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Operator thermalisation in d > 2 Huygens or resurgence Engelsoey, Julius; Larana-Aragon, Jorge; Sundborg, Bo; Wintergerst, Nico

Published in: Journal of High Energy Physics

DOI: 10.1007/JHEP09(2020)103

Publication date: 2020

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Citation for published version (APA): Engelsoey, J., Larana-Aragon, J., Sundborg, B., & Wintergerst, N. (2020). Operator thermalisation in d > 2: Huygens or resurgence. Journal of High Energy Physics, 2020(9), [103]. https://doi.org/10.1007/JHEP09(2020)103

Download date: 04. okt.. 2021 JHEP09(2020)103 2 d > b Springer , July 2, 2020 : August 16, 2020 : September 16, 2020 : Received Accepted Published and Nico Wintergerst a : Huygens or 2 Published for SISSA by https://doi.org/10.1007/JHEP09(2020)103 d > [email protected] Bo Sundborg , a . 3 Expansion, AdS-CFT Correspondence, Holography and condensed [email protected] , Jorge Larana-Aragon, 2007.00589 N a The Authors. 1/ c

Correlation functions of most composite operators decay exponentially with , [email protected] free fields subjected to a singlet constraint. This study in dimensions N AlbaNova, 106 91 Stockholm, Sweden The Niels Bohr Institute,Blegdamsvej University 17, of 2100 Copenhagen, Copenhagen Ø,E-mail: Denmark [email protected] The Oskar Klein CentreStockholm for University, Cosmoparticle Physics & Department of Physics, b a Open Access Article funded by SCOAP ArXiv ePrint: power of resurgence technology wealthough still these find arguments support are for incomplete. thermalisation in odd dimensions, Keywords: matter physics (AdS/CMT), Quantum Dissipative Systems Furthermore, Huygens’ principle, valid for wavedifferences equations only in in thermalisation. even dimensions, It leadsbut works to thermalisation straightforwardly when is Huygens’ morefind principle a applies, elusive link if to resurgence itperturbative theory corrections does by to noting not an that apply. exponential asymptotic relaxation perturbation Instead, is expansion. analogous in to Without odd non- applying dimensions the we Abstract: time at non-zero temperature, evenin in free an field OTH theories. (operatorlarge This thermalisation insight was hypothesis). recently codified Wemotivates reconsider technical modifications an of early the example, original with OTH to allow for generalised free fields. Julius Engelsöy, Operator thermalisation in resurgence JHEP09(2020)103 9 10 15 ] and 10 composite op- 5 d ], characterised by chemical poten- 9 ], correlation functions in free QFTs – 6 – 7 , 3 4 in any 4 6 R . G – 1 – 9 H H T 3 7 thermalisation 11 T 2 T < T d > 10 3 1 = 4 14 d 13 3.2.1 3.2.2 General 2.1.1 Partial2.1.2 operator thermalisation Non-thermalisation2.1.3 and the thermalisation hypothesis Thermalisation and resurgence ] and their late time behaviour is therefore unlikely to approach ensemble averages, . Such operators can in fact interact with the thermal bath and as such exhibit 2 , Nonetheless, it is known that nontrivial interference effects can effectively mimic equi- 1 3.1 Low temperatures: 3.2 High temperatures: 2.1 Operator thermalisation 2.2 Thermal singlet models tials for all conserved chargestemperature. which in Similarly, free nontrivial QFTs time iserators dependence equivalent arises to when a considering momentum-dependent a range of phenomenainstance, that their are correlation usually functions attributed can to exhibit their exponential interacting decay counterparts. at late For times [ in free QFTssis fail [ to satisfytantamount the to requirements the absence of of the eigenstate thermalisationlibration. hypothe- For example, afterapproach quantum those quenches of [ a generalised Gibbs ensemble [ Free field theories arequantum field the theories (QFTs), simplest rendered andof exactly solvable conserved most by charges. the prominent existence examplesconstrained A of an time direct of infinite evolution consequence set (super-)integrable even of in the thermal presence backgrounds. of such In charges particular, is simple a operators severely 1 Introduction 4 Discussion 5 Conclusions A Thermal contribution to light cone 3 Thermalisation in singlet models Contents 1 Introduction 2 Preliminaries JHEP09(2020)103 ], 10 ], reminiscent of 11 , 10 ], a simple criterion has of a given operator, charac- 14 , 13 confinement/deconfinement phase transition. – 2 – N . To leading order, it obeys our generalisation of the absence of thermalisation /N adjoint scalars, plays the role of a generalised free field ]. N 12 )), built of x 2. This decay is directly related to the thermalisation later extracted )Φ( x d > singlet constraint. These theories have received widespread attention in the N , in complete accordance with the OTH. Below the phase transition, a composite ] by arguments which however do not resolve a difference between even and odd In the low temperature phase we observe the absence of thermalisation to leading order Thus, we will study OTH in a particular class of free field theories, namely those with This note aims to shed light on several remaining puzzles. First, in singlet models [ Clearly, the composite nature of an operator is a necessary condition for its effective /N 13 operator tr(Φ( with interaction strength of ordernon-thermalisation 1 condition, confirmed by thebutions absence to of its temperature responseis dependent functions generic contri- and to in all particular QFTs the that lack admit of a exponential description damping. in This terms of generalised free fields and the singlet models thus providesgenerally, insight however, they on allow putative one black tofrom holes disentangle those generic in properties particular higher of to spinworkings composite gravity. strong of operators gauge/gravity More coupling, duality. thereby teaching valuable lessons onin the 1 inner structure on compact spacesIn with ordinary a AdS/CFT, large this transitionPage transition is from also thermal present AdS to andphase, the can large thermalisation be AdS in black mapped holographic hole to in gaugehole the the theories bulk. formation Hawking- is In and in the deconfined direct equilibration correspondence in with the black bulk. Understanding thermal properties of free results on response functions in different phases. a large- context of gauge/gravity dualityan as infinite the tower holographic of duals massless of fields gravitational of theories higher with spin. They exhibit an interesting thermal learn about thermalisation or singlet models,developments in of fact both, concepts from indeed a leads closerthe study. to singlet a The requisite model more precise studyexhibit formulation demonstrated of different how the relaxation phases OTH. properties,functions below Second, diagnose most or thermalisation, above plainly previous a for singlet critical model auto-correlators. studies temperature should Since be response extended with the calculated correlation functionsdimensions were observed toin display [ exponential decaydimensions. in Since even theexponentially, odd-dimensional the singlet results model appear to correlation be functions in do tension not with decay each other. There is something to been formulated that guarantees the terised by a lack ofaddition, exponentially it decaying was contributions conjectured to that itsfails a linear this converse response non-thermalisation statement function. condition can In be inas fact made the and thermalises. Operator any This Thermalisation operator conjecture Hypothesis that was (OTH). introduced collision-less Landau damping [ thermalisation, since only thenture does dependence it couple of to itsa a response sufficient thermal functions. bath, condition indicated On is by the less a other understood. tempera- hand, to In which recent extent work it [ is also their spectral densities have support in the deeply off-shell regime [ JHEP09(2020)103 ] 13 and a 3.2 ] and consid- thermalisation 13 ]. They discuss ]). By explicitly 2 , we also introduce 15 1 , we then deduce and 2.2 3.1 , we introduce the concept of op- . 2.1 5 2 OTH on a firmer footing. This light cone – 3 – d > leads up to our conclusions 4 2. ]. That cautionary observation aside, our scrutiny of OTH permits us to , when it decays exponentially, in line with the retarded Green’s function d > 19 – 16 Our paper is organised as follows. In section The general lessons from our study concern details of the formulation of OTH, and spectrum, as done bythermalisation the in eigenstate terms thermalisation of hypothesis, antion exponentially ETH values in fast [ response return to to athe equilibrium perturbation of retarded by the operator Green’s operator expecta- function in ofof question. the a More operator perturbation precisely, in in [ question is taken to define 2 Preliminaries 2.1 Operator thermalisation Sabella-Garnier et al. formulated the operatorered thermalisation the hypothesis thermalisation in [ propertiesrather of operator than correlation operator functions in expectation a values fixed in background the presence of assumptions on the energy discuss absence of exponentialtemperature relaxation phase. in singlet Themensions model high and response temperature appears functions to phase,discussion in allow which in for the section displays it low relaxation in in odd even dimensions, di- is analysed in section condition in all erator thermalisation, non-thermalisation andgeneralised the free role fields. stable Before thermalthe going quasi-particles potential into relation and basics of of thermalisation singlet to models resurgence. in In section involved in isolating exponentially decaying termsin in odd the response dimensions. functions We becomebut describe critical also the indications difficulties and thatresurgence find the [ some resolution support requires forgive more a thermalisation, more powerful precise tools formulation of from both the the hypothesis theory and the of converse non-thermalisation focusing on evaluating correlators closeodd to and the even light dimensions, conebetween we and different reduce identify the dimensions, the difference tolimit damped between notwithstanding, put quantities crucial that differences between continueOTH even analytically and can odd be dimensions remain. confirmed While straightforwardly in even dimensions, we observe that subtleties the source is retainedare to effectively a lost larger in degree exponentially damped than terms. in standardthe thermalisation, difference although parts between eventerior and of odd the light dimensions.and cone odd plays Indeed, dimensions a it (cf fundamentally is Huygen’s different principle well role and known in Hadamard’s that wave problem the propagation [ in ineven dependence becomes significantly richer.dent Response and functions are become characterisedplane. by temperature non-analyticities depen- They off describe the aexponentially damped real response decaying axis contributions. to in sources, The the withpredicted latter complex a by contribution frequency power the is law OTH. tail the The and exponential sub-leading presence damping of the power law tail implies that information about thermal version of this concept will be presented below. At high temperatures, the time JHEP09(2020)103 2 In was 1 ). ] and the 13 2.1.3 do not even ] to state that 13 thermalisation, and state a non-thermalising operators partial ] are aware of the need for some refinements. 13 induced by a perturbation by the same to mean that: the retarded Green’s func- O O ] to argue that they describe a sector of the 21 ]. While pure exponential decay occurs in free – 4 – ]. We will see that these operators 13 13 , 11 , 10 , of an operator 6 . In essence, these non-thermalising operators are generalisations of free operator thermalisation hypothesis. Our arguments are essentially copied . The requirement of exponential decay for a perturbation to thermalise corre- partial thermalisation contains terms with exponentially damped factors at late times. This definition O partial ], and the “partial” qualifier only indicates a slight shift of definitions. The new 2, as well as multiplied by inverse powers of time. We will provide such examples O 13 We interpret the operator thermalisation hypothesis proposed in [ Crucially, partial operator thermalisation captures the observation that conservation A motivation behind the operator thermalisation hypothesis, and one of its strengths, d > Identifying sub-leading exponentials isIn subtle private and discussions, we requires have special found that attention the (provided authors in of [ 1 2 any other localThis operator, is not the converse representing of a the stable above non-thermalisation quasi-particle condition. field, For it will to thermalise. hold, the notion they correspond to stable thermal quasi-particlewhich fields are having clear-cut permitted dispersion relations, the to Narnhofer-Requardt-Thirring differ theorem from [ thermal those system of which free isrelations. completely relativistic free The particles. from theorem permits interactions, Onequasi-particles. except other sectors, for may but modified invoke they dispersion are completely decoupled from the all other operators thermalisecharacterised partially. by A Sabella-Garnier special et classthermalise al. of partially [ fields which satisfy a sharp dispersionconditions relation on relating the energy operators to momentum. are given Formally, the by the mathematical descriptions below. Physically, such cases, time evolution stillClearly, partial “forgets” thermalisation part includes of themore the thermalisation conventional initial notion notion discussed perturbation, of in but approach [ not to a all thermal of ensemble, it. butlaws it prevent is some a operators broader in concept. free or integrable theories to thermalise, but that almost 2.1.1 Partial operatorWe thermalisation define tion of allows for leading power-law decay, and exponential terms which are only sub-leading. non-thermalisation condition which excludes this kindconverse of from [ definitions are important forapproximately the consistency same. with the examples we discuss, but the idea is is that it can beand used free to or study integrable surprising systemssystems similarities in [ between contrast ordinary to interacting naive expectations, systems for it is generally masked by leadingbelow. power law In decay reviewing the operatorterminology thermalisation hypothesis, which we precisely will captures therefore these introduce new features. In effect, we demonstrate an operator operator sponds to the intuitionlate that times. a The thermalising latter means perturbationreconstruct that the is exponential source “forgotten” precision from by would the be the response required system of in at the order medium. to fully encoding the linearised response of the operator JHEP09(2020)103 , . Q 2 2 i| i| . To (2.1) (2.2) (2.3) (2.4) (2.5) n n Q , | Q | | Q |

i m ) to distin- m n | |h |h ) t, x o ( x · t,x i ) Q ( )) n p Q n − p ) + 0) m m , − p ( E i (0 m − p − t ) ( |Q β n n x · n i E − h E iP s − k 0)] 2 e ], expressed in our terminology. ( m , + ( E 1) δ 13 ( (0 iHt i ω − e − s ( Q s 2 e , 2 ) 1) 0) i− 1) ], and also play a natural role in AdS/CFT as , n − | t, x − ( 20 : any local operator not representing what (0 ( ( ). By definition it has a definite dispersion 0) can mediate between momentum states de- − Q Q , − [ x · t, x h Q x (0 i ( · ) ) iP t )) – 5 – Q O n or a generalised quasi-particle field, thermalises − ) m p e ) + Θ( 3 p − i n t,x m − iHt − E ( . Making use of translations p ( e ] still apply. i n Q − p |Q + 13 t ( ) = n ) m ) = h n − E E t, x k n ( − t, x ( ( + ( R m δ βE E Q G ω ( − i e − e n n n X βE n ) ) − t ) β ) ) e ( βE t β β ( i − Θ( ( Z e i Θ( Z m,n X Z i − − × − m,n X 0 is infinitesimal. ) = × is the spin of the operator ) = ) = partial operator thermalisation hypothesis > s t,x ( t, x The retarded thermal Green’s function is Consider a stable thermal quasi-particle operator, which we denote ω, k R ( Generalised free fields were originally introduced in [ ( 3 G R R G G where boundary duals of free fields in AdS. In Fourier space and inserting a complete set of states where with volume. which can be expanded in a sum of expectation values termine the simultaneous transitions betweenif energy there eigenvalues. We is do a notor single need if functional to there know relation are betweenreproduce several momentum branch branches and cuts of energy which solutions for canto to be the be the found, operator important dispersion for example, to relations in allow coupling singlet for to models, a it growth will turn of out the number of solutions to dispersion relations We now proceed to essentially repeat the arguments ofguish [ it from more generalrelation. local operators In finitemomentum, this volume means and that in the transitions a basis which simultaneously diagonalises energy and precise we call a thermalpartially. generalised Note free that field, we stillAll have the not plausibility proven arguments this in hypothesis, [ though we2.1.2 find it reasonable. Non-thermalisation and the thermalisation hypothesis of thermalisation has to be weakened to partial thermalisation. Thus, we propose a more JHEP09(2020)103 by i (2.7) (2.8) (2.9) n | Q | , m 2 h i| . n ) ] remain, and this i | Q | 13 ) grows without bound limits, it may become m k ) + |h lead to a proof of non- k − N M )) − ( β, Q m ( Q j p Q j − H + Ω n , p ) ω ( n s and defining the residue functions 2 p − are due to transitions between states 1) k Q − ( Q − δ ( ) (2.6) m n p k ( ( − βE Q j Q j i − e ) ) – 6 – = Ω = Ω n ) + p n k ω − ( β, k E ( m Q j p Q j ( − Ω Q j H m limit, branch cuts may also arise, but they will be on the − X the number of different branches of solutions E =Ω ω n N ) has to be real by definition, the retarded Green’s function E ( k M ( − m Q j =1 M j X E , the singularities are manifestly poles. If ) β M i ( ) = to the dispersion relations for Z − ω, k ( R ) = G ,...,M . For finite β, k ). Noting that Ω = 1 ( Q Q j j ). For a more detailed analysis of thermodynamic and large 2.8 H 2.6 The OTH in our version becomes: all local operators which are not generalised quasi- The converse of the original non-thermalisation result would be that only stable quasi- Now, the special properties of the quasi-particle operator . This generalisation replaces quasi-particles with generalised quasi-particles or thermal expansions containing both inverse powerstems, and they force damped us exponentials toisation, of consider time. which the allows In partial, for and freeUnfortunately, the more sys- the general, possibility price version for that of the exponentialtion. operator generalisation damping thermal- It is terms is another required are level for of sub-leading. a mathematical physical sophistica- reason: only under very special circumstances, e.g. when particles thermalise partially. Theadjusted original version plausibility survives arguments all of tests [ we have considered. 2.1.3 Thermalisation andBelow resurgence we will introduce examples of retarded Green’s functions with asymptotic late time particle operators are non-thermalising. Allowingpower for partial law thermalisation, fall-offs which includes relatedbranch to cuts branch also cuts in onM the the non-thermalisation real results. axis,generalised it Thus free seems we fields. judicious are to led consider to allow unbounded operator in the thermodynamic or large real axis. There will stillreal not axis even can be produce partial exponentially thermalisation, decaying since terms. only singularities off the whose energies and momenta differ by amountseq. related ( by the allowed dispersion relationsuseful in to allow an effective temperatureto dependence eq. in ( the dispersion relationsonly contributing has singularities on the real axis, which is tantamount to non-thermalisation of the The frequencies and wave numbers above are related to the matrix elements signifying that all contributions from the operator we find thermalisation. Denoting by labeled JHEP09(2020)103 1 − d ) or S of the N × R R of fields. A N complementing power series 2 /g 1 − , the inverse of the radius e , /R symmetry group, for example U( t/β − ]. N β/t e 23 → → ] can be systematised in non-perturbative tech- – 7 – 2 2 g 22 /g 1 is typically asymptotic rather than convergent, and ], would lend itself nicely to an extended definition − e 19 t/β – 4 in the limit of vanishing gauge coupling, where only the 17 µ A that imposes the Gauss’ law constraint remains. We will focus 0 is inverse temperature, we are alerted to the possibility that ther- A β 1 − d S R . Substituting g ∼ . To make the distinction sharper we consider a large number α 1 − d will then allow for qualitatively different limits for physics below and above the is time and S t N ). Projection onto the singlet sector is achieved by weakly gauging the symmetry, The beautiful idea that there is a relation between the form of non-perturbative terms This discussion is parallel to the potentially more familiar discussion about prescription Such a subtle definition may seem outlandish, but is apparently useful in the mathematical description N 4 O( i.e. introducing a gaugezero field mode of related physics, namely classical Landau damping [ leading to a characteristicsphere temperature scaling as 1 large characteristic temperature. Weusually consider fundamental a or scalar adjoint, field of transforming some in large a representation, of partial thermalisation.the The series response contained function exponentials. would be said2.2 to thermalise partially Thermal if singletIn models order to distinguish low and high temperatures, we consider free field theories on niques like Borel resummation, but doesbe not clear, always for yield thermal a unique Green’sunambiguous. answer functions A for in series the free representation series. does fieldresponse To not theory, function. improve the the integral However, already representations a complete are double well-defined encoding series representation of expansion of a with the inverse result powersframework and of of exponentials the resurgence as integral above, theory in a [ trans-series a in the functions. Cases withbe terminating useful or in at practice, andexceptional. least would a allow convergent unambiguous identification (inverse) of power exponentials, series but would are and the divergence of asymptotic series [ where malisation, signalled byperturbative exponential effects. damping The at analogytime expansion late indeed in holds times, inverse powers for of canexponential standard terms Green’s be can functions: sometimes analogous be their to extracted late from non- integral representations of the Green’s There, one encounters non-perturbativein exponentials a coupling sub-leading exponentials from morecircumstances important can we inverse have powers. ain chance Only principle. to under resolve and these observe special the damped exponentials, even dependence of non-perturbative terms in quantum mechanics and in quantum field theory. an asymptotic series of inverse powers terminates, is it possible to operationally separate JHEP09(2020)103 , or (1), O (2.11) (2.10) dτα ). It is 1 . This is S ). π R . 1 number of T < 2 i t, x λ ( ( e δ N G #) ]. This is read- ) = 2 , → . At very high 2 − λ P ∼ 24 d ( ) 2 )Im ) can formally be N ) ρ t λ θ ! ( 2 ρ 2.10 2 − 5 cos d ) − , corresponding to times θ ) are real. An imaginary ) ) = Θ( 2 m R ρ cos . Eq. ( t, x 2.10 0 iβm ( 0 − R + N ) t G imλ e ). At low temperatures, iβm λ (cos( ( + ρ . At intermediate temperatures whose " 0 t N Re i is the polar angle on the sphere. The entire ]. The distribution of the large =1 (cos( ∞ X θ m 25 (0) – 8 – # ], there are distances and times where inside correlation −∞ O 2 ∞ ) = 2 X − 26 2 , d m − ) x, t 10 d ), which appear as eigenvalue densities in the more θ

( ) , where λ ) θ ( R λ 1 hO cos ρ ( ) 1 cos − d 2 t π 2 − − ] dλ ρ t d +2 Rθ ( π π 10 d 2 (cos − 2 ] corresponding to the Polyakov loop operator, Γ Z " (cos | ≈ ] 24 x = ≡ | )) for scalars in the adjoint representation, whose time ordered two- 10 ) , Im x R +4Im t, x ( t, ( ) ( t 2 G , the model can be solved in a saddle point approximation [ t N tr(Φ 1 ) = Θ( N t,x ( ) = R Correlation functions of singlet operators can be constructed through finite temper- At large At finite temperature, the integral over the gauge field can be recast into a unitary ma- There is a subtlety here. As shown in [ G scaling depends on the representation under consideration, there is a transition to a t, x 5 ( functions, the finite widththe of regimes the considered eigenvalue here. distribution always matters. This is however unimportant for simple to see thatpart the can purely thus thermal contributions ariseExplicitly, one to only finds eq. from [ ( the vacuum piece, and the mixed thermal-vacuum term. points to zero. The pre-factor hasis been normalised chosen to such simplify that thederived expression, its while using two-point the operator function the is aforementionedmatrix of Wick is order contraction, represented as inGreen’s the well scalar function as kinetic the can term fact be like that a extracted temporal the using gauge unitary its field. definition, The retarded where we have used rotational and time translational invariance to set one of the insertion ature Wick contractions.O For simplicity,point we function focus is given here by on [ the scalar singlet primary, the dominant saddle corresponds tothe a confined constant phase, eigenvalue with distribution, N a free energy ofdeconfined order phase, characterisedtemperatures, by the a eigenvalue free distribution becomes energy a that delta-function, is extensive in trix model, where the projection ontoover singlets unitary results matrices from the [ integralgauge over the holonomy gauge around group the thermalmatrix circle eigenvalues [ then controls the thermal behaviour. ily achieved by introducing the eigenvalue density and distances difference between low andencoded high completely temperature in physics in functions detailed effectively description flat in space the can next then two be paragraphs. attention on correlation functions on scales much smaller than JHEP09(2020)103 2. (3.3) (3.1) (3.2) d > ) fails to t, x ( . The low and O 4 considerations. At low . In odd dimensions the N , 2.1 . , 2 n = 0. For the retarded Green’s − − d = 4 2 ) d 6=0 θ N k become equal, which is the case in ρ in ∼ m cos 1 ρ − 2 2 below, even for response functions of conn t ω , the eigenvalue distribution at low tem- ) is analytic everywhere off the real line i 2 ) -point functions − µ n d > 3.2 n = 1 and 2.2 2 (cos -th Fourier cosine coefficient of the eigenvalue , x k 0 k n ρ t – 9 – ( log O ) Im − t ... H ) ). We note that the infinite series in the second term ) = 1, and Fourier transform, thus for example obtaining 1 kλ , and thus ) = Θ( , x on the real line, representing a continuum of physical ex- scalar field, when all 1 ω, k π 1 t ( 2 2 ( t, x T < T R t, θ ( ) cos( G R hO > k λ ) = ( G λ 2 ( ω ρ λ ρ ) involves power-law fall-off, and we will find similar fall-offs to be )). d R 3.1 t, x fundamental ( = 2 limit. From the explicit lack of exponentials we see that k ), this implies ρ N tr(Φ corresponding to 1 N 2.11 is a renormalisation scale. In this Lorentz invariant expression, there is only one R µ ) = The physical origin of non-thermalisation is clear from large t, x ( , one finds for the connected components of general consequences of freeoperators field that dynamics display in relaxation after long time in exponentiallyT decaying terms. corresponds one-to-one with thespace lacks fact exponentially that decaying the contributions.tend corresponding the We see notions expression explicitly of in thatto the configuration it cuts, non-thermalisation as is condition long useful beyond asdynamic to poles they ex- limit in are the of on frequency ( the plane real axis. In position space, the corresponding thermo- where branch cut located at citations on top of the vacuum state. That ( in the large thermalise at low temperatures.of For large completeness, let us take the “thermodynamic limit” 3.1 Low temperatures: As noted in the thermalperatures singlet is model constant, section function ( high temperature phases of theseparately singlet below, models with are qualitatively equationsboth different specialised and cases, are to the discussed scalarsO concrete in operator the under adjoint study representation. is In the lowest dimension singlet operator 3 Thermalisation in singletA models number of challengesIn to this the section, OTH wedimensions, may given describe the be adjustments our we tested have technical introducedinterpretation in in of results, section thermal results which is singlet support intricate, modes and the in will hypothesis be in deferred even to the discussion distribution, captures all temperature dependence, andpropagator in of fact the is preciselythe that high of temperature the thermal limit. Feynman where in the last line we have introduced the JHEP09(2020)103 . . i /N = 4, (0) (3.5) (3.6) (3.4) -point d O ) n y ). ( ..., O momentum ) t, x + ( x O ( # O ) 2 k hO . 2 − ) d − ) p ky 2 #) t + − 1 1. In this regime the 2 ( px 2 − − ( d O i 2 ) ) ) x 2 k ( y e ) ( β d 3.5 " O d ) x p iβm 1 d ( corrections will reveal nontrivial d + O 1 and ) t R + 4 Im ( /N ≡ − p,k 2 ]. ) are then parametrically close to being ( θ 2 − ) 1 d 27 x − 3 . ) ( t, x 1, 2 ( p, k t " 1 ( k G O 3 = 1 − d , as long as background energy densities are not Re G d k 2 t – 10 – ρ x =1 /N ( Z ∞ X m and 1 N H × i T Im )+ ( (0) p ) t ) becomes upon insertion of ( − O ( ) T Θ( x O 2 ( ) 2.11 − p d ( hO 1 − ipx G plays the role of a coupling constant. Thus, individual ) = 2 ) x e p d ( t, x . Composite operators like d , here corresponding to /N ( O R R N R . Up to cubic order, and for simplicity displayed in flat space where trans- G p O ≡ d d ) = 4 p analysis, and is therefore beyond the scope of this work, even the leading order ( Z d G ∼ N By the above argument, taking into account 1 eff S contributing to the response function of the3.2.1 quadratic composite operator To get a better understandingsince of generalisation the precise to dynamics, higher we even will confine dimensions ourselves is to simple once the basic ingredients are Clearly, the operator now respondsIn to fact, the the thermal second term bath, represents which the may cross induce term between thermalisation. vacuum and thermal propagation Note that one shouldlimit” only of really large expectretarded effective Green’s thermalisation function in ( the “thermodynamic 3.2 High temperatures: At very high temperature,function. the One thus eigenvalue obtains distribution for can the be Fourier cosine approximated coefficients by a delta- finite behaviour can change drastically onceof occupation order numbers of in the the inversehigh thermal coupling. temperature background phase. Indeed, are as we will show now, this is what happens in the In consequence, quasi-particles remainrather intact superficial, to but thisaction, can order. for be example Of made using collective course, more field this precise theory argument by [ is properlyfeatures in constructing the the response effective functions even below the phase transition. While this requires a lation invariance constrains its form, it reads where Here, the conservation of individual momentum modes is explicit to zeroth order in 1 modes are conserved to leading ordermacroscopic in in 1 generalised quasi-particles, and should notcriterion. thermalise according To to see the this,functions non-thermalisation of consider the effective action that generates the connected In other words, 1 JHEP09(2020)103 . than (3.9) (3.7) (3.8) 2.1.2 expan- ensures , N O . interactions    ]). However, k k smaller 0   . 28 − + ) ) to the branch N | x ) ω ω counting, which is 3.8 β −| t ], ( N log π πt β 10 suppression of interac- 2 k ω N ) assumes the vanishing of coth | − x coth | − 3.4 β π ) πi β | 2 ) thus probes indirectly. kβ ). 2 x | β . Only large + + t ), yielding [ 3.6 3.2 ( t limit, due to order 2 − π 3.1 3.8 − counting, thanks to the thermal con- ! N )) )) ) coth N | k k π x + − on the real line and the subdominant expo- + − | ω ω , which invalidates the argument that individ- k 2 ( ( t ( x π π N – 11 – iβ iβ δ 4 4 2 − and − − ) 2 | ) + t k | Γ( Γ( x x −  

) − | | − | . Above the phase transition, the equilibrium background t x log t ( β ( δ 0 − | t 1 δ πi t kβ hOi 4 ( ( δ t 2 1 t ∂ + 2 + 1 2 2 −    ) = ) t ) = t, x d > ( Θ( R ]. It can be contrasted with that of eq. ( t 2 ω, k G above the critical temperature, which eq. ( ( 29 plane located between R ) = O ω G t, x does not represent a generalised quasi-particle, by the arguments of subsection ( momentum modes are conserved in the large R O Let us now return to how thermalisation could be consistent with the effective action O G 3.2.2 General The above retarded Green’s function ofinto an a operator vacuum-vacuum term quadratic and in a free mixedalso fields vacuum-thermal clearly term. decompose separates Higher analogously. powers of free (Purely fields thermal terms will not contribute to the retarded with the background. Generalised quasi-particlession, are although then their not intact response inthat functions the are large well-defined. TheAs thermalisation explained of above, thisdensate is of consistent with large the same at high temperature, seemedtions to is be indeed the important. same The asthermal large at one-point low temperature, functions but the actionexpectation ( value is non-zero andual of order This expression allows us tocut map in the the late-time dominantnential behaviour decay of to ( thediscussed branch in cuts [ located off the real line.arguments Similar presented in analytic the low structures temperature are discussion may be further illuminated by Fourier transforming ( Evidently, the Green’s function fallsin off vacuum. as a This powerprevalent is law, in with in generic a thermal fact systems power a in thatjudging the manifestation is by high the of temperature coth the limit term (see effective there e.g. dimensional are [ sub-leading reduction exponentially that decaying is contributions. This which can be simplified to understood. There, JHEP09(2020)103 (3.12) (3.10) (3.11) . . A ) terms produce t, x ( m 0 is given by d ˜ S ≥ ) t n , Θ( ) 2 − , d t/β ) | 2 . For all even dimensions the 2 , the temperature dependent m d x | ≡ A π − # for integer 2 2 (2 t 2 − 3 ( d 2 − δ +1) ) d 3 2 n n ( 2 ) ) − ! K d 2 − t 3 n ) ) 2 ∂ − 2 ) is then ( d t iβm | 1 ) are discussed in the appendix t/β t − ( 3.6 = + m ( | 2 d t 3 ( – 12 – x S 2 − +1) d − n ( π 2 ¯ t − x ) ( 2 mπ t " 2 − − e Re 2 2 x } ) now leads to an asymptotic expansion in inverse powers for =1 0 ∞ X m \{ Im( ) Z t 3.12 X ∈ m Θ( 2 , while square root branch cuts ensures support also inside the light − d d ) as an expansion in modified Bessel functions t ( ) and its light cone limit d = 2 S . This is a known property of the wave equation, which evidently is inherited lc t, x d ) ( ]. terms in the series above explicitly yield exponentially decaying terms, which d ˜ S 30 t, x ( m , which does not terminate. Hence, it is far from clear what significance to attach R G m The odd-dimensional case is significantly more subtle. A similar treatment of the Bessel In even dimensions the calculation confirms thermalisation on the light cone, essentially The light-cone factor isolated from eq. ( function terms in series ( each to exponentially small terms.larger If than the the asymptotic exponentials series we isare have truncated, used extracted, the [ even error if terms exponentially will improved expansions be positive are of a formpower that law cannot fall-off. cancel with Thus,cone other the in exponentials. response even functions The dimensions. signal negative partial thermalisation on the light each of which equalsfor a even decaying dimensions exponential (i.e.finiteness times of half-integer a the orders expressions terminating of are sum also the of given Bessel inverse in function). the powers appendix Details concerning functions by expressing This expression lends itself tosion, a and comparatively it uniform treatment measures independently the of effect dimen- of the heat bath on the light cone in position space. The term of the response functionlight is entirely cone determined divergence. by the We factorbehaviour thus which of factor multiplies what out a might simple its beof singular called terminology. the light position cone space behaviour “residue” and of study the the singularity, by abuse light cone for even cone for odd by thermal systems probedthe by behaviour composite in operators thefunctions built than interior of the of powers retarded the of Green’s light free function, cone fields. and is for While odd interesting, both for other correlation even dimensions. The imaginary part of ( which demonstrates that the support of the retarded Green’s function is confined to the propagator.) Now, the vacuum factors differ significantly in behaviour between odd and JHEP09(2020)103 ] , d 13 which is too 0 N 2 treatment of [ , they have in many 1 d > operator thermalisation. 2. The ). These examples clearly show partial , to be valid at leading order in A.3 d > 2.1.2 ) from the high temperature phase of ) and ( 3.12 3.8 )–( – 13 – 3.11 = 2 discussion, and we were able to resolve new d ) lacks exponentially decaying terms and the individual 2 motivate us to consider 3.1 = 4 expressions ( d d > ). connects to the intuition that such operators should not thermalise. 2.9 3 limit, where there is a phase transition, and below the critical temper- N . To the extent that the generalised quasi-particle picture holds, we can rely on /N To conclude the match of the OTH and singlet model response functions we should Above the critical temperature, the response functions develop exponentially decaying In the large Known thermal behaviour of singlet models motivated a closer study of response func- now argue that theabove operators the critical we temperature. considerwe fail have Without found to going a be deeply suitabletemperature into generalised mechanism, phase the namely quasi-particle physics that modifies operators of themuch the background singlet for propagation condensate models, a in of generalised the free perturbations high field, at unless there order is extreme fine-tuning. which generally appear in This partial thermalisation concept alsothermal simplifies the quasi-particles, non-thermalisation results by forparagraph allowing stable after branch eq. cuts ( in the thermodynamic limit as in the non-thermalisation. Indeed, the ideato ideal that cases the makes general the results theory much applies more parametrically powerful. close In thisterms, example, as we for see example in how the ithow works. power law tails and damped exponentials combine non-trivially. Indeed, such terms ature, the response functionmomentum ( modes are independentlythe conserved. presence This of means generalised non-thermalisation quasi-particleseral non-thermalisation and of results. definite also momenta, Clearly, we asoperator can expected thermalisation only from theory, expect the described a gen- small precise in 1 match section to the general the singlet model responseit functions. proved important To to get focus expressionsthe on which support the depend light for cone. analytically theHuygens’ on Thereby, the Green’s principle qualitative were functions difference factored in between out. the light cone, related to the absence or not of any other operators thermalise partially. tions in order to compareis with somewhat the OTH less ineven/odd detailed dimensions dimension than differences the in eqs. ( for operators which areerators non-linear belonging in to free this fields.instances class, been and Composite shown as to operators display described are theof in decay regularised the which the op- we OTH introduction take is to definethermalise that thermalisation. partially the would The implication have idea to could be go generalised in quasi-particle the operators. other Or direction: equivalently, operators that do not Our description of an important classoperators of in non-thermalising section operators as generalisedGenerally, free however, field naive intuition isvation treacherous, that and free our field study equations is of founded motion on do the not generally obser- guarantee absence of relaxation 4 Discussion JHEP09(2020)103 ) ) A.12 πt/β 4 − . At half 3 2 − ) and ( d A.9 is divergent and the are integers, but the n β/t ), where , now fails to terminate. Instead β/t πnt/β 4 − = 2, operator thermalisation is generally d – 14 – singlet models, we have found both non-thermalising and N vanishes. This enhances the scope of our analysis. /N . The further sums in the thermal response functions primarily gives terminates and the damped exponential terms can be distinguished from β/t β/t The analysis is comparatively straightforward in even dimensions, where Huygens’ In our model system, large Tentatively, the Borel summability of modified Bessel asymptotic expansion indicates In odd dimensions Huygens’ principle does not apply and some of the induced thermal- Partial thermalisation diffuses the dichotomy between thermalising and non-thermalis- haviour above the critical temperature.turn Importantly, out the operator to thermalisation belimit, concepts applicable in this to case operators when which 1 only satisfy theprinciple holds theoretical and conditions ensures in that a the thermalised responses induced by a heat bath are lo- ing terms at late times.in free This finding field establishes theories: thethe while intermediate more exponential nature unyielding relaxation of time is thermalisation dependence ubiquitous, expected it from typically the coexists presencethermalising with of behaviour of conservation laws. the sameexponential operator: relaxation generalised below quasi-particle behaviour the without critical temperature, and thermalising exponential be- 5 Conclusions We have refined the operatoreralised thermalisation free concept fields. and the Exceptincomplete OTH, in and and partial, the related since special it there case are to power gen- law tails that dominate exponentially decay- that there are nosuggest exponential that correction the terms in exponentialon its terms the asymptotic we other expansions, actually hand which findorder error would in as terms the the of appendix sub-leading even are exponential doubly not terms. improved masked, asymptotic but series are of the same exponential terms can only beresurgence ascertained of within asymptotic a series. larger Ining framework, such to such a partial as framework thermalisation the one of studybut should composite of a be operators firm able in conclusion to odd-dimensional is assign free a beyond field mean- the theories, scope of the present work. isation diffuses into the interior ofthe the light light cone, cone. which The corresponds seriesit to encoding the produces the polynomial thermalisation an in on infinitemodified Bessel asymptotic functions expansion of controlled integertask order. by to The identify the resulting sub-leading asymptotic series exponentials expansion in becomes of quite subtle. The potential meaning of expansion in the resulting polynomial. Thermalisationthan can be if confirmed power although law with tailsin a had even bit dimensions, not more when been work the present.light whole cone This effect by comparatively of Huygens’ simple the principle. procedure induced works thermalisation is confined to the ing operators to somedemonstrate degree, the but general in structureinteger even from order dimensions the modified calculations resulting Bessel functionsand like functions are ( powers of simple of polynomials order rise of to exponentials an exp infinite ( series of higher order terms exp ( JHEP09(2020)103 (A.3) (A.1) (A.2) ! ] which 2 t 31 1 − 2 x − 2 ) . . iβm 2 # t + 1 2 d t 2 1 − − ( d 2 ) − x . 2 ) 2 2 − t x 1 8 in any iβm ) 1 − x R −∞ + ∞ = t − X πt β β G ( t 2 m ( −

π 2 1 2 x coth ( – 15 – = " coth βt π 4 − 2 Re ) ) x =1 ) = ∞ t X m iβm + ( β 4 t + 1 ( S t ) = π ( − t, x = 4 we have After we received the proofs we noted the new paper [ ( 2 d coth d x ˜ S π βx Re ), the temperature-dependent part of the retarded Green’s function is 2 =1 3.6 the retarded Green’s function only has support on the light cone we will ∞ X 1 2 m d = ) = t, x ( 4 ˜ S Finally, we find it inspiring to contemplate other conformal or integrable systems, in Some properties of singlet models that are highlighted by our study generate further Since in even only evaluate the sum there. We have For later comparison, in From equation ( determined by the sum work of BS was supportedthat by the of Swedish NW Research by Council FNU contract DNR-2018-03803 grant and number DFF-6108-00340. A Thermal contribution to light cone Note added in proof. has interesting overlap with our work. Acknowledgments It is a pleasure to thank P. Sabella-Garnier, K. Schalm and J. Zaanen for discussions. The tives? There are also holographiclimits gravity of duals large to N these gauge questions, theories, since some singlet models of are whichparticular are in conformal. odd dimensions, where resurgence appears to be fundamental. questions. For example,elusive. an efficient We expect descriptionrepresent that of generalised all the standard quasi-particles. high compositemodynamics temperature operators of The phase the will fundamental remains high thermalisephysical free temperature and singlet fields “deconfined” states. no Φ limit longer Do describe well, they but the provide they the ther- do best not description, or represent are there better alterna- intricate due to their distributionWe over refrain the from forward light formulating conea a and deeper definite the conceptual conclusion whole of analysis. inverse its powers, odd and interior. The dimensions, decaying importance exponentials, since of ofbetween time, we thermalisation suggests simultaneous believe of resurgent infinite integrable analysis. in systems expansions A and connection in resurgence may in- find further applications. calised to the light cone. In contrast, the thermalised responses in odd dimensions are quite JHEP09(2020)103 (A.6) (A.7) (A.8) (A.4) (A.5) ! ) ¯ t πm . s cosh(2 s d d 2 s r ) m r d 2 ! # π ) s 2 ) d − itβm r 2 e s r s ) πm − 2 2 ) =1 tβsm + ∞ 1) d ¯ t m X ( m . Using the modular transfor- 2 1) d r − β iβm − ) − + + s . e t t/β cos(2 2 r d ) ( t s − 2 1 + 2 − − 2 r . = # ys 2 2 −∞ sβ

d m 2 x ¯ ∞ t x − ¯ t, e = ( 2 ( X 2 − − 2 e r 4 ¯ t β d − − m 1 ( r 2 − ) e − e d 5 − ϑ − 2 4 e 4 2 ( α − ) r e 2 2 − − d 4 s d d tβs, e r π =1 2 r − ( ∞ ∞ s s d X ∞ 0 m ϑ iβm 0 r r ∞ ( Z 1 ∞ ∞

0 Z 4 – 16 – 0 0 ) + ∞ Z 2 = 4 − ) Z 2 ) yields Z 0 d t 2 − ) 2 . α − ) ( d Z s 1 2 d 2 2 s 2 ) 2 ) r − A.6 Γ( 2 − π − ∞ 2 Re d 1 d Γ( 1 0 r ∞ 2 2 − − πm 2 π 0 = Z x d Γ( =1 − ( Γ( Z + ∞ ) 2 √ ¯ t d X α ¯ t, e m ( 1 2 ) " " Γ( 1 − β y r ) 2 2 2 d − iπ 2 − 2 2 d − − 1 β Re Re 1 e 2 d d − 2 1 − d β ϑ =1 =1 Γ( 2Γ( 2 2 ∞ ∞ −∞ ¯ t X X Γ( m m ∞ r ) = = X = = t − m ( = = e ) = , we obtain d t ) = r r π π ϑ ( S d ) can be rewritten using the formula r r S t, x ( d = A.1 ˜ S ) = r ) is the third Jacobi theta function and − z, q ¯ t, e ( r ϑ ( ϑ The sum ( The last term isIn divergent order but to will allow for be such canceled a by cancellation a we regularise divergence the stemming integral from by the introducing an sum. expo- Substituting for this in the integral ( where mation property of Restricting ourselves to the light cone we obtain We thus obtain JHEP09(2020)103    1 (A.9) R   (A.12) (A.11) (A.10) 2 + β R 2 . 2 ¯ t # − q 2 | 2 . − 2 − ¯ t   d m d ) | β ¯ t α π πm R n 2 ( 4 − 2 z n 3 1 − R − 4 a − e 3 + d ! n 2 # − 2 ¯ t d − 2 =0 ¯ t ∞ 1 4 R X n q β − R   K 2 − z 2 1 d 3 2 R + − 2 − 2 β R + d e πm ¯ t − m 2 2 | 2 ¯ t z π − ! 2 =1 m q − ∞ e | ¯ t X 1 m r 3 2 9) r   d 2 ) − =1 ∞ d πm n 3 − − X r ≡ 2 m ) R 2 − 1 π ¯ R t 2 2 − ¯ t d − -behaviour we employ the asymptotic 1 ¯ 2 t π R e + 2 α ( t ) 1 ) 2 ) yields π n n ¯ t mπ z (2 4 + 1 ... 2 − r R a 1 q πm 4 | 2 − 1)(4 A.9 2 ¯ t + − − e + + m =0 2!(8 d ¯ t ∞ | 2 ( ). We obtain X n − 2 q π r + R ¯ t } ¯ m t 2 3 2 0 ! − 25) 2 − 2 m α + − e q 2 + A.3 d \{ d =1 ¯ t – 17 – ¯ 5 t ∞ − ¯ t Z ¯ t (4 1 π X π 2 − m X π 2 ∈ 2 d 4 πm 2 # πm − + α m + r 4 4 r − e ) − − n + 1 d 4 − ∞ 3 } 3 e ¯ t e ) 0 0 2 3 − +2 r 9)(4 R − − 3 d ) Z taken to inﬁnity after performing the integral. We have z d 3 d \{ ) =1 the series terminates at a ﬁnite number of terms. This 2 πm − z − 8 2 ¯ =1 t ∞ − Z − ∞ 4 3 d X e X m d X R Γ( 1 , α m R ∈ 2 2 ¯ 4 t − − α −∞ 4 ¯ t ∞ 1 d 2 2 e ¯ t m − 1 Γ(

α = 3!(8 X + t d 1 ( ) 2 8 m n r    + + ¯ 2 t =1 + ∞ ) with a ) 1 + 2 X − ¯ ¯ t t m 1 1 ∞ 1)(4 − ¯ t 2 2 1 d n 0 2 2 ¯ t − − ) 1     ). π Z z −

d d r/R π πt Γ( β . Substituting for this in ( − ) 2 2 2 3. As we are interested in large 2 2 + 2 √ π π − d e 2 Γ( = 4 to compare with ( (2 Γ( β β π π α β e π − A.3 ¯ t 2 2 2 √ z √ − 2 2 d 1 2 d π 2 d , 2 (4 − − β =0 ∞     " d > d d X n 2

coth    β Γ( 1 β r + 3 2 " 2 2 2 2 2 2 − − 2 β βt →∞ →∞ →∞ π − π − d "    d d d 2 4 R R R ) = π β →∞ β R z 2 2 R + = lim = = = lim ( →∞ →∞ α ) = lim R R − − t ( K ) = lim = lim t 4 ( ) = lim S d t ( S d S which agrees with ( As a check we set We note that forcorresponds half-integer to even which is valid for expansion nential suppression JHEP09(2020)103 J. , D (A.13) 2 singlet 0706 ]. ] (1991) N . 43 (2020) 064 . ! 8 SPIRE ) IN 1 2 ]. A Bound on ]. ][ + n ) n ¯ t J. Stat. Mech. ( , π ζ (1994) 888 SPIRE SPIRE (2 ]. IN Phys. Rev. A IN (0) 50 , arXiv:1903.03559 ][ n ][ ]. [ SciPost Phys. a . , ]. Black holes from large SPIRE =0 ∞ X n IN SPIRE ¯ t ][ 1 IN = 3 even though the expression is √ arXiv:1602.08547 SPIRE ][ Relaxation in a completely integrable [ d IN + Phys. Rev. E ¯ t ][ , 1 2 (2019) 111602 (2007) 050405 hep-th/0205051 + πm n 4 [ ) for 98 − ]. 123 cond-mat/0601225 m e [ (2016) 164 Quantum Quenches in Free Field Theory: – 18 – A.11 =1 ∞ 05 X m SPIRE arXiv:1903.12242 n IN ) (2002) 042 [ ¯ t (0) ][ arXiv:1507.07266 π n [ Generalized Eigenstate Thermalization Hypothesis in 09 JHEP a (2 arXiv:1712.06963 , Minkowski space correlators in AdS/CFT correspondence: ]. ), which permits any use, distribution and reproduction in Time-dependence of correlation functions following a quantum [ (2006) 136801 Phys. Rev. Lett. Quantum Quenches in Extended Systems =0 3 because of the logarithmic divergence in the ﬁrst term. ∞ Phys. Rev. Lett. , X n , 96 ¯ JHEP t (2019) 139 SPIRE 1 , d > √ IN 08 ][ (2018) 075 (2016) 023103 CC-BY 4.0 arXiv:0704.1880 03 Quantum statistical mechanics in a closed system This article is distributed under the terms of the Creative Commons Γ(0) + [ JHEP Chaos and quantum thermalization ,

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