li/Ricorresponds to the equilibrium density matrix of theL/Rreservoir associated with the i-th rotor. The system of rotors being prepared at very high temperature, its initial density matrix is the identity in the rotors space:

̺(t= 0)∝I . (4.10)

All these density matrices are normalized to be of unit trace. Thet > 0evolution of the whole system plus environment is encoded in

̺_tot(t) =U(t,0)̺_tot(0) [U(t,0)]^†, (4.11) where the unitary evolution operator is given byU(t,0)≡Te^{−~i R0tdt′Htot(t′)}withH_tot = H+H_L+H_R+H_int andTthe time-ordering operator (see App.4.A). We analyze the non-equilibrium dynamics using the Schwinger-Keldysh formalism (see [253, 254] for a modern review) that we briefly introduce in the following lines.

4.2.2 Schwinger-Keldysh formalism

The Suzuki-Trotter decomposition of the two unitary evolution operators that appear in Z ≡ lim

τ→∞TrU(τ,0)̺tot(0) [U(τ,0)]^†= 1, (4.12) yields a path-integral involving two sets of fields with support on two different branches.

The first ones are time-integrated on a forward branch fromt= 0to+∞. In the following, these fields carry a+superscript. The other ones are time-integrated on a backward branch from+∞to0and carry a−superscript. These two branches constitute the Keldysh contour C, see Fig.4.5. The identity (4.12) can now be expressed as a path integral,

Z = Z

cD[s^±,ψ^±,ψ¯^±] e^~iStot hs⁺(0),ψ¯⁺(0)|̺tot(0)|s⁻(0),ψ⁻(0)i, (4.13) where we collected all thes^µa_i fields into the notations^a, and all the fermionic fieldsψ^a_αi and their Grassmannian conjugates intoψ^aandψ¯^a(witha=±).

hs⁺(0),ψ¯⁺(0)|̺_tot(0)|s⁻(0),ψ⁻(0)i is the matrix element of the density matrix which has support at timet= 0only. The actionS_totis a functional of all these fields:

S_tot= X LLandLR are the Lagrangians of the free fermions in theLandR reservoirs. The index

‘c’ at the bottom of the integral sign in eq. (4.13) is here to remind us that the integration is performed over fields satisfying the constraint that each rotor has a fixed unit length:

s^a_i(t)² = 1 ∀a, i, t. The path-integral formalism gives a nice way to restore an uncon-strained integration over all fieldss^a_i by the introduction of Lagrange multipliersz^a_i:

Z where we used the integral representation of the delta function (see App.4.A) and collected the new auxiliary real fieldsz^a_i into the notationz^a. In terms of a Lagrangian, this gives rise to the new term

Figure 4.5: The Keldysh contourCgoes from0to+∞and then back to0. The Keldysh action involves forward fields (that live on the+branch ofC) that are time-integrated from0to+∞and backward fields (that live on the−branch ofC) and are time-integrated from+∞to0.

4.2.3 Macroscopic observables

We are interested in the macroscopic dynamics of the rotors after an infinitely rapid quench and we wish to give an answer to the following questions (among others). Does the system reach a steady state? Does a steady state current establish? What are the long-time dynamics? We first obtain an effective generating functional for the rotors by expanding the system-drive interaction up to second order in the coupling, integrating away the fermionic degrees of freedom, and averaging over the disorder distribution.

Introducing the external real fieldsh^a_iµ(t)that we collect in the notationh^a(t)(a=±), the generating functionalZ[h^±]reads

Z[h^±] ≡ Z

D[s^±,z^±,ψ^±,ψ¯^±] e^{~iStot[s±,z±,ψ±,ψ¯±,h±]}

×hs⁺(0),ψ¯⁺(0)|̺_tot(0)|s⁻(0),ψ⁻(0)i, (4.20) where we introduced the source term

S_tot7−→S_tot+~ i

X

a=±

Z

dt X

i

X

µ

s^µa_i (t)h^µa_i (t). (4.21) The generating functional obeys the normalization propertyZ[h^± =0] =Z = 1which is a fundamental feature of the Keldysh formalism in this setup (see eq. (4.12) and Sec.4.4.1).

One has

hs^µa_i (t)i= 1 Z

δZ[h^±] δh^µa_i (t)

h^±=0

, (4.22)

where we introduced the notation h · · · i ≡

Z

D[s^±,z^±,ψ^±,ψ¯^±] · · · e^~iStoths⁺(0),ψ¯⁺(0)|̺_tot(0)|s⁻(0),ψ⁻(0)i.(4.23) Notice that one can distinguish this bracket notation from the quantum statistical average that we denote similarly by the occurrence of Keldysh indices inside the brackets. However, they coincide in the case of one time observables, e.g.

hs^µ_i(t)i=hs^µa_i (t)i, (4.24) witha= +or−equivalently if the observable is time-reversal invariant.

Keldysh Green’s functions

We introduce the two-time Green’s functionsG_ijµν^ab (t, t^′), defined on the Keldysh con-tour (a, b=±), as s^µa_i being real fields, one has the following time-reversal property

G_ijµν^ab (t, t^′) =G_jiνµ^ba (t^′, t). (4.26) In the operator formalism, the Keldysh Green’s functions read

i~G_ijµν^ab (t, t^′) =Tr

T_C s^µ_iH(t, a)s^ν_jH(t^′, b)̺_tot(0)

, (4.27)

wheres^µ_iH(t, a)denotes the Heisenberg representation of the operator s^µ_i at timetand on thea-branch of the Keldysh contour. T_C is the time-ordering operator acting with respect to the relative position of(t, a)and(t^′, b)on the Keldysh contourC(see App.4.A).

We define the macroscopic Keldysh Green’s functions by summing over theN rotors and each of theirncomponents

G^ab(t, t^′)≡ 1 From the identity (4.27), one establishes two relations between the four Green’s functions

G⁺⁺(t, t^′) = G⁻⁺(t, t^′)Θ(t−t^′) +G⁺⁻(t, t^′)Θ(t^′−t),

We define the macroscopic two-time correlation as C(t, t^′) ≡ 1 it is one at equal times:C(t, t) = 1. The two-time correlation function is the simplest non-trivial quantity giving information on the dynamics of a system. In particular, a loss of its time translational invariance is a signature of aging.

Self linear response

The response at timetof the observables^µ_i to an infinitesimal perturbation performed at a previous timet^′ on an observablef_i^µlinearly coupled tos^µ_i is defined as

Causality ensures that the response vanishes if t < t^′. We define the macroscopic linear response as The functional derivative with respect tof_i^µ(t^′)in eq. (4.33) can be written in terms of the source fieldsh^µ_i^±(t^′)sincef_i^µappears to play a similar role in the action functional:

Therefore we obtain a Kubo relation, stating that the response can be expressed in terms of two-time Green’s functions:

where we made use of the relations (4.29).

Finally the four Keldysh Green’s functionsG^ab(t, t^′)can be re-expressed in terms of a couple of physical observables (namely correlation and response):

i~G^ab(t, t^′) =C(t, t^′)−i~ 2

aR(t^′, t) +bR(t, t^′)

. (4.38)

Keldysh rotation

The Keldysh rotation of the fields is a change of basis that simplifies the expressions of the physical observables such as the correlationCand the responseRin terms of Green’s functions. One introduces new fields as

( 2s⁽¹⁾_i ≡ s⁺_i +s⁻_i ,

~s⁽²⁾_i ≡ s⁺_i −s⁻_i , (4.39)

and the inversion relation

s^a_i =s⁽¹⁾_i +a~

2s⁽²⁾_i . (4.40)

We define the Green’s functions of these new fields asi~G^rs(t, t^′) ≡ 1/N PN

i=1hs^r_i(t)· s^s_i(t^′)iwithr, s= (1),(2). We have

i~G⁽¹¹⁾(t, t^′) =C(t, t^′), i~G⁽¹²⁾(t, t^′) =−iR(t, t^′),

i~G⁽²¹⁾(t, t^′) =−iR(t^′, t), i~G⁽²²⁾(t, t^′) = 0. (4.41) The fact that G⁽²²⁾ vanishes identically is very general and can be tracked back to be a consequence of causality. The unit length constraint imposed on the rotor coordinates, s^a_i(t)·s^a_i(t) = 1, becomes an orthogonality constraint between the fields in the new basis, s⁽¹⁾_i (t)·s⁽²⁾_i (t) = 0, and a relation between their norms:s⁽¹⁾_i (t)²+^~₄²s⁽²⁾_i (t)² = 1.

After the Keldysh rotation, the connection with the classical MSRJD generating func-tional presented in Chapter2 is straightforward [253,254,259,260]. Indeed, comparing the relations (4.41) with eqs. (2.27) and (2.34) reveals a very strong resemblance between the fieldss⁽¹⁾_i andψon the one hand, and betweenis⁽²⁾_i andψˆon the other hand. We shall come back to this connection in Sec.4.4.5.

Bosonic FDT

When the system of rotors is in equilibrium at a given temperatureβ⁻¹, the fluctuation-dissipation theorem holds (in its bosonic version) giving an extra relation between the Green’s functions. In Fourier space (see App.4.Afor our Fourier conventions) it reads

C(ω) =~ coth (β~ω/2) ImR(ω). (4.42) For completeness, we derive this theorem in App.4.D.2.