Hydrodynamical phase transition for domain-wall melting in the XY chain
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Hydrodynamical phase transition for domain-wall melting in the XY chain
Viktor Eisler and Florian Maislinger
Institut f¨ur Theoretische Physik, Technische Universit¨at Graz, Petersgasse 16, A-8010 Graz, Austria
We study the melting of a domain wall, prepared as a certain low-energy excitation above theferromagnetic ground state of the XY chain. In a well defined parameter regime the time-evolvedmagnetization profile develops sharp kink-like structures in the bulk, showing features of a phasetransition in the hydrodynamic scaling limit. The transition is of purely dynamical nature and canbe attributed to the appearance of a negative effective mass term in the dispersion. The signaturesare also clearly visible in the entanglement profile measured along the front region, which can beobtained by covariance-matrix methods despite the state being non-Gaussian.
Uncovering the mechanism of phase transitions belongsto one of the most spectacular achievements of statisticalphysics. The abrupt changes in the properties of matter,in response to the tuning of a control parameter, couldbe understood through simple concepts such as order pa-rameter, symmetry breaking, or free energy. While thetheory is well established for systems in thermal equilib-rium, and can even be extended to quantum phase tran-sitions at zero temperature [1], it is far from obvious howthese concepts generalize to the nonequilibrium scenario.Due to this ambiguity, there has been various attemptsto lift the definition of a phase transition into the dy-namical regime. In the particular context of quantumquenches [2, 3], dynamical quantum phase transitions(DQPT) were introduced by analogy, via the definitionof a dynamical free energy density [4]. It is simply givenvia the overlap between initial and time-evolved states,and DQPT manifests itself in the nonanalytic real-timebehavior of this return probability, see [5] for a recentreview. Despite not being a conventional observable, thereturn probability and the signatures of a DQPT coulddirectly be detected in a recent experiment [6].On the other hand, in a number of approaches thedefinition of dynamical phases is based on the time-asymptotic behavior of an order parameter that showsabrupt changes when crossing the phase boundaries. Dy-namical phase transitions based on a suitable order pa-rameter have been identified for quench protocols of vari-ous closed many-body systems [7–9] and the studies haveeven been extended to the open-system scenario [10, 11].Furthermore, connections between the different conceptsof a DQPT, based on dynamical free energy vs. orderparameter, have recently been pointed out [12, 13].Here we shall address the question whether a phasetransition in simple quantum chains might occur due tothe presence of initial spin gradients, which drive the sys-tem towards a nonequilibrium steady state (NESS). Inthe context of Markovian open system dynamics, such anexample was found earlier for a boundary driven open XYspin chain, where the emergence of long range order wasobserved in the NESS below a critical value h < h c of amodel parameter [14]. Although the phenomenon seemsrobust enough against the details of incoherent driving -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 γ =0.5 M n ( t ) (n-n +1/2)/t h=0.9h=0.75h=0.5 FIG. 1: Normalized magnetization profiles (symbols) at t =200 compared to the hydrodynamic solution (red solid lines)in (10). The phase transition is located at h c = 1 − γ = 0 . [15], no counterpart of the phase transition under closedunitary dynamics has been found so far.To mimic the effect of gradients imposed at the bound-aries in the open system setup, here we prepare instead adomain-wall initial state and then let the system evolveunder its own unitary dynamics. The domain wall iscreated as a simple low-lying excitation above the fer-romegnatic (symmetry-broken) ground state of the XYchain. Our main result is illustrated on Fig. 1, wherethe qualitative change in the time-evolved and properlynormalized magnetization profiles is clearly visible. Thephase transition point h c exactly coincides with the onefound in Ref. [14], and is signalled by an infinite slopein the center of the profile, whereas kinks are develop-ing in the bulk for h < h c . The nonanalytical behaviorappears only in the hydrodynamical limit, shown by thesolid lines in Fig. 1. However, in contrast to Ref. [14],our results on the correlations indicate that the NESSitself is similar to the symmetry-restored ground state ofthe chain and does not show any criticality around h c .Hence we use the term hydrodynamical phase transitionto distinguish between the two behaviors.The Hamiltonian of the XY chain is given by H = − N − X n =1 (cid:18) γ σ xn σ xn +1 + 1 − γ σ yn σ yn +1 (cid:19) − h N X n =1 σ zn (1)where σ αn are Pauli matrices on site n , γ is the anisotropyand h is a transverse magnetic field. The XY modelcan be mapped to a chain of free fermions via a Jordan-Wigner (JW) transformation, by introducing the Majo-rana operators a j − = j − Y k =1 σ zk σ xj , a j = j − Y k =1 σ zk σ yj , (2)satisfying anticommutation relations { a k , a l } = 2 δ k,l .While the open boundaries in Eq. (1) are most suitablefor numerical investigations of the dynamics on finite sizechains, for the analytical treatment one should imposeantiperiodic boundary conditions σ x,yN +1 = − σ x,y on thespins, such that H can be brought into a diagonal formby a Fourier transform and a Bogoliubov rotation [16].We focus on the parameter regime 0 < γ ≤ ≤ h <
1, where the model is in a gapped ferromagneticphase, with magnetic order in the x direction. In par-ticular, in the limit N → ∞ , the ground state is twofolddegenerate, with | i NS and | i R located in the Neveu-Schwarz (NS) and Ramond (R) sectors, correspondingto ± P = Q Nk =1 σ zk ,which commutes with the Hamiltonian [ H, P ]=0. Sinceboth of the ground states are parity eigenstates, theirmagnetization is vanishing. However, starting from thesymmetry-broken ground state | ⇑ i , a domain wall ini-tial state can be prepared via a JW excitation, i.e. actingwith a single Majorana operator as | JW i = a n − | ⇑ i , | ⇑ i = | i NS + | i R √ . (3)In numerical calculations we always consider domainwalls localized in the middle of the chain, n = N/ − iHt | JW i = | φ t i NS + | φ t i R √ | φ t i NS = 1 √ N X q ∈ NS e − iǫ q t e − iq ( n − e iθ q / | q i NS , (5)where the single-particle dispersion ǫ q and the Bogoli- ubov phase θ q are given by ǫ q = q (cos q − h ) + γ sin q , e i ( θ q + q ) = cos q − h + iγ sin qǫ q . (6)The result for | φ t i R is completely analogous to (5), withthe sum running over momenta p ∈ R . In turn, thenormalized magnetization can be cast in the form M n ( t ) = h JW | σ xn ( t ) | JW ih ⇑ | σ xn | ⇑ i = Re R h φ t | ˆ M n | φ t i NS , (7)where, in the limit N ≫
1, the form factors read [17, 18] R h p | ˆ M n | q i NS = − iN ǫ p + ǫ q √ ǫ p ǫ q e i ( n − / q − p ) sin q − p . (8)Combining the results (5)-(8) and considering the ther-modynamic limit, one ends up with a double integralformula for the magnetization [16]. Interestingly, this isexactly the same expression as the one found earlier forthe transverse Ising (TI) chain [19], except that the formof the dispersion and the Bogoliubov angle (6) are nowmore general. In fact, it is the very presence of the XYanisotropy that will give rise to a peculiar dynamical be-havior. The hydrodynamical phase transition is encodedin the q ≪ ǫ q ≈ ∆ + h − h c q + c q , (9)where ∆ = 1 − h is the excitation gap and h c = 1 − γ isa critical field. The coefficient c has a lengthy expressionin terms of h and γ , satisfying c > h < h c .In contrast, the mass term in Eq. (9) becomes negativebelow the critical field. -1-0.5 0 0.5 1 -3 -2 -1 0 1 2 3v q q h=0.9h=0.75h=0.5 ǫ q FIG. 2: Single-particle velocities v q and dispersion ǫ q (inset). While a negative effective mass has no effect on theground-state properties, it will play a crucial role in thedynamics. Indeed, in a well-defined limit, the shape ofthe melting domain wall is entirely determined by thegroup velocities v q = d ǫ q d q . These are shown on Fig. 2for γ = 0 .
5, and three different magnetic fields above,below and at the critical value h c . In case h < h c , thenegative slope of v q around q → ν = ( n − n + 1 / /t , the magnetizationprofile reads M n ( t ) = 1 − Z π − π d q π Θ( v q − ν ) , (10)where Θ( x ) is the Heaviside step function. The result(10) follows rigorously from a stationary-phase analysis[16] of the integral representation of M n ( t ), and hasa clear physical interpretation. Namely, each single-particle excitation carries a spin-flip [20–23] and thusthe magnetization along a fixed ray follows from the in-tegrated density of excitations whose speed exceeds ν .Hence, for h < h c the nonanalytical behavior of thedensity is a consequence of the new branch of solutionsaround the local maximum for negative momenta.The comparison between the profiles and the hydrody-namic scaling function is shown on Fig. 1. The magne-tization at t = 200 and various h were calculated for anopen chain of size N = 400 using the Pfaffian formalismdescribed in [19]. One has an excellent agreement withclear signatures of the developing kink for h < h c . Thehydrodynamic profile in general depends on the detailsof the dispersion and is hard to obtain analytically, sincethe solution of v q = ν leads to a fourth-order equation.Nevertheless, one expects a universal behavior to emergearound the edge of the front [24]. Indeed, the stationaryphase calculation around v q ∗ = v max can be extended tocapture the fine structure of the front [25–28], suggestingthe following choice for the scaling variable X = ( n − n + 1 / θ ′ q ∗ / − v q ∗ t ) (cid:18) | v ′′ q ∗ | t (cid:19) / . (11)In turn, the edge magnetization is given by [16] M n ( t ) = 1 − (cid:18) | v ′′ q ∗ | t (cid:19) / ρ ( X ) , (12)where ρ ( X ) = (cid:2) Ai ′ ( X ) (cid:3) − X Ai ( X ) is just the diagonalpart of the Airy kernel [29].The edge scaling (12) is tested against numerical cal-culations for h = γ = 0 . t = 50 is due tothe presence of the kink in the profile. In fact, one couldask whether zooming on around the kink would yield asimilar universal fine structure as for the edge. However, γ =0.5 ρ ( X ) X t=50t=100t=200
FIG. 3: Edge scaling of the magnetization profile, with thescaling variable X and function ρ ( X ) defined by Eqs. (11)and (12), respectively. in the latter case the density has a nonuniversal bulk con-tribution superimposed, which spoils the step structure.It is also worth noting that the edge scaling (12) for theXY chain can not be derived from a simple higher-orderextension of the hydrodynamical picture [30].The signatures of the hydrodynamical phase transi-tion are also visible on the entanglement profiles, as mea-sured by the von Neumann entropy between the segment A = [1 , N/ r ] and B the rest of the system. Al-though the XY chain maps to free fermions, extractingthe entropy via covariance-matrix techniques for Gaus-sian states [31, 32] requires some additional care. In-deed, the initial state is excited from the symmetry-broken ground state of the model, which is inherentlynon-Gaussian [33]. This difficulty can, however, be over-come by the following considerations. Let us denote by ρ ⇑ the reduced density matrix (RDM) arising from thetime evolved state (4) after tracing out the degrees offreedom in B . The arrow indicates the choice of thesymmetry-broken ground state in (3) and the entropyof the RDM is given by S ( ρ ⇑ ) = − Tr ρ ⇑ ln ρ ⇑ . In fact,one could equally well have defined ρ ⇓ starting from thespin-reversed initial state, with the entropies of the twoRDMs satisfying S ( ρ ⇑ ) = S ( ρ ⇓ ) due to obvious symme-try reasons. The main trick is now to consider the convexcombination ρ G = ρ ⇑ + ρ ⇓ , (13)which removes all the parity-odd contributions from theRDMs, albeit still mixing parity-even terms from the twosectors NS and R. However, in the thermodynamic limitall the expectation values of local operators become equalin both sectors [33], hence ρ G is equivalent to a GaussianRDM where the excitation is created upon the parity-symmetric ground state | i NS .Due to its Gaussianity, the entropy of ρ G can now beobtained by applying the covariance-matrix formalism asshown in Ref. [34]. Indeed, the effect of the Majoranaexcitation can be represented in a Heisenberg picture a ′ k = a n − a k a n − = N X l =1 Q k,l a l , (14)as an orthogonal transformation on the Majoranas, withmatrix elements Q k,l = δ k,l (2 δ k, n − − a ′ k ( t ) = e iHt a ′ k e − iHt = N X l =1 R k,l a ′ l , (15)with matrix elements R k,l given as in Ref. [19]. Hence ρ G corresponds to a RDM associated to the Gaussian statewith covariance matrix˜Γ = R Q Γ Q T R T , (16)where i Γ k,l = NS h | a k a l | i NS − δ k,l . Note that the matrix˜Γ is exactly the one that appears in the Pfaffian by thecalculation of the magnetization [19].Although the entropy of ρ G follows simply via theeigenvalues of the reduced covariance matrix ˜Γ A [31, 32],one still has to relate it to the entropy of the non-Gaussian RDM ρ ⇑ that we are interested in. To thisend, one can make use of the inequality for convex com-binations of density matrices [35, 36] S ( X i λ i ρ i ) ≤ X i λ i S ( ρ i ) − X i λ i ln λ i . (17)Furthermore, it is also known that the inequality is satu-rated if the ranges of ρ i are pairwise orthogonal. Apply-ing it to Eq. (13), the orthogonality condition is clearlysatisfied due to h⇑ | ⇓i = 0 and hence one arrives at S ( ρ ⇑ ) = S ( ρ G ) − ln 2 . (18)The entropy can thus be exactly evaluated using Gaus-sian techniques.The result for the profile ∆ S , measured from the t = 0value, is shown on Fig. 4 at time t = 200, against therescaled cut position. The parameters are chosen to beidentical to Fig. 1, and a kink for h = 0 . r/t equal to the local maximum of the ve-locity v q . Furthermore, the entropy growth for the half-chain ( r/t = 0) clearly converges towards the value ln 2,which can be interpreted as a restoration of the spin-flipsymmetry in the NESS. Note also the light dip in themiddle for h = h c = 0 .
75, which is the consequence of amuch slower convergence towards the NESS at criticality.The entropy profiles obtained by the Gaussian techniquehave also been compared to the results of density-matrixrenormalization group [37] calculations, finding an excel-lent agreement and thus justifying the result in Eq. (18). γ =0.5 ∆ S r/th=0.9h=0.75h=0.5 FIG. 4: Entanglement profiles as a function of the rescaleddistance r of the cut from the middle of the chain. The en-tropy difference ∆ S from the initial state value is shown at t = 200 for the same parameter values as in Fig. 1. We finally consider the normalized equal-time spin-correlation functions C m,n ( t ) = NS h φ t | ˆ M m ˆ M n | φ t i NS which can be studied via the form-factor approach byinserting a resolution of the identity between the oper-ators. Although in general all the multi-particle formfactors are nonvanishing, the dominant contribution tothe correlations comes from the single-particle terms C m,n ( t ) ≃ X p NS h φ t | ˆ M m | p i R R h p | ˆ M n | φ t i NS . (19)The above expression can again be evaluated in the hy-drodynamic scaling limit and for m < n yields [16] C m,n ( t ) ≃ − Z π − π d q π Θ( v q − µ )Θ( ν − v q ) , (20)where µ is defined analogously to ν . The integral in (20)gives the number of excitations with velocities betweenthe rays defined by µ and ν , and has again a simple inter-pretation. In fact, it is directly related to the differenceof the magnetizations along those rays and thus showssimilar nonanalytical behavior for h < h c .In the NESS limit t → ∞ with m, n fixed, Eq. (20)predicts long-range magnetic order C m,n ( t ) →
1. To-gether with M n ( t ) →
0, this behavior is characteristic ofthe ground state | i NS at large separations n − m ≫ C m,n ( t ) converges towards the proper ground-state valueeven for small separations of the spins. Indeed, in theferromagnetic regime the normalized correlators deviatefrom unity by a term decaying exponentially with thedistance [38]. The source of the discrepancy is the ap-proximation in (19), which neglects the contribution ofthe multi-particle form factors. A detailed analysis of thecorrelations will be presented elsewhere [39].In conclusion, our studies of domain-wall melting inthe XY chain have revealed a phase transition, manifestin the emergence of kinks in the profiles of various observ-ables. While the critical point h c = 1 − γ coincides withthe one found earlier for open-system dynamics [14], thetransition exists only in the hydrodynamic regime, anddoes not survive the NESS limit. In contrast, the latterone seems to be given by the parity-symmetric groundstate, which does not show any criticality around h c .Although demonstrated on a simple free-fermion exam-ple, there is good reason to believe that this phenomenoncarries over to generic integrable systems, where theproper hydrodynamic description has only recently beenidentified [40, 41] and applied to initial states with do-main walls [42, 43]. In particular, the emergence of kinksin the magnetization profile has been observed for theXXZ chain at large anisotropies, resulting from the ve-locity maxima of the various quasiparticle families thatgovern the hydrodynamics [42]. While the mechanismseems to be closely related to the one presented here,it is unclear whether a hydrodynamical phase transitionpoint exists in the XXZ case, since all the profiles con-sidered in [42] belong to the kink phase.Finally, it remains to be understood whether the fi-nite increase of entropy after the JW excitation couldbe interpreted within a framework similar to the one in-troduced for local operator insertions in conformal fieldtheories [44]. While the results have been checked againstthe lattice equivalent of local primary excitations for thetransverse Ising chain in Ref. [34], it would be interest-ing to see whether the field theory treatment could begeneralized to include the massive case and the non-localoperators considered here.We thank H. G. Evertz and M. Fagotti for discussions.The authors acknowledge funding from the Austrian Sci-ence Fund (FWF) through Project No. P30616-N36, andthrough SFB ViCoM F41 (Project P04). [1] S. 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FERMIONIZATION OF XY HAMILTONIAN
In order to obtain the many-body eigenstates of the XY chain, it is useful to consider periodic H + or antiperiodic H − chains, instead of the open one in Eq. (1). These are given by H s = − N X n =1 (cid:18) γ σ xn σ xn +1 + 1 − γ σ yn σ yn +1 + hσ zn (cid:19) , (S1)where the boundary conditions are σ xN +1 = sσ x and σ yN +1 = sσ y for s = ± . Since H s commutes with the parity P ,it can be written in a block-diagonal form H s = 1 − sP H R + 1 + sP H NS , P = N Y n =1 σ zn . (S2)The parity subspaces are the Ramond (R) and Neveu-Schwarz (NS) sectors, defining two different Hamiltonians. Interms of Majorana operators, obtained via the Jordan-Wigner transformation (2), both of them can be brought intothe quadratic form H R / NS = i N X j =1 (cid:18) γ a j a j +1 − − γ a j − a j +2 + ha j − a j (cid:19) , (S3)where the two Hamiltonians differ only in the boundary conditions a N +1 = ± a and a N +2 = ± a being periodicfor R and antiperiodic for the NS sector. Each sector can be simultaneously diagonalized by a joint Fourier andBogoliubov transformation a j − = 1 √ N X q ∈ R / NS e − iq ( j − e iθ q / ( b † q + b − q ) , a j = − i √ N X q ∈ R / NS e − iqj e − iθ q / ( b † q − b − q ) , (S4)where the allowed values of the momenta are q k = πN k for R and q k = πN ( k + 1 /
2) for NS, respectively, with k = − N/ , . . . , N/ −
1. Note that the site index j in the Fourier transformation is shifted by one for odd Majoranaoperators. This is a dual representation in terms of which the Bogoliubov angle must satisfye i ( θ q + q ) = cos q − h + iγ sin qǫ q , ǫ q = q (cos q − h ) + γ sin q . (S5)In fact, the above definition ensures that θ q is a continuous and smooth function in its full domain q ∈ [ − π, π ], forarbitrary parameters 0 < γ ≤ ≤ h < H R / NS = X q ∈ R / NS ǫ q b † q b q + const , | q , q , . . . , q m i R / NS = m Y i =1 b † q i | i R / NS . (S6)Finally, it should be pointed out that the boundary condition on the spins selects the parity of the many-particlebasis: m = 2 ℓ is even for the spin-periodic Hamiltonian H + , and m = 2 ℓ + 1 is odd for the spin-antiperiodic one H − . FORM FACTOR APPROACH
To calculate the time evolution of the magnetization, one also needs the corresponding form factors of the σ x operator. In fact, it is more convenient to consider the matrix elements normalized by the equilibrium magnetization,which in the large N limit reads [17, 18] R h p | ˆ M n | q i NS = R h p | σ xn | q i NSR h | σ xn | i NS = − iN cosh ∆ p − ∆ q sinh ∆ p +∆ q p sinh ∆ p sinh ∆ q e i ( n − / q − p ) sin q − p . (S7)The above definition of the form factors is well-suited for the parameter regime p − γ < h <
1, i.e. in thenon-oscillatory ferromagnetic phase [38], where the auxiliary parameter ∆ q is defined viasinh ∆ q = p − γ γ p γ + h − ǫ q . (S8)In the oscillatory phase 0 < h < p − γ the form factors can be obtained by analytic continuation [17], i.e. byintroducing the variable ˜∆ q = ∆ q + iπ/
2. In fact, the form-factor formula (S7) can even be further simplified bymaking use of the identity cosh ∆ p − ∆ q p + ∆ q p + sinh ∆ q ) . (S9)Substituting (S8) and (S9) into (S7), one obtains immediately Eq. (8). Using these form factors and taking thethermodynamic limit, Eq. (7) for the magnetization can be written out as a double integral M n ( t ) = Im Z π − π d p π Z π − π d q π ǫ p + ǫ q √ ǫ p ǫ q e i ( n − n +1 / q − p ) sin q − p e i ( θ q − θ p ) / e i ( ǫ p − ǫ q ) t . (S10)Using the properties ǫ − q = ǫ q and θ − q = − θ q , the above expression can be written as M n ( t ) = M en ( t ) + M on ( t ) withonly two nonvanishing contributions M en ( t ) = Z π − π d p π Z π − π d q π ǫ p + ǫ q √ ǫ p ǫ q cos [( n − n + 1 / q − p )]sin q − p sin θ q − θ p ǫ p − ǫ q ) t , M on ( t ) = Z π − π d p π Z π − π d q π ǫ p + ǫ q √ ǫ p ǫ q sin [( n − n + 1 / q − p )]sin q − p cos θ q − θ p ǫ p − ǫ q ) t . (S11)Hence the magnetization is the sum of an even and an odd function M e,on ( t ) = ±M e,o n − − n ( t ) under reflections withrespect to the initial domain wall position. Note that, in general, the even term has a contribution of much smallermagnitude, and it vanishes completely in the hydrodynamic scaling limit. Moreover, in the limit γ = 1 of a transverseIsing chain, the even part M en ( t ) = 0 vanishes identically even for finite times.The normalized correlation functions C m,n ( t ) = NS h φ t | ˆ M m ˆ M n | φ t i NS can also be studied through the form factorapproach. The standard trick is to insert an identity between the two operators, written in terms of the eigenbasis X p | p ih p | + X p ,p | p , p ih p , p | + X p ,p ,p | p , p , p ih p , p , p | + . . . (S12)Thus, in contrast to the magnetization which could be exactly evaluated using only single-particle form factors, thesituation for the correlations is much more complicated as an infinite series of many-particle matrix elements appear.Nevertheless, it is reasonable to expect that the dominant contribution to the correlations still comes from the single-particle sector. Hence, we will consider this approximate expression, given by (19) in the main text, which for N → ∞ can be converted into the integral form C m,n ( t ) ≃ Z d q π Z d q π e − i ( θ q − θ q ) / e i ( ǫ q − ǫ q ) t Z d p π ǫ p + ǫ q √ ǫ p ǫ q ǫ p + ǫ q √ ǫ p ǫ q e − i ( m − n +1 / q − p ) sin q − p e i ( n − n +1 / q − p ) sin q − p . (S13) STATIONARY PHASE CALCULATIONS
The profiles in the hydrodynamic scaling limit can be obtained by stationary phase arguments, and their derivationclosely follows the lines of Refs. [25–28]. Let us consider first the magnetization as given by Eq. (S10). In the limit n − n ≫ t ≫
1, the integrand is highly oscillatory and thus the main contribution comes from around thepoints q s where the stationarity condition is satisfied v q s t = n − n + 1 / θ ′ q s / , v q = d ǫ q d q . (S14)The stationary phase condition for the integral over p is exactly the same. Moreover, the integrand has a pole at p = q which suggests the change of variables Q = q − p and P = ( q + p ) /
2. In the new variables, the stationaritycondition is Q s = 0 for arbitrary values of P . One shall thus expand the integrand in (S10) around Q = 0, setting ǫ p + ǫ q √ ǫ p ǫ q ≈ , sin q − p ≈ Q , (S15)to arrive at 2 Re Z π − π d P π Z ∞−∞ d Q πi e i ( n − n +1 / θ ′ P − v P t ) Q Q . (S16)To carry out the integration around the pole, we use a formal identity in complex analysis as well as the integralrepresentation of the Heaviside theta function1 Q = iπδ ( Q ) + lim δ → Q + iδ , Θ( x ) = − lim δ → Z ∞−∞ d Q πi e − iQx Q + iδ . (S17)In the hydrodynamic regime one can neglect the term θ ′ P and introduce the scaling variable ν = ( n − n + 1 / /t ,which brings us to the result (10) in the main text.In general, the hydrodynamic profile is found by solving the equation v q = ν . Special attention is needed aroundthe maximum v q ∗ = v max of the velocities, where the solutions coalesce at momentum q ∗ . To get the fine structureof the front edge, one has to expand the dispersion around q ∗ as ǫ q ≈ ǫ q ∗ + v q ∗ ( q − q ∗ ) + ǫ ′′′ q ∗ q − q ∗ ) . (S18)Furthermore, one can introduce the following rescaled variables X = ( n − n + 1 / θ ′ q ∗ / − v q ∗ t ) (cid:18) − ǫ ′′′ q ∗ t (cid:19) / , Q = (cid:18) − ǫ ′′′ q ∗ t (cid:19) − / ( q − q ∗ ) , P = (cid:18) − ǫ ′′′ q ∗ t (cid:19) − / ( p − q ∗ ) . (S19)Substituting (S18) and (S19) into (7), one arrives at the following integral (cid:18) − ǫ ′′′ q ∗ t (cid:19) / Im Z d P π Z d Q π e iX ( Q − P ) e i ( Q − P ) / ( Q − P ) / . (S20)Using the integral representation of the Airy kernel K ( X, Y ) = lim δ → Z d P π Z d Q π e − iXP e − iP / e iY Q e iQ / i ( P − Q − iδ ) = Ai( X )Ai ′ ( Y ) − Ai ′ ( X )Ai( Y ) X − Y , (S21)one recovers (12) of the main text, where the diagonal terms of the Airy kernel [29] can be obtained as ρ ( X ) = lim Y → X K ( X, Y ) = (cid:2) Ai ′ ( X ) (cid:3) − X Ai ( X ) . (S22)The stationary phase calculation for the approximation of the correlation function in (S13) is very similar to thatfor the magnetization. Indeed, introducing the new set of variables Q = q − p, Q = q − p, P = q + p , (S23)and expanding around Q = 0 and Q = 0, one obtains C m,n ( t ) ≃ Z d P π Z d Q π e − i ( m − n +1 / θ ′ P − v P t ) Q Q Z d Q π e i ( n − n +1 / θ ′ P − v P t ) Q Q . (S24)Applying (S17) in both the Q and Q integrals, the result can again be written with the help of step functions. Usingthe property Θ( x − x )Θ( x − x ) = Θ( x − max( x , x )), and introducing the scaling variable µ = ( m − n + 1 / /t analogously to ν , one arrives at C m,n ( t ) ≃ − Z π − π d P π Θ( v P − µ ) + 2 Z π − π d P π Θ( v P − ν ) , (S25)where we assumed µ < νµ < ν