Nonequilibrium dynamical mean-field theory based on weak-coupling perturbation expansions:Application to dynamical symmetry breaking in the Hubbard model

Naoto Tsuji1 and Philipp Werner21Department of Physics, University of Tokyo, 113-0033 Tokyo, Japan

2Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland(Dated: June 2, 2014)

We discuss the general formalism and validity of weak-coupling perturbation theory as an impurity solverfor nonequilibrium dynamical mean-field theory. The method is implemented and tested in the Hubbard model,using expansions up to fourth order for the paramagnetic phase at half filling and third order for the antiferro-magnetic and paramagnetic phase away from half filling. We explore various types of weak-coupling expansionsand examine the accuracy and applicability of the methods for equilibrium and nonequilibrium problems. Wefind that in most cases an expansion of local self-energy diagrams including all the tadpole diagrams with respectto the Weiss Greens function (bare-diagram expansion) gives more accurate results than other schemes suchas self-consistent perturbation theory using the fully interacting Greens function (bold-diagram expansion). Inthe paramagnetic phase at half filling, the fourth-order bare expansion improves the result of the second-orderexpansion in the weak-coupling regime, while both expansions suddenly fail at some intermediate interactionstrength. The higher-order bare perturbation is especially advantageous in the antiferromagnetic phase near halffilling. We use the third-order bare perturbation expansion within the nonequilibrium dynamical mean-fieldtheory to study dynamical symmetry breaking from the paramagnetic to the antiferromagnetic phase inducedby an interaction ramp in the Hubbard model. The results show that the order parameter, after an initial expo-nential growth, exhibits an amplitude oscillation around a nonthermal value followed by a slow drift toward thethermal value. The transient dynamics seems to be governed by a nonthermal critical point, associated with anonthermal universality class, which is distinct from the conventional Ginzburg-Landau theory.

PACS numbers: 71.10.Fd, 64.60.Ht

I. INTRODUCTION

The study of nonequilibrium phenomena in correlatedquantum systems is an active and rapidly expanding field,which is driven by the progress of time-resolved spec-troscopy experiments in solids15 and experiments on ul-tracold atoms trapped in an optical lattice.68 Recentstudies are revealing ultrafast dynamics of phase transi-tions and order parameters, which include the melting ofcharge density waves (CDW),911 nonequilibrium dynamicsof superconductivity,12 photoinduced transient transitions tosuperconductivity,13,14 and the observation of the amplitudemode in CDW materials1517 and the Higgs mode in an s-wavesuperconductor.18 Such experiments offer a testing ground forthe study of dynamical phase transitions and dynamical sym-metry breaking19,20 in real materials. They also raise impor-tant theoretical issues related to the description of nonequilib-rium phenomena in correlated systems. One is the possibleappearance of nonthermal quasistationary states that are in-accessible in equilibrium, such as prethermalized states,2123

which can be interpreted as states controlled by nonthermalfixed points.2426 For example, it has been suggested that asymmetry-broken ordered state can survive for a long timein a nonthermal situation in which the excitation energy cor-responds to a temperature higher than the thermal criticaltemperature.25,26 Such a state does not exist in equilibrium,so that the concept of nonthermal fixed points drastically ex-tends the possibility for the presence of long-range order. An-other aspect is the long-standing theoretical issue of how tocharacterize a nonequilibrium phase transition and its criticalbehavior.27,28

Since the dynamical phase transition that we are interestedin occurs very far from equilibrium, where the temporal varia-tion of the order parameter is not particularly slow, we need atheoretical description of nonequilibrium many-body systemsbased on a microscopic theory, without employing a macro-scopic coarsening or a phenomenological description (e.g., thetime-dependent Ginzburg-Landau equation). The nonequilib-rium dynamical mean-field theory (DMFT)2931 is one suchapproach, which has been recently developed. It is a nonequi-librium generalization of the equilibrium DMFT32 that mapsa lattice model onto an effective local impurity problem em-bedded in a dynamical mean-field bath. It takes account ofdynamical correlation effects, while spatial correlations areignored. The formalism becomes exact in the large dimen-sional limit.33 Furthermore, it can describe the dynamics ofsymmetry-broken states with a long-range (commensurate)order.25,26 Since DMFT is based on a mean-field description,it allows to treat directly the thermodynamic limit (i.e., thecalculations are free from finite-size effects).

To implement the nonequilibrium DMFT, one requires animpurity solver. Previously, several approaches have been em-ployed, including the continuous-time quantum Monte Carlo(QMC) method,3437 the noncrossing approximation and itsgeneralizations (strong-coupling perturbation theory),38 andthe exact diagonalization.39,40 In this paper, we explore theweak-coupling perturbation theory as an impurity solverfor the nonequilibrium DMFT. Our aim is to establish amethod that is applicable to relatively long-time simulationsof nonequilibrium impurity problems in the weak-couplingregime, where a lot of interesting nonequilibrium physics re-mains unexplored. In particular, our interest lies in simulating

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2dynamical symmetry breaking toward ordered states such asthe antiferromagnetic (AFM) phase. QMC is numerically ex-act, but suffers from a dynamical sign problem,35 which pro-hibits sufficiently long simulation times. An approximate dia-grammatic approach, such as the weak-coupling perturbationtheory, allows one to let the system evolve up to times whichare long enough to capture order-parameter dynamics.

Perturbation theory is a standard and well-known diagram-matic technique4143 to solve quantum many-body problemsin the weak-coupling regime. It has been successfully appliedto the study of the equilibrium Anderson impurity model.44,45

Although it is an expansion with respect to the ratio (U/pi)between the interaction strength U and the hybridization toa conduction bath , it has turned out to be a very good ap-proximation up to moderate U/pi. Later the weak-couplingperturbation theory was employed as an impurity solver forthe equilibrium DMFT.4650 Especially the bare second-orderperturbation [which is usually referred to as the iteratedperturbation theory (IPT)] was found to accidentally repro-duce the strong-coupling limit and the Mott insulator-metaltransition.32,47 It was also applied to nonequilibrium quantumimpurity problems.51,52 The nonequilibrium DMFT has beensolved by the second-order perturbation theory in the param-agnetic (PM) phase of the Hubbard model at half filling5356

and by the third-order perturbation theory in the AFM phase.26

However, a thorough investigation of weak-coupling pertur-bation theory, including higher orders, as a nonequilibriumDMFT solver has been lacking so far.

The paper is organized as follows. In Sec. II, we give anoverview of the nonequilibrium DMFT formalism, putting anemphasis on the treatment of the AFM phase. In Sec. III,we present a general formulation of the nonequilibrium weak-coupling perturbation theory following the Kadanoff-Baym57

and Keldysh58 formalism. We discuss various issues of theperturbation theory related to bare- and bold-diagram expan-sions, symmetrization of the interaction term, and the treat-ment of the Hartree diagram. After testing various implemen-tations of the perturbation theory for the equilibrium phases ofthe Hubbard model in Sec. IV, we examine the applicabilityof the method to nonequilibrium problems without long-rangeorder in Sec. V. Finally, in Sec. VI, we apply the third-orderperturbation theory to the nonequilibrium DMFT to study dy-namical symmetry breaking to the AFM phase of the Hubbardmodel induced by an interaction ramp. By comparing the re-sults with those of the phenomenological Ginzburg-Landautheory and time-dependent Hartree approximation, we findthat the order parameter does not directly thermalize but istrapped to a nonthermal value around which an amplitudeoscillation occurs. We show that the transient dynamics of theorder parameter is governed by a nonthermal critical point,26

and we characterize the associated nonthermal universalityclass. In the Appendix A, we provide details of the numeri-cal implementation of the nonequilibrium Dyson equation thatmust be solved in the nonequilibrium DMFT calculations.

0 tmax

-i

FIG. 1. The Kadanoff-Baym contour C.

II. NONEQUILIBRIUM DYNAMICAL MEAN-FIELDTHEORY FOR THE ANTIFERROMAGNETIC PHASE

We first review the formulation of the nonequilibriumDMFT including the antiferromagnetically ordered state.25,26

It is derived in a straightforward way by extending the ordi-nary nonequilibrium DMFT for the PM phase to one havingan AB sublattice dependence. The general structure of the for-malism is analogous to other symmetry-broken phases with acommensurate long-range order. For demonstration, we takethe single-band Hubbard model,

H(t) =k

k(t)ckck

i

ni + U(t)i

nini, (1)

where k is the band dispersion, ck (ck) is the creation (anni-

hilation) operator, ni = cici is the density operator, is the

chemical potential, and U is the on-site interaction strength.k and U may have a time dependence. Let us assume that thelattice structure that we are interested in is a bipartite lattice,which has an AB sublattice distinction.

In the DMFT construction, one maps the lattice model (1)onto an effective single-site impurity model. In principle, onehas to consider two independent impurity problems dependingon whether the impurity site corresponds to a lattice site onthe A or the B sublattice. The impurity action for the a = A, Bsublattice is defined by

S impa [] =CdtCdt

d(t)a(t, t)d(t)

Cdt

n(t) +Cdt U(t)n(t)n(t). (2)

Here d (d) is the creation (annihilation) operator for the im-purity energy levels, a(t, t) is the hybridization functionon the a sublattice, which is self-consistently determined inDMFT, n = dd, and C is the Kadanoff-Baym contour de-picted in Fig. 1. The contour runs in the time domain fromt = 0 to tmax, up to which the system time evolves, comesback to t = 0, and proceeds to i, which corresponds to theinitial thermal equilibrium state with temperature 1. Us-ing the impurity action (2), one can define the nonequilibrium

3Greens function as

Gimpa (t, t) = iTr

(TC eiS impa []d(t)d(t)

)/Zimpa , (3)

where TC time orders the operators along the contour C repre-sented by the arrows in Fig. 1, and Zimpa = Tr TC eiS impa [].

On the other hand, one has the lattice Greens function,

Glati j,(t, t) = iTr

(TC eiS latci(t)cj(t)

)/Zlat, (4)

with S lat = C dt H(t) and Zlat = Tr TC eiS lat . The hybridiza-tion function a is implicitly determined such that the localpart of the lattice Greens function (4) coincides with the im-purity Greens function (3),

Glatii,[](t, t) = Gimpa (t, t

) (i a), (5)where i a means that the lattice site labeled by i belongs tothe a = A, B sublattice. The essential ingredient of DMFT isthe approximation that the lattice self-energy is local in space,based on which one requires the local lattice self-energy to beidentical to the impurity self-energy,

lati j,(t, t) = i jimpa (t, t

) (i a). (6)With this condition, the self-consistency relation between thelattice and impurity models is closed, and the nonequilibriumDMFT for the AFM phase is formulated. In the following, weomit the labels lat and imp thanks to the identifications(5) and (6).

To implement the self-consistency condition in practice,one uses the Dyson equation. In solving the lattice Dysonequation, it is efficient to work in momentum space, wherethe lattice Greens function is Fourier transformed to

Gabk(t, t) = N1

ia, jb

eik(RiR j)Glati j,(t, t), (7)

with N the number of sublattice sites. Then the lattice Dysonequation reads(it + A kk it + B

)(GAAk G

ABk

GBAk GBBk

)=

(C 00 C

).

(8)

Here represents a convolution on the contour C, k(t, t) =k(t)C(t, t), and C(t, t) is the function defined on C. Thelocal Greens function is obtained from a momentum summa-tion. If the system has an inversion symmetry (we consideronly this case here), the off-diagonal components of the localGreens function vanish, and we have

k

Gabk Gaab. (9)

The local Greens function satisfies the Dyson equation for theimpurity problem,

Ga = G0,a + G0,a a Ga, (10)

where

G0,a = (it + a)1 (11)is the Weiss Greens function. Thus we obtained a closed setof nonequilibrium DMFT self-consistency relations: (8), (9),and (10), for a, G0,a (or a), and Ga. The calculation ofGa from G0,a is the task of the impurity solver.

Before finishing this section, let us comment on how tosolve the lattice Dyson equation (8). Due to the existence ofthe AFM long-range order, it has a 2 2 matrix structure, i.e.,consists of coupled integral-differential equations. However,as we show below, it can be decoupled to a set of integral-differential equations of the form

[it (t)]G(t, t)Cdt (t, t)G(t, t) = C(t, t), (12)

and integral equations of the form

G(t, t)Cdt K(t, t)G(t, t) = G0(t, t). (13)

To see this, let us denote the lattice Greens function Gaak (a =A, B) for k = 0 by ga. It satisfies

(it + a) ga = C(t, t), (14)which is in the form of Eq. (12). Using ga, we can explicitlywrite the solution for Eq. (8),

GAAk = (1 gA k gB k)1 gA, (15)GBBk = (1 gB k gA k)1 gB, (16)GABk = G

AAk k gB, (17)

GBAk = GBBk k gA. (18)

By substituting F = ga k ga k (a denotes the sublatticeopposite to a) and Q = ga, we have (1F)Gaak = Q, whichis exactly of the form of Eq. (13). Since Eqs. (12) and (13) areimplemented in the standard nonequilibrium DMFT withoutlong-range orders, one can recycle those subroutines to solveEq. (8).

For the case of the semicircular density of states (DOS),D() =

42 2/(2pi2), and k(t) = k (time indepen-

dent), one can analytically take the momentum summation forthe lattice Greens function, resulting in the relation

a(t, t) = v2Ga(t, t). (19)

Thus, instead of solving Eq. (8), one can make use of

(it + v2Ga) G0,a = C(t, t) (20)as the DMFT self-consistency condition. The DMFT calcu-lations in the rest of the paper are done for the semicircularDOS, and we use v (v1 ) as a unit of energy (time). Withthe symmetry Ga = Ga in the AFM phase, it is sufficient toconsider the impurity problem for one of the two sublatticesso that we can drop the sublattice label a.

4In a practical implementation of the nonequilibrium DMFTself-consistency, what one has to numerically solve are ba-sically equations of the forms (12) and (13). These areVolterra integral(-differential) equations of the second kind.Various numerical algorithms for them can be found in theliterature.5961 Here we use the fourth-order implicit Runge-Kutta method (or the collocation method). The details of theimplementation are presented in the Appendix A.

III. WEAK-COUPLING PERTURBATION THEORY

In this section, we explain the general formalism of theweak-coupling perturbation theory for nonequilibrium quan-tum impurity problems. It is explicitly implemented up tothird order for the AFM phase at arbitrary filling and fourthorder for the PM phase at half filling. We discuss varioustechnical details of the perturbation theory, including the sym-metrization of the interaction term, bare and bold diagrams,and the treatment of the Hartree term.

A. General formalism

To define the perturbation expansion for the nonequilib-rium impurity problem, we split the impurity action (2) into anoninteracting part S imp0 and an interacting part S imp1 (S imp =S imp0 + S imp1 ),

S imp0 =CdtCdt

d(t)(t, t)d(t)

Cdt

[ U(t)] n(t), (21)

S imp1 =Cdt U(t)

(n(t)

) (n(t)

). (22)

Here we have introduced auxiliary constants to symmetrizethe interaction term. Accordingly, the chemical potential inS imp0 is shifted, and the Weiss Greens function is modifiedinto

G0, = (it + U )1. (23)

As a result, the self-consistency condition for the case of thesemicircular DOS is changed from Eq. (20) to

(it + U v2G) G0, = C . (24)

Physical observables should not, in principle, depend on thechoice of , whereas the quality of the approximation madeby the perturbation theory may depend on it. Such pa-rameters have been used to suppress the sign problem in thecontinuous-time QMC method.53,62,63

The weak-coupling perturbation theory for nonequilibriumproblems is formulated in a straightforward way as a general-ization of the equilibrium perturbation theory in the Matsub-ara formalism.41,64 We expand the exponential in Eq. (3) into

a Taylor series with respect to the interaction term,

G(t, t) = (i) 1Zimpn=0

(i)nn!

Cdt1 dtn

Tr(TC eiS

imp0 H1(t1) H1(tn)d(t)d(t)

),

(25)

where H1(t) = U(t)(n )(n ). The linked clustertheorem ensures that all the disconnected diagrams that con-tribute to Eq. (25) can be factorized to give a proportionalityconstant Zimp/Zimp0 with Z0 = Tr TC eiS

imp0 . As a result, the

expansion can be expressed in the simplified form

G(t, t) = (i)n=0

(i)nC,t1tn

dt1 dtn

TC H1(t1) H1(tn)d(t)d(t)conn.0 , (26)

where 0 denotes Tr (TC eiS imp0 )/Zimp0 , and conn.means that one only takes account of connected diagrams.The factor n! is canceled by specifying the contour ordering ast1 tn (t1 comes first and tn last). Owing to Wicks the-orem, one can evaluate each term in Eq. (26) using the WeissGreens function,

G0,(t, t) = iTC d(t)d(t)0. (27)

In the standard weak-coupling perturbation theory, one usu-ally considers an expansion of the self-energy (t, t) insteadof the Greens function. This is because one can then take intoaccount an infinite series of diagrams for the Greens functionby solving the Dyson equation. The self-energy consists ofone-particle irreducible diagrams of the expansion (26), i.e.,the diagrams that cannot be disconnected by cutting a fermionpropagator. Figure 2 shows examples of Feynman diagramsfor the self-energy. In addition, we have tadpole diagrams.Since the quadratic terms in H1 [U(t)n] play the role ofcounterterms to the tadpoles, each tadpole diagram amountsto n0,(t) , where n0,(t) = iG

5H2L H3aL H3bL

H3cL H3dL H3eL

FIG. 2. The self-energy diagrams up to third order (except for theHartree diagrams). The solid lines represent the fermion propagator,while the dashed lines are interaction vertices.

One notices that, if the weak-coupling perturbation theory isemployed as an impurity solver, does not explicitly appearin the DMFT calculation. Instead, G0, represents the dynam-ical mean field.

We show several examples of the application of the Feyn-man rules above in Sec. III D (third order) and III E (fourthorder).

B. Self-consistent perturbation theory

Instead of expanding the self-energy diagrams with respectto the Weiss Greens function G0,, one can also expand itwith respect to the fully interacting Greens function G. Inthis expansion, each bare propagator G0,(t, t) (depicted by athin line) is replaced by the dressed propagator G(t, t) (boldline). Since G itself already contains an infinite numberof diagrams, which is recursively generated from the DysonEq. (10), one can take account of many more diagrams than inthe expansion with respect to G0,. To avoid a double count-ing of diagrams in this expansion, we take the skeleton di-agrams of the self-energy, i.e., two-particle irreducible di-agrams that cannot be disconnected by cutting two fermionpropagators, which reduces the number of diagrams to be con-sidered.

At first, G is not known, so that one starts with an ini-tial guess of G (which is usually chosen to be G0,). Usingthe perturbation theory, one evaluates the self-energy fromG. Plugging into the Dyson Eq. (10), one obtains a newG, which is again used to evaluate the self-energy. One it-erates this procedure until G and converge. In this way,G and are determined self-consistently within the pertur-bation theory (hence named the self-consistent perturbationtheory).

In the self-consistent perturbation theory, there is a shortcut in implementing the DMFT self-consistency. Since theself-energy is determined from the local Greens function G,the Weiss Greens function G0, does not explicitly appear inthe calculation. Thus, one can skip the evaluation of G0, with

the impurity Dyson Eq. (10). For the case of the semicircularDOS, one can eliminate G0, from Eqs. (10) and (20) to obtain

G = (it + U v2G )1, (28)which defines the DMFT self-consistency condition.

Let us remark that the self-consistent perturbation theoryis a conserving approximation,65 i.e., it automatically guar-antees the conservation of global quantities such as the totalenergy and the particle number. The perturbation theory de-fines the self-energy as a functional of G, = [G], whichis a sufficient condition to preserve the conservation laws. Itis important that the conservation law is satisfied in a simu-lation of the time evolution to obtain physically meaningfulresults. However, this does not necessarily mean that the self-consistent perturbation theory is superior to a nonconservingapproximation (such as the expansion with respect to G0,).As we see in Sec. IV and V, under some conditions the non-conserving approximation (despite small violations of conser-vation laws) reproduces the correct dynamics more accuratelythan the conserving approximation.

One can also consider a combination of bare- and bold-diagram expansions. An often used combination is to take thebold diagram for the Hartree term (Fig. 3) and bare diagramsfor the other parts of the self-energy. This kind of expansion isnecessarily a nonconserving approximation. We examine thistype of approximations in Sec. IV.

C. Treatment of the Hartree term

There is a subtle issue concerning the treatment of theHartree term in the self-energy diagrams. The Hartree termis the portion of the self-energy (t, t) that is proportional toC(t, t). Let us denote it by

Hartree (t, t) = h(t)C(t, t). (29)

The corresponding diagram, summed up to infinite order in U,is given by the bold tadpole shown in Fig. 3. It reads

h(t) = U(t)(n(t) ), (30)where n(t) = iG

6H1L H2L H3aL

H3bL H3cL

FIG. 4. An explicit expansion of the Hartree term in the bare Greensfunction G0, up to third order.

expansion of the Hartree diagram with respect to G0,. Weshow the resulting diagrams up to third order in Fig. 4. Inthis bare-diagram expansion, a lot of internal tadpoles aregenerated in the Hartree diagrams. Each tadpole gives a con-tribution of n0,(t) instead of n(t). The question iswhich is the better approximation. [Note that Eq. (30) itself isexact, but if it is combined with other diagrams, it becomes anapproximation.] The bold Hartree term (Fig. 3) includes manymore diagrams than the bare expansion (Fig. 4); however, theanswer is not a priori obvious.

Moreover, we have the freedom to choose the constant .At half-filling and in the PM phase, it is natural to take =12 because of the particle-hole symmetry. It cancels all thetadpole diagrams since n = n0, = 0. Due to theparticle-hole symmetry [i.e., G0,(t, t) = G0,(t, t)], all theodd-order diagrams vanish as well. On the other hand, whenthe system is away from half filling or is spin-polarized (i.e.,n , n), the particle-hole symmetry (for each spin) is lost,and we do not have a solid guideline to choose the value of. can be n, n0,, or some other fixed values (such as 12 ).

Later, in Sec. IV, we examine these issues for the Hartreeterm by considering five representative cases summarized inTable I. It might look better if one sets = n for the bold

I II III IV V

nontadpole bare bare bare bare boldtadpole bare bare bold bold bold 1/2 n0, 1/2 n

contribution oftadpole n0, 1/2 0 n 1/2 0 n

TABLE I. Choices of the nontadpole and tadpole diagrams (bare orbold) and the constant . The bottom row shows the contribution ofeach tadpole diagram. For the case (V), can be arbitrary.

diagrams or = n0, for the bare diagrams, since all thetadpole diagrams are then shifted into the propagator G orG0,. However, it turns out that this choice is not a particu-larly good approximation. Due to cancellations among differ-ent diagrams, the naive expectation that more diagrams meansbetter results is misleading.

D. Third-order perturbation theory

In the case of the spin-polarized phase or away fromhalf filling, when the particle-hole symmetry [G(t, t) =G(t, t)] is lost, it becomes important to take into accountthe odd-order diagrams. Here we consider the third-orderweak-coupling perturbation theory. First, we look at the bare-diagram expansion. We have shown topologically distinctFeynman diagrams of the self-energy up to third order inFig. 2 and the bare Hartree diagrams up to third order in Fig. 4.

Using the Feynman rules presented in Sec. III C, we canexplicitly write the contribution of each diagram. The self-energy at second order [Fig. 2(2)] is given by

(2) (t, t) = U(t)U(t)G0,(t, t)G0,(t, t)G0,(t, t), (31)

and the first two of the self-energy diagrams at third order[Figs. 2(3a) and 2(3b)] are given by

(3a) (t, t) = iU(t)U(t)G0,(t, t)

Cdt U(t)G0,(t, t)G0,(t, t)G0,(t, t)G0,(t, t), (32)

(3b) (t, t) = iU(t)U(t)G0,(t, t)

Cdt U(t)G0,(t, t)G0,(t, t)G0,(t, t)G0,(t, t). (33)

To write the rest of the self-energy diagrams at third or-der [Figs. 2(3c)-2(3e)], it is convenient to define a contour-ordered function,

(3)1,(t, t)

Cdt U(t)

(n(t)

)G0,(t, t)G0,(t, t), (34)which takes care of the internal tadpoles. With (3)1,, we canwrite the self-energy diagrams [Figs. 2(3c)-2(3e)] as

(3c) (t, t) = U(t)U(t)G0,(t, t)G0,(t, t)(3)1,(t, t), (35)

(3d) (t, t) = U(t)U(t)G0,(t, t)G0,(t, t)(3)1,(t, t), (36)

(3e) (t, t) = U(t)U(t)G0,(t, t)G0,(t, t)(3)1,(t, t). (37)

In the bare-diagram expansion, we need to evaluate theHartree diagrams (Fig. 4). To this end, we define anothercontour-ordered function,

(3)2,(t, t) =

Cdt U(t)G0,(t, t)G0,(t, t)G0,(t, t)G0,(t, t).

(38)

7With (3)1, and (3)2,, each Hartree diagram in Fig. 4 reads

h(1) (t) = U(t)(n0,(t) ), (39)h(2) (t) = iU(t)1,(t, t), (40)h(3a) (t) = U(t)

Cdt U(t)G0,(t, t)G0,(t, t)(3)1,(t, t), (41)

h(3b) (t) = iU(t)Cdt U(t)(n0,(t) )G0,(t, t)(3)1,(t, t),

(42)

h(3c) (t) = iU(t)Cdt U(t)G0,(t, t)(3)2,(t, t). (43)

One can see that in the third-order perturbation the calcu-lation of each self-energy diagram includes a single contourintegral at most. This means that the computational cost ofthe impurity solution is of O(N3) with N the number of dis-cretized time steps. It is thus of the same order as solving theDyson equation [in the form of Eq. (12) or Eq. (13)] or cal-culating a convolution of two contour-ordered functions. Thismeans that the impurity problem can be solved with a costcomparable to the DMFT self-consistency part, which is cru-cial for simulating the long-time evolution.

In the self-consistent version of the third-order perturba-tion theory, we consider the two-particle irreducible diagrams.The diagrams of Figs. 2(2), 2(3a), and 2(3b) are two-particleirreducible, whereas the others in Fig. 2 are reducible. TheHartree term is given by the bold diagram (Fig. 3). The equa-tions to represent each diagram are the same as those for thebare diagrams, except that all G0, are replaced by G.

E. Fourth-order perturbation theory for the paramagneticphase at half filling

For the PM phase at half filling, we consider the fourth-order perturbation theory. At fourth order, the number of dia-grams that we have to consider dramatically increases, so thatwe restrict ourselves to the case where the particle-hole sym-metry holds. In this case, the odd-order diagrams disappear.We take = 12 to cancel all the tadpoles and the Hartreeterm. What remains are the second-order diagram [Fig. 2(2)]and 12 fourth-order diagrams,44,4850 as shown in Fig. 5.

We classify the 12 diagrams in four groups [Figs. 5(4a)-5(4d)], each of which contains three diagrams. Using theparticle-hole symmetry G0,(t, t) = G0,(t, t), one can showthat those three classified in the same group give exactly thesame contribution.49 Thanks to this fact, it is enough to con-sider one of the three for each group. In total, the number ofdiagrams to be computed is reduced to four. We represent thissimplification by writing the fourth-order self-energy as

(4)(t, t) = 3[(4a)(t, t) + (4b)(t, t) + (4c)(t, t) + (4d)(t, t)],(44)

where we have omitted the spin label . We can explicitlyevaluate each contribution of the self-energy diagrams to ob-

H4aL

H4bL

H4cL

H4dL

FIG. 5. The nonvanishing self-energy diagrams at fourth order forthe PM phase at half filling.

tain

(4a)(t, t) = U(t)U(t)G0(t, t)CdtCdtU(t)U(t)

[G0(t, t)G0(t, t)G0(t, t)]2 , (45)(4b)(t, t) = U(t)U(t)

[G0(t, t)]2CdtCdtU(t)U(t)G0(t, t)G0(t, t)

[G0(t, t)]3 , (46)(4c)(t, t) = U(t)U(t)

CdtCdtU(t)U(t)

G0(t, t)G0(t, t)G0(t, t)[G0(t, t)G0(t, t)]2 , (47)

(4d)(t, t) = U(t)U(t)G0(t, t)CdtCdt U(t)U(t)

G0(t, t)G0(t, t)G0(t, t)G0(t, t)[G0(t, t)]2 . (48)

Note that they involve double contour integrals. However, for(4a) and (4b), we can decouple the integrals by defining thecontour functions

(4a)(t, t) =Cdt U(t)[G0(t, t)G0(t, t)]2, (49)

(4b)(t, t) =Cdt U(t)[G0(t, t)]3G0(t, t)], (50)

which involve single contour integrals. With these, (4a) and

8(4b) can be rewritten as

(4a)(t, t) = U(t)U(t)G0(t, t)Cdt [G0(t, t)]2(4a)(t, t), (51)

(4b)(t, t) = U(t)U(t)[G0(t, t)]2Cdt G0(t, t)(4b)(t, t), (52)

which again involves only single integrals. Unfortunately, thiskind of reduction is not possible for (4c) and (4d). Hence, thecomputational cost for the fourth-order diagrams is O(N4),which is one order higher than the calculation of the third-order diagrams or solving the DMFT self-consistency. Themaximum time tmax up to which one can let the system evolveis therefore quite limited compared to the third-order pertur-bation theory.

For the fourth-order self-consistent perturbation theory, weonly take the two-particle irreducible diagrams among Fig. 5,which are those grouped in (4a), (4c), and (4d).48 The dia-grams in (4b) are two-particle reducible, and are not consid-ered in the self-consistent perturbation theory.

IV. APPLICATION TO EQUILIBRIUM PHASES

To establish the validity of the weak-coupling perturbationtheory as an impurity solver for DMFT, we first apply it to theequilibrium phases of the Hubbard model. In particular, wefocus on the PM phase (Sec. IV A) away from half filling andthe AFM phase (Sec. IV B), where the conventional second-order perturbation theory fails.32 There has been a proposal toimprove it for arbitrary filling by introducing control param-eters in such a way that the perturbation theory recovers thecorrect strong-coupling limit.66 However, it is not known atthis point how to generalize this approach to nonequilibriumsituations. Here we explore the different types of perturba-tion theories that have been overviewed in Sec. III and clarifywhich ones improve the quality of the approximation com-pared to previously known results.

A. Paramagnetic phase

Let us consider the PM phase of the Hubbard model (1),and first look at the half filled system. In Fig. 6, we show theresults for the double occupancy,

d = nn = 12U

( G)M(0), (53)

given by DMFT with various perturbation expansions. d isa good measure of correlation effects. As is well known, thesecond-order perturbation theory with bare diagrams (whichis often referred to as the IPT) works remarkably well over theentire U regime.32 In particular, it captures the Mott metal-insulator transition. Quantitatively, deviations from QMC(exact) start to appear around U 3 in the weak-couplingregime. The fourth-order bare expansion improves the resultsup to U 4, just before the Mott transition occurs (U 5). It

QMC2nd order, bare2nd order, bold4th order, bare4th order, bold

0 1 2 3 4 5 60.00

0.05

0.10

0.15

0.20

0.25

U

d

FIG. 6. (Color online) The double occupancy for the Hubbard modelin equilibrium at half filling with = 16 calculated from DMFT withvarious impurity solvers.

quickly fails to converge at U & 4.3 (convergence is not recov-ered by mixing the old and new solutions during the DMFT it-erations). The bold diagrams (self-consistent perturbation the-ory) give worse results than the bare expansions (Fig. 6). Thesecond-order bold expansion deviates from QMC at U 2,and it does not converge at U > 3. The fourth-order bolddiagram improves the second-order bold results for U . 2,but it fails to converge at U > 2. Hence, at half filling thefourth-order bare expansion gives the best results in the weak-coupling regime (U 4).

Away from half filling, we calculate the density per spin,n = GM(0), as a function of U for a fixed chemical poten-tial U/2 = 0.5, 1, 2. The results obtained by DMFT withQMC, the Hartree approximation, and the second-order per-turbation theories are shown in Fig. 7, while the results fromthe third-order perturbation expansions are shown in Fig. 8.We consider five types of perturbation expansions (I)-(V) asindicated in Table I. The QMC results indicate that there areMott transitions in the strong U regime (e.g., Uc 6 forU/2 = 1), where the density n approaches 0.5. The Hartreeapproximation (dashed line in Fig. 7), which only includes theHartree term (30) as the self-energy correction, deviates fromQMC already at relatively small U ( 1). Among the var-ious second-order expansions, type (IV) (bare second-orderand bold Hartree diagrams with = n) seems to be closestto the QMC result up to U 2.5 for U/2 = 0.5, 1 andU 4 for U/2 = 2. However, this approach, as wellas types (III) and (V), leads to a convergence problem in theDMFT calculation as one goes to larger U [which is why thelines for types (III)-(V) in Figs. 7 and 8 are terminated]. Onthe other hand, type (I) easily converges even for large U.

By comparing the second- (Fig. 7) and third-order (Fig. 8)perturbation theories, we see a systematic improvement of theresults in most cases. In particular, the third-order type (I)becomes better than type (IV) and gets closest to QMC for U/2 = 2. It agrees with QMC up to U 4. Again

9QMCHartree2nd order HIL2nd order HIIL2nd order HIIIL2nd order HIVL2nd order HVL

-U2=0.5

-U2=1

-U2=2

0 1 2 3 4 5 60.5

0.6

0.7

0.8

0.9

1.0

U

n

FIG. 7. (Color online) The density for the Hubbard model in equi-librium with = 16 calculated by DMFT with QMC, Hartree ap-proximation, and the second-order perturbation theories. The labels(I)-(V) correspond to the classification in Table I.

QMC3rd order HIL3rd order HIIL3rd order HIIIL3rd order HIVL3rd order HVL

-U2=0.5

-U2=1

-U2=2

0 1 2 3 4 5 60.5

0.6

0.7

0.8

0.9

1.0

U

n

FIG. 8. (Color online) The density for the Hubbard model in equi-librium with = 16 calculated by DMFT with the third-order pertur-bation theories. The labels (I)-(V) correspond to the classification inTable I.

type (I) shows excellent convergence for the entire U range,in contrast to the other approaches. For U/2 = 1, theresults of type (I) are not improved from second to third order,while other types fail to converge around U 2.5. Thus,it remains difficult to access the intermediate filling regime(0.1 < |n 0.5| < 0.2,U > 2) using these weak-couplingperturbation expansions. If one goes far away from half filling(dilute regime), the system effectively behaves as a weaklycorrelated metal, and the perturbative approximations becomevalid.

Let us remark that it was pointed out earlier by Yosida andYamada44 that the bare weak-coupling perturbation theory is

QMCHartree2nd order HIL2nd order HIIL2nd order HIIIL

0 1 2 3 40.00

0.05

0.10

0.15

0.20

U

T

FIG. 9. (Color online) The AFM transition temperature for the Hub-bard model at half filling derived from DMFT with various impuritysolvers. The region below (above) the curves represents the AFM(PM) phase. The labels (I)-(III) correspond to the classification inTable I. The QMC data are taken from Ref. 67.

well behaved for the Anderson impurity model if it is ex-panded around the nonmagnetic Hartree solution. This cor-responds to the expansion of type (IV) ( = n) in our clas-sification (Table. I). Thus, their observation is consistent withour conclusion that type (IV) is as good as type (I) and is betterthan the other schemes. The difference is that when type (IV)expansion is applied to DMFT, it suffers from a convergenceproblem in the intermediate-coupling regime.

B. Antiferromagnetic phase

Next, we test the validity of the perturbative impuritysolvers for the equilibrium AFM phase of the Hubbard modelat half filling. We show the AFM phase diagram in the weak-coupling regime obtained from DMFT with QMC, the Hartreeapproximation, and the second-order perturbation theories inFig. 9. We also depict the phase diagram covering the entire Urange in Fig. 10. QMC provides the exact critical temperatureTc, which in the small-U limit behaves as Tc ve1/D(F )U[similar to the BCS formula for the superconducting phase;D(F) is the DOS at the Fermi energy]. Tc takes the maxi-mum value at U 4 and slowly decays as Tc v2/U inthe strong-coupling regime. This is analogous to the BCS-BEC crossover for superconductivity which is often discussedin the context of cold-atom systems. Here, it corresponds toa crossover from the spin density wave in the weak-couplingregime to the AFM Mott insulator with local magnetic mo-ments in the strong-coupling regime.

The Hartree approximation (dashed curve in Fig. 9) cor-rectly reproduces the weak-coupling asymptotic form, Tc ve1/D(F )U , but starts to deviate from the QMC result alreadyat U 0.5. The second-order perturbation theories of types(I)-(III) give better results than the Hartree approximation asshown in Fig. 9. However, quantitatively the agreement withQMC is still not so good for U > 1. This problem was pre-viously pointed out for the second-order perturbation expan-

10

QMC2nd order HIL

PM

AFM

0 2 4 6 8 0.1 0.05 00

0.1

0.2

U 1U

T

FIG. 10. (Color online) The AFM phase diagram of the Hubbardmodel covering the weak-coupling and strong-coupling limits. TheQMC data are taken from Ref. 67.

sion of type (III).32 We have not plotted Tc estimated from thesecond-order perturbations of types (IV) and (V), since type(IV) gives a discontinuous (first-order) phase transition whichis not correct for the AFM order, and type (V) yields a patho-logical discontinuous jump of the magnetization as a functionof temperature within the ordered phase which is physicallyunreasonable.

Type (I) continues to converge in the strong-couplingregime, in contrast to other second-order approaches that failto converge at some point. What is special about this weak-coupling expansion is that it qualitatively captures the BCS-BEC crossover; i.e., the critical temperature scales appropri-ately both in the weak- and strong-coupling limits (Fig. 10).Quantitatively, the value of Tc given by the type (I) second-order scheme is roughly a factor of 2 lower than the QMCresult in the large-U regime. If one restricts the DMFT so-lution to the PM phase, it is known that the second-orderbare-diagram expansion (IPT) reproduces the correct strong-coupling limit. The AFM critical temperature, on the otherhand, depends on the treatment of the Hartree term (note thateven in the PM phase the evaluation of the spin susceptibilitymay depend on the choice of the Hartree diagram since it en-ters in the vertex correction), and only the approach of type (I)among the various methods that we tested survives for largeU. It would be interesting to compare the situation with theT -matrix approximation68,69 that is often adopted in the studyof the attractive Hubbard model. It takes account of a series ofladder diagrams for the self-energy and similarly reproducesthe BCS-BEC crossover for Tc.

We also plot the staggered magnetization m =

n = G

M (0) for the ordered state evaluated by QMC and

the second-order perturbation theory of type (I) in the weak-coupling (the top panel of Fig. 11) and strong-coupling (bot-tom panel) regimes. For small U, the second-order perturba-tion theory gives a smooth curve for the magnetization as afunction of T . As one increases U, there emerges a kink inthe magnetization curve for U 4 (Fig. 11), which is an ar-tifact of the perturbation theory as confirmed by comparison

QMC2nd order HIL

U=1

U=2

U=3

U=4

0.00 0.05 0.10 0.15 0.200.0

0.2

0.4

0.6

0.8

1.0

T

m

QMC2nd order HILU=5U=6

U=8

U=10

0.00 0.05 0.10 0.15 0.200.0

0.2

0.4

0.6

0.8

1.0

Tm

FIG. 11. (Color online) The staggered magnetization for the Hubbardmodel at half filling evaluated by DMFT with QMC and the second-order perturbation theory of type (I) in the weak-coupling (top panel)and strong-coupling (bottom panel) regimes. The QMC data for U =2, 10 are taken from Ref. 67.

to the QMC results. Hence, although Tc behaves reasonablyin the large U regime, it is unlikely that the second order per-turbation of type (I) correctly describes the strong-couplingstate.

Figure 12 plots the spectral function A() = 1pi ImGR()obtained from the second-order perturbation of type (I). Theweak-coupling regime (U = 3, top panel of Fig. 12) showscoherence peaks separated by the AFM energy gap and ac-companied by the Hubbard sidebands. However, as onegoes to the strong-coupling regime (U = 6, bottom panel ofFig. 12), the coherence peaks are rapidly shifted away fromthe Fermi energy, and two additional bands appear around = 1.5. This is quite different from the result of the non-crossing approximation25 (dashed lines in the bottom panelof Fig. 12), which is supposed to be reliable in the strong-coupling regime, and shows spin-polaron peaks on top of theMott-Hubbard bands with a large energy gap. Therefore, weconclude that the second-order perturbation of type (I) doesnot correctly describe the AFM state in the strong-couplingregime.

The phase diagram derived from DMFT using variousthird-order perturbation theories is shown in Fig. 13. Againwe do not draw the Tc curve for type (V), since it has a dis-continuous jump of the magnetization as a function of tem-

11

U=3

-5 0 50.0

0.2

0.4

0.6

0.8

1.0

AHL

U=6

-5 0 50.0

0.2

0.4

0.6

0.8

1.0

AHL

FIG. 12. (Color online) The spectral function of the majority (mi-nority) spin component [blue (red) solid curve] for the AFM phaseof the Hubbard model at half filling calculated by DMFT with thesecond-order perturbation theory of type (I). The dashed lines showthe spectral function calculated by DMFT with the noncrossing ap-proximation. The parameters are U = 3, = 20 (top panel) andU = 6, = 16 (bottom).

perature in the ordered state. We find that the third-order per-turbation of type (I) (all the diagrams including the Hartreeterm are bare) reproduces Tc very accurately up to U = 3. Acomparable accuracy cannot be obtained with the other third-order expansions. To establish the validity of this approach,we calculate the staggered magnetization below Tc, which isillustrated in Fig. 14. By comparing the results with those ofQMC, we can see that the type (I) third-order approach pre-dicts not only accurate Tc but also correct magnetizations forU 2.5.26 When U becomes larger than 2.5, deviations fromthe exact QMC results start to appear, and the curvature of themagnetization curve gets steeper. Thus, the third-order per-turbation of type (I) is the method of choice for studying theAFM phase in the weak-coupling regime (U < 3). This isagain consistent with the observation of Yosida and Yamada44

that the bare weak-coupling expansion works well if expandedaround the nonmagnetic Hartree solution (i.e., = 12 ).

V. INTERACTION QUENCH IN THE PARAMAGNETICPHASE OF THE HUBBARD MODEL

Having examined the performance of the weak-couplingperturbation theories for the equilibrium state of the Hubbardmodel, we move on to studying the validity of the perturba-tive methods for nonequilibrium problems. In this section, we

QMC3rd order HIL3rd order HIIL3rd order HIIIL3rd order HIVL

0 1 2 3 40.00

0.05

0.10

0.15

0.20

U

T

FIG. 13. (Color online) The AFM transition temperature for the Hub-bard model at half filling derived from DMFT with various third-order perturbation impurity solvers. The region below (above) thecurves represents the AFM (PM) phase. The labels (I)-(IV) corre-spond to the classification in Table I.

QMC3rd order HIL

U=1U=1.5 U=2 U=2.5 U=3

0.00 0.05 0.10 0.15 0.200.0

0.2

0.4

0.6

0.8

1.0

T

m

FIG. 14. (Color online) The staggered magnetization of the Hubbardmodel at half filling evaluated by DMFT with QMC and the third-order perturbation theory of type (I).

focus on the interaction quench problem for the PM phase ofthe Hubbard model; i.e., we consider the Hamiltonian (1) withthe interaction parameter abruptly varied as

U(t) =

{Ui t = 0

U f t > 0. (54)

The interaction quench problem for the Hubbard modelhas been previously studied using the flow equation andunitary perturbation theory,22,70 nonequilibrium DMFT,23,53

time-dependent Gutzwiller variational method,71,72 general-ized Gibbs ensemble,73 equation-of-motion approach,74,75 andquantum kinetic equation.76 In the weak-coupling regime, thephysics of prethermalization21 has been discussed.

Here we take the parameters, Ui = 0 (noninteracting initialstate) and the initial temperature T = 0, to allow a systematiccomparison with these previous results. We use the second-order and fourth-order perturbation theories for the half-filling

12

case in Sec. V A and the third-order perturbation theory forcalculations away from half filling in Sec. V B. We restrictourselves to the PM solution of the nonequilibrium DMFTequations throughout this section.

A. Half filling

To study the relaxation behavior of the Hubbard model afterthe interaction quench, we calculate the time evolution of thedouble occupancy d(t) = n(t)n(t) within nonequilibriumDMFT via the formula,

d(t) = i2U

( G)

13

0 1 2 3 4 5

0.10

0.15

0.20

0.25

t

d

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

t

Dn

FIG. 16. (Color online) Time evolution of the double occupancy (toppanel) and the jump of the momentum distribution function (bottom)after the interaction quenches U = 0 4, 5, 6, 8 (from bottom to topin the first minima of d, and from top to bottom in the initial decreaseof n) in the PM phase of the Hubbard model at half filling calcu-lated by the nonequilibrium DMFT with QMC (circles, taken fromRef. 23), the bare second-order (dashed curves), and bare fourth-order (solid curves) perturbation theories.

turbation theories (both second and fourth order) correctly re-produce the short-time dynamics of these collapse-and-revivaloscillations in n with period 2pi/U. Especially, they capturethe sharp qualitative change of the short-time behavior of n.In the second-order perturbation theory the oscillations of nalways damp faster than those given by the fourth-order cal-culation. Similarly to the double occupancy, the n obtainedfrom the perturbation theories fail to reproduce the QMC re-sults after the second- and fourth-order calculations start todeviate with each other.

The quality of the bare-diagram perturbation theory can bejudged by looking at the evolution of the total energy (Fig. 17).The total energy of the Hubbard model is given by

E(t) =k

knk(t, t) + U(t)(d(t) 1

4

). (57)

For the semicircular DOS in the PM and AFM phases, thekinetic-energy term can be rewritten in terms of the local

bare 2nd order Ht=5Lbare 2nd order Ht=10L

bare 4th order Ht=5Lbare 4th order Ht=10L

0 1 2 3 4

-0.2

-0.1

0.0

0.1

0.2

U f

EHtL-

EH0+ L

FIG. 17. (Color online) The drift of the total energy E(t) E(0+)(measured at t = 5, 10) in the simulation of the interaction quenchU = 0 U f using the bare second-order and bare fourth-orderperturbation solvers.

Greens functions ask

knk(t, t) = i

( G) 2 (Figs. 15 and 16). Let us remark that thisdoes not necessarily mean that the bare perturbation theoryalways fails to describe the dynamics of the Hubbard modelwith U > 2. It all depends on how the system is perturbed(interaction quench, slow ramp, electric-field excitation, etc.),the initial state (noninteracting or interacting), and other de-tails of the problem. The general tendency is that the total en-

14

U = 016U = 024U = 032

0 1 2 3 4 50.18

0.19

0.20

0.21

0.22

0.23

0.24

0.25

t

d

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

t

Dn

0 1 2 3 4 5-1.0

-0.9

-0.8

-0.7

-0.6

t

E

FIG. 18. (Color online) Time evolution of the double occupancy(top), the jump of the momentum distribution function (middle), andthe total energy (bottom) after the interaction quenches U = 0 16, 24, 32 (from long- to short-time data) in the PM phase of theHubbard model at half filling calculated by nonequilibrium DMFTwith the bare second-order perturbation theory.

ergy is conserved when the excitation energy is small and/orthe interaction strength is weak.

If we further increase U f , the second-order bare perturba-tion theory again starts to work reasonably well. In Fig. 18, weplot d, n, and E for quenches U = 0 16, 24, 32. The sim-ulation is numerically stable within the accessible time range,and the observables do not diverge as time grows. The resultsnicely show the coherent collapse-and-revival oscillations ofperiod 2pi/U, which are characteristic of the atomic limit. Wealso observe that the envelope curve of rapidly oscillating nis universal; i.e., it is almost invariant against large enough

U. In contrast, the fourth-order bare perturbation theory failsto produce physically reasonable results for these U. Theseeming success of the second-order bare perturbation the-ory (IPT) for very large U appears to be related to the factthat IPT reproduces the correct atomic limit of the Hubbardmodel in equilibrium.32 However, it is a priori not obviousthat IPT also describes the correct nonequilibrium dynamicsnear the atomic limit, since the dynamics here starts from thenoninteracting state which is very far from the atomic limit,and errors can accumulate in the strong-coupling regime asthe system time evolves. Indeed, if one looks at the total en-ergy E (bottom panel of Fig. 18), there is a non-negligibleenergy drift whose magnitude ( 10% of the absolute valueof E) is roughly independent of U. Despite the violation ofenergy conservation, IPT seems to work surprisingly well outof equilibrium near the atomic limit.

We have also tested the second-order and fourth-order self-consistent perturbation theories (bold-diagram expansions).The results for the double occupancy and the jump in the mo-mentum distribution for U f 3 (U f 4) are shown in the topand bottom panels of Fig. 19 (Fig. 20), respectively. By com-paring with QMC, one can see that the self-consistent pertur-bation theories are not particularly good. Although we have aslight improvement from the second-order to the fourth-orderexpansion, a deviation from the QMC results still remains,even in the short-time dynamics. The detailed dynamics ofd and n in the transient and long-time regimes is not cor-rectly reproduced. The damping of the double occupancy istoo strong in both the weak-coupling and the strong-couplingregimes (see also Ref. 23). This may be due to a too-tightself-consistency condition, i.e., the self-consistency within theperturbation theory and the DMFT self-consistency. [Flaws inthe (second-order) self-consistent perturbation theory, whenapplied to the equilibrium DMFT,77 were pointed out alreadyin Ref. 46. In particular, it was noted that it does not re-produce the high-energy features (Hubbard sidebands) of thespectral function.] For n, the height of the prethermalizationplateau is not correctly reproduced for U f 3. On the strong-coupling side, n relaxes monotonically without showing anyoscillation. This evidences that the self-consistent perturba-tion theory cannot describe the dynamical transition found inthe interaction-quenched Hubbard model at half filling.23,71

Hence, even though the self-consistent perturbation theory isa conserving approximation, it is not the impurity solver ofchoice for nonequilibrium DMFT.

B. Away from half filling

When the filling is shifted away from half filling, theparticle-hole symmetry is lost, and odd-order diagrams startto contribute in the calculation. Here we consider the interac-tion quench problem for the PM phase of the Hubbard modelat quarter filling, i.e., n = 1/4, and apply the second-orderand third-order perturbation theories. We adopt the type (I)and type (IV) approaches in the classification of Table I, i.e.,the bare-diagram expansions having the bare tadpole diagramwith = 1/2 and bold tadpole with = n, since they have

15

0 2 4 6 8 10

0.10

0.15

0.20

0.25

t

d

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

t

Dn

FIG. 19. (Color online) Time evolution of the double occupancy (toppanel) and the jump of the momentum distribution function (bottom)after the interaction quenches U = 0 0.5, 1, . . . , 3 (from top tobottom) in the PM phase of the Hubbard model at half filling calcu-lated by the nonequilibrium DMFT with QMC (circules, taken fromRef. 23), the bold second-order (dashed curves), and the bold fourth-order (solid curves) self-consistent perturbation theories.

been shown to be relatively good approximations away fromhalf filling in Sec. IV A.

We plot the results produced by the type (I) and type (IV)expansions in Figs. 21 and 22, respectively. In Fig. 21(a), weshow the time evolution of the double occupancy calculatedby the type (I) scheme. The noninteracting initial state hasd(0) = 1/4 1/4 = 1/16 = 0.0625. The second-order andthird-order perturbations give quantitatively different evolu-tions after the quench. When U f is small enough (U f 1),the double occupancy quickly damps to a thermal value. Asone increases U f , an enhanced oscillation starts to appear inboth the second-order and the third-order calculations. InFig. 21(b), we plot the time evolution of the jump n at theFermi energy in the momentum distribution. Initially, the sys-tem has a Fermi distribution with T = 0, so that n(0) = 1.The second-order calculations (dashed curves in the bottompanel of Fig. 21) show that after a rapid decrease, n stabilizesat an intermediate value for a certain time and then slowly de-cays to zero. This behavior (prethermalization) is quite simi-lar to what we have seen in the case of half filling. If we usethe third-order perturbation theories, however, the results dif-fer from those of the second order for U f 1.5. In particular,at U f = 2 the jump n starts to oscillate and finally exceeds

0 1 2 3 4 5

0.10

0.15

0.20

0.25

t

d0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

tDn

FIG. 20. (Color online) Time evolution of the double occupancy (toppanel) and the jump of the momentum distribution function (bot-tom) after the interaction quenches U = 4, 5, 6, 8 (from bottom totop in the long-time limit of d, and from top to bottom in n) inthe PM phase of the Hubbard model at half filling calculated by thenonequilibrium DMFT with QMC (circules, taken from Ref. 23), thebold second-order (dashed curves), and the bold fourth-order (solidcurves) self-consistent perturbation theories.

1, implying that the third-order calculation gives physicallyunreasonable results.

To examine the validity of the perturbation theories, weshow the density and total energy as a function of time inFigs. 21(c) and 21(d). They should be conserved through-out the time evolution. The results suggest that the total en-ergy and density (n = 1/4) are reasonably conserved whenU f 1 in both the second-order and the third-order pertur-bations. Only in this parameter regime, the simulation is re-liable. This limitation is more severe than in the half-fillingcase, where the total energy is sufficiently conserved up toU f = 2.

We also investigated the type (IV) approach in Table Iand show the results in Fig. 22. The behavior of d and n[Figs. 22(a) and 22(b)] looks qualitatively similar to the resultof the type (I) expansion, while there are quantitative differ-ences such as the value of d after relaxation and the plateauheight for n in the prethermalization regime. The simulationwith the third-order expansion of type (IV) becomes particu-larly unstable at U f = 2.5, showing rapid oscillations in d andan irregular evolution in n. If one looks at the density and

16

HaL

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

t

d

HcL0 2 4 6 8 10

0.20

0.22

0.24

0.26

0.28

0.30

t

n

HbL0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

t

Dn

HdL0 2 4 6 8 10

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

tEHtL-

EH0+

LFIG. 21. (Color online) Time evolution of the double occupancy (a), the jump of the momentum distribution function (b), the density (c), andthe total-energy drift (d) after interaction quenches in the PM phase of the Hubbard model at quarter filling calculated by the nonequilibriumDMFT with the type (I) second-order (dashed curves, U = 0 0.5, 1, . . . , 3), and type (I) third-order (solid curves, U = 0 0.5, 1, 1.5, 2)perturbation theories.

total energy given by the type (IV) perturbation [Figs. 22(c)and 22(d)], the conserving nature is somewhat improved withrespect to the type (I) calculation. The conservation starts tobreak down earlier in U f in the third-order expansion com-pared to the second-order one. Thus, we do not see a sys-tematic improvement away from half filling by proceeding tohigher-order perturbation expansions. It should be noted thatalso the weak-coupling QMC method can only reach timeswhich are about a factor of two shorter than in the case ofhalf-filling, because the odd-order diagrams contribute to thesign problem. Hence, the development of a useful impuritysolver for nonequilibrium DMFT calculations away from halffilling in the weak-coupling regime remains an open issue.

VI. DYNAMICAL SYMMETRY BREAKING INDUCED BYAN INTERACTION RAMP IN THE HUBBARD MODEL

So far, we have considered the interaction quench dynamicsof the Hubbard model without any long-range order. Since theformalism of the nonequilibrium DMFT has been generalizedto the AFM phase in Sec. II, we can apply the perturbativeimpurity solvers to the dynamics of such an ordered state.

In this section, we study dynamical symmetry breaking inthe Hubbard model induced by an interaction ramp by meansof the nonequilibrium DMFT with the third-order perturba-tion theory of type (I) (Table I). This impurity solver correctly

reproduced the AFM phase diagram (Fig. 13) and the magne-tization (Fig. 14) in the weak-coupling regime. We begin withthe PM initial state in thermal equilibrium, and then changethe interaction parameter continuously (ramp) as

U(t) =

{Ui + (U f Ui)t/tramp 0 t tramp,U f t > tramp,

(59)

where tramp is the ramp time, to go across the phase transitionline in the phase digram (Fig. 13). We consider an interactionramp (tramp > 0) rather than a quench (tramp = 0) to reduce theincrease of the energy, but it turns out that the results do notsignificantly depend on tramp.

In order to trigger the symmetry breaking, we introduce atiny staggered magnetic field h in the initial state. We assumethat the seed field h is uniform in space, so that the order pa-rameter (staggered magnetization m) grows uniformly. Froma large-scale point of view, this assumption is probably notappropriate, since the direction of symmetry breaking is ran-dom at each position, which leads to domain structures andtopological defects in between (Kibble-Zurek scenario19,20).However, our interest here lies in the fast microscopic dynam-ics of the order parameter, where our set up can be justified.For convenience, we ramp off the seed field in the following

17

HaL

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

t

d

HcL0 2 4 6 8 10

0.20

0.22

0.24

0.26

0.28

0.30

t

n

HbL0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

t

Dn

HdL0 2 4 6 8 10

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

tEHtL-

EH0+

LFIG. 22. (Color online) Time evolution of the double occupancy (a), the jump of the momentum distribution function (b), the density (c), and thetotal-energy drift (d) after interaction quenches in the PM phase of the Hubbard model at quarter filling calculated by the nonequilibrium DMFTwith the type (IV) second-order (dashed curves, U = 0 0.5, 1, . . . , 3), and type (IV) third-order (solid curves, U = 0 0.5, 1, . . . , 2.5)perturbation theories.

way:

h(t) =

{h(1 t/tramp) 0 t tramp,0 t > tramp.

(60)

A. Nonequilibrium DMFT results

In Fig. 23, we show the evolution of the staggered magne-tization after the interaction ramp obtained by the nonequilib-rium DMFT. The parameters are chosen such that Ui = 1.75, = 11, h = 104, and tramp = 10. The initial state is inthe PM phase and is quite close to the AFM phase boundary(solid red circle in Fig. 24). We fix the initial state and sys-tematically change U f to perform a series of interaction-rampsimulations. The initial magnetization is very small but finitedue to the presence of the staggered magnetic field h. Afterthe interaction ramp, the PM state becomes unstable, and theorder parameter starts to grow exponentially (m et/i withi the initial growth rate). It is followed by an amplitude os-cillation and a gradual relaxation toward the final state. Herethe oscillation is not as coherent as in the case of a ramp outof the symmetry-broken phase,26 and one can see a softeningof the amplitude mode in Fig. 23.

In the long-time limit, the system finally thermalizes in thenonintegrable Hubbard model. We can estimate the final tem-perature by searching for the equilibrium thermal state with

effective temperature Teff that has the same total energy as thetime-evolving state, since the total energy should be conservedafter the interaction ramp. In Fig. 25, we plot the total en-ergy for the interaction ramps that correspond to Fig. 23. ForU f . 2.1, the total energy is nicely conserved after the interac-tion ramps. As U f is further increased, there emerges a smallenergy drift during the symmetry breaking (10 t . 40). Af-ter the symmetry breaking, the conservation of the total energyis recovered. Thus, we have a slight inaccuracy in the simu-lation of the interaction ramps for larger U f . We use the finalvalue of the total energy, Etot(t = 100), to extract the effec-tive temperature of the thermal states reached in the long-timelimit. In Fig. 24, we indicate the final thermalized states inthe phase diagram by open circles. As we increase U f , Teffincreases in the vicinity of the phase boundary. Similarly tothe case of the dynamical phase transition out of the AFMphase,26 it seems to trace more or less the constant entropycurve,78 although the interaction ramps that we consider hereare not at all adiabatic processes.

The arrows in Fig. 23 indicate the thermal value of the orderparameter (mth) that is realized in the long-time limit. We no-tice that there is a large deviation between the transient mag-netization and the thermal values mth for 1.9 U f 2.2. Es-pecially, the center of the oscillation of the amplitude mode isdifferent from the long-time limit mth, so that the evolution ofthe order parameter is a superposition of a damped oscillationand a slow drift. This reminds us of the behavior of the or-

18

0 20 40 60 80 1000.0

0.1

0.2

0.3

0.4

0.5

t

m

FIG. 23. (Color online) Time evolution of the staggered magnetiza-tion after the interaction ramps U = 1.75 1.8, 1.9, . . . , 2.6 (frombottom to top) in the Hubbard model. = 11, h = 104, andtramp = 10. The arrows indicate the corresponding thermal valuesreached in the long-time limit.

AFM

PM

0 1 2 3 40.00

0.05

0.10

0.15

0.20

U

T

FIG. 24. (Color online) The initial (solid circle, Ui = 1.75) and final(open circles, U f = 1.9, 2, . . . , 2.6 from left to right) thermal statesin the simulation of the dynamical symmetry breaking in Fig. 23.The labels PM and AFM indicate the paramagnetic and antifer-romagnetic phases, respectively.

der parameter seen in the dynamical phase transition from theAFM to PM phase induced by an interaction ramp,26 where mdoes not decay immediately after the ramp but is trapped toa nonthermal value for a long time. It has been shown for thatcase that on a relatively short time scale the order-parameterdynamics is governed by the presence of a nonthermal criti-cal point, in the vicinity of which the period of the amplitude

0 20 40 60 80 100-1.00

-0.98

-0.96

-0.94

-0.92

-0.90

t

E

FIG. 25. (Color online) Time evolution of the total energy for theinteraction ramps that correspond to Fig. 23.

mode diverges.One may define two time scales that characterize the trap-

ping of the nonthermal fixed point, namely the approachtime to and the escape time from the nonthermal fixedpoint. The escape time is determined by U f . In the U f 0limit it diverges to infinity, while for U f 3 it becomes quiteshort (Fig. 23) since thermalization is accelerated. This is con-sistent with the previous observation of fast thermalization inthe PM phase of the Hubbard model.23 On the other hand, thecharacterization of the approach time is unclear because it de-pends on the definition. If one considers the initial exponen-tial growth of the order parameter as part of the nonthermalfixed-point behavior, then the approach time is very short anddoes not significantly depend on Ui and U f . Indeed, as wesee in Sec. VI C, this exponential growth exists in the Hartreesolution, which characterizes the nonthermal fixed point. Theweak dependence of the approach time on U is consistent withRefs. 22 and 23. However, if one interprets the approach timeas the time necessary for the order parameter to enter the co-herently oscillating regime, then it is roughly determined byi.

To analyze the critical behavior of the dynamical symme-try breaking near the phase transition point, we plot severalrelevant quantities in Fig. 26. mth (the thermal value reachedin the long-time limit) vanishes at the thermal critical point(U f = U thc ) as

mth (U f U thc )12 . (61)

This is consistent with the mean-field prediction mth (U f U thc )

with the mean-field critical exponent = 12 . i is thetime constant of the initial exponential growth (m et/i ),which diverges as

i (U f U thc )1. (62)Note that these exponents are universal; i.e., they do not de-pend on details of the problem (the initial condition, perturba-tion of the system, etc.). We also measured the maximum ofthe first peak (mmax) and the minimum of the first dip (mmin)of the amplitude oscillation, and we plot these quantities in

19

mmaxmminmthi-1H4L

Ucth

1.6 1.8 2.0 2.2 2.40.0

0.1

0.2

0.3

0.4

U f

FIG. 26. (Color online) Various quantities that characterize the crit-ical behavior of the dynamical symmetry breaking with Ui = 1.75:mmax and (mmin) is the maximum (minimum) of the first cycle of theoscillation in m(t), mth is the thermal value taken in the long-timelimit, and i is the rate of the initial exponential growth. The dashedline is an extrapolation of the middle points of mmax and mmin.

Fig. 26. mmax and mmin characterize the trapping of theorder parameter in the transient regime. They behave differ-ently from mth: mmax and mmin are always smaller than mth.The middle point mnth (mmax + mmin)/2 (nonthermal mag-netization) seems to depend linearly on U f , which must becontrasted with the square-root dependence for mth (61). Wesee in Sec. VI C that this linear scaling can be justified in theweak-correlation limit. The linear extrapolation of the middlepoints (dashed line in Fig. 26) implies that the trapped orderparameter vanishes at a certain point U f = Unth , which is dif-ferent from the thermal critical point (U f = U thc ), as

mnth (U f Unth )1. (63)As we see later in Sec. VI C, this nonthermal critical behav-ior becomes exact in the small U regime (where the Hartreeapproximation is applicable) with Unth identical to U

thc . There

are several possible interpretations of the behavior (63) forlarger U: One is that the nonthermal critical point is shiftedfrom U f = U thc to U

nth due to correlation effects. Another in-

terpretation is that the nonthermal critical point still exists atU f = U thc , but mmax and mmin are lifted up due to thermaliza-tion toward mth. In any case, it is likely that the qualitative fea-tures of the nonthermal critical point survive to some extent inthe moderate U regime, so that it affects the order-parameterdynamics during the dynamical symmetry breaking.

If we start with a smaller Ui, the amplitude mode inducedby the interaction ramp becomes more coherent. In Fig. 27,we show the time evolution of m for Ui = 1.25, U f = 2, = 22, h = 104, and tramp = 10 (blue curve). We canclearly see many oscillation cycles. Again the oscillation cen-ter slowly drifts to the thermal value mth (arrow in Fig. 27).Figure 28 illustrates the corresponding evolution of the dou-ble occupancy. After the ramp (t = 10), the double occu-pancy quickly approaches the thermalized value within thePM phase indicated by the arrow on the left in Fig. 28. This

DMFTGL H=0.07LGL H=0.13L

0 20 40 60 80 100 1200.0

0.1

0.2

0.3

0.4

0.5

t

m

FIG. 27. (Color online) Comparison between the DMFT result ofm(t) for the quench U = 1.25 2, = 22 with the phenomenolog-ical Ginzburg-Landau theory. The arrow indicates the thermal valueof m reached in the long-time limit.

0 20 40 60 80 100 1200.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

t

d

FIG. 28. (Color online) The time evolution of the double occupancyfor the quench U = 1.25 2, = 22. The arrow on the leftindicates the thermal value of m for the PM phase, while the one onthe right shows the value in the AFM phase.

suggests that the system prethermalizes within the PM phasebefore the dynamical symmetry breaking occurs. After theorder parameter m starts to grow, the double occupancy alsooscillates along with the amplitude oscillation of m. In thesame way as m, the double occupancy slowly approaches thethermal value for the AFM phase (the arrow on the right inFig. 28).

The dynamics of the order parameter is reflected in thetime-resolved spectral function A(, t), which is defined bythe retarded Greens function,

A(, t) = 1pi

Im

0dt eitGR(t + t/2, t t/2). (64)

This function represents the single-particle spectrum at timet. Since the range of the time arguments is limited ( tmax),we have to introduce a cutoff in the semi-infinite integral inEq. (64). As a result, the energy resolution is restricted(energy-time uncertainty). Here 2pi/tmax 0.06, which

20

FIG. 29. (Color online) The time-resolved spectral function A(, t)of the minority spin component for the quench U = 1.25 2, =22.

is fine enough to resolve the AFM energy gap.In Fig. 29, we depict A(, t) for the interaction ramp

(U = 1.25 2) which corresponds to the blue magnetiza-tion curve in Fig. 27. At first, the system is noninteracting, sothat A() =

4v2 2/(2piv2) (noninteracting DOS). Af-

ter the interaction ramp, an energy gap is dynamically gener-ated at the Fermi energy ( = 0) in the spectral function. Oncethe gap has opened, the magnitude of the gap [the distancebetween the coherence peaks in A()] stays nearly constantin time. On the other hand, there is a coherently oscillatingspectral-weight transfer between the lower ( < 0) and higher( > 0) energy region, consistent with the time-evolution ofthe order parameter m. The drift of the oscillation center isalso reflected in A(, t). Therefore, the spectral function cap-tures the characteristic behavior of the order-parameter dy-namics. Experimentally, it is not easy to observe the timeevolution of the staggered magnetization directly. However,the change of the spectral function can be detected by time-resolved photoemission spectroscopy and pump-probe opticalspectroscopy. These techniques thus provide a way of track-ing the evolution of the staggered magnetization.

B. Comparison to the phenomenological Ginzburg-Landauequation

To analyze the behavior of the order parameter after theinteraction ramp, we compare the nonequilibrium DMFT

results with the phenomenological Ginzburg-Landau (GL)equation.7981 The GL equation has been widely used to de-scribe the order-parameter dynamics in superconductors andother ordered phases. It is justified when the quasiparticle en-ergy relaxation time is much longer than the time scale of theorder-parameter dynamics.82

Here we adopt a phenomenological description, assumingthat the motion of the order parameter is governed by thefree-energy potential of the final thermal state [Fth(m)] afterthe interaction ramp; i.e., the initial free energy is suddenlyquenched to the final one (sudden approximation). Our equa-tion reads

2tm + tm = Fth(m)m

= 2athm 4bm3, (65)

where is a friction constant, and the free energy of the fi-nal thermal state is expanded as Fth(m) = athm2 + bm4. Todistinguish the coefficient a of the thermal free energy fromthat for the nonthermal potential that will be defined later, weput the subscript th. We can freely rescale both sides ofEq. (65), so that we choose the coefficient of 2tm to be unity.By taking the final free energy, we can guarantee that the or-der parameter converges to the thermal value of the final statein the long-time limit. Of course, the transient state right af-ter the interaction ramp is far from equilibrium, so one cannotexpect that the whole dynamics is reproduced by this suddenapproximation. Here we use the phenomenological approachto demonstrate to what extent the order parameter behaves dif-ferently from the conventional GL picture.

In equilibrium, the order parameter takes the thermal value

mth =ath

2b. (66)

Initially, the order parameter grows exponentially, m et/i .Since the order parameter is small at the initial stage, one canneglect the second term on the right-hand side of Eq. (65).Substituting m et/i in Eq. (65), one obtains the relation

2i + 1i = 2ath. (67)

mth and i can be directly measured. If we fix one parameter(say ), we can identify the other parameters ath and b usingEqs. (66) and (67).

In Fig. 27, we plot the solution of the time-dependent GLEq. (65) for = 0.07 and = 0.13 on top of the nonequi-librium DMFT result. We have agreement in the initial expo-nential growth and the final value, whereas the transient dy-namics of the GL calculations looks quite different from theDMFT result. The GL equation cannot describe the trappingeffect of the order parameter, i.e., the center of the amplitudeoscillation is fixed to mth from the beginning. The amplitude,damping rate, and phase shift of the oscillation are not cor-rectly captured by the GL equation, no matter how the valueof the free parameter is chosen. If we try to fit the frequencyof the amplitude mode ( = 0.13), the damping is too strong.If we try to fit the amplitude of the oscillation ( = 0.07),we have a phase mismatch. Furthermore, the GL equationdoes not capture the softening of the amplitude mode. Thus,

21

we conclude that the DMFT order-parameter dynamics whichshows a softening amplitude mode and a trapping by a non-thermal critical point is out of the adiabatic regime, so the GLdescription is not applicable.

C. Comparison to the time-dependent Hartree approximation

Finally, let us compare the nonequilibrium DMFT resultswith the Hartree approximation, which may be valid in theopposite limit, where the order parameter changes fast com-pared to the quasiparticle scattering time in the weak-couplingregime. In the Hartree approximation, one takes the tadpolediagram (Fig. 3) as the self-energy,

a(t, t) = U(t)naC(t, t). (68)

In the AFM phase the local density is

nA(t) = n +12m(t), (69)

nB(t) = n12m(t), (70)

where n is the average density per spin, and =, = . Athalf filling, n = 12 .

As shown in the Supplemental Material of Ref. 26, theDyson equation (8) and its conjugate equation can be writtenin the form of a Bloch equation for spin precession,

t f k(t) = bk(t) f k(t). (71)Here we use a vector representation f k = ( f xk , f

yk , f

zk) for

the momentum distributions, analogous to Andersons pseu-dospin representation for superconductivity.83 The compo-nents are defined by

f xk (t) =12

[nBAk(t) + nABk(t)], (72)

f yk (t) =i2

[nBAk(t) nABk(t)], (73)

f zk(t) =12

[nAAk(t) nBBk(t)], (74)

where nabk(t) iGab

22

From Eqs. (84) and (86), we have

2 f1(k) = U f f2(k), (87)f3(k) = 2k f2(k), (88)

with U(t > 0) = U f . Substituting these equations intoEq. (85), we get

f2(k)2tm(t) = [(2k)2 f2(k) + U f f0(k)]m(t)U2f2

f2(k)m(t)3.(89)

Let us assume that f2(k) , 0. Then, we can divide the aboveequation by f2(k) to obtain

2tm(t) =[(2k)2 + U f f0(k)f2(k)

]m(t) U

2f

2m(t)3. (90)

This should hold for arbitrary k, which suggests that the coef-ficient of m(t) on the right-hand side must be independent ofk. This motivates us to set it to a k-independent constant,

(2k)2 + U f f0(k)f2(k) anth, (91)

or

f2(k) = U ff0(k)

(2k)2 + anth. (92)

If such a constant anth(> 0) exists, m(t) satisfies the followingGL-like equation,

2tm(t) = Fnth(m)m

, (93)

Fnth(m) = 12anthm2 +

U2f8m4, (94)

where Fnth(m) is a nonthermal free-energy potential. Notethat Fnth(m) , Fth(m). In particular, terms with orders higherthan four are absent in Fnth(m). Now the mean-field condition(83) becomes

U fk

2k(2k)2 + anth

f0(k) = 1. (95)

f0(k) is determined from the initial condition. Let us assumethat the initial magnetization m(0) induced by the seed mag-netic field is very small and tm(t) = 0, which leads to

f xk (0) = f (k i + Uin) f (k i + Uin) + O(m(0)2),(96)

f yk (0) = 0, (97)f zk(0) = O(m(0)), (98)

with i the chemical potential of the initial state. Here we haveused the noninteracting equilibrium distribution function,

nBAk = nABk = T

n

ein0+ k

(in + i Uin)2 2k=

12

[ f (k i + Uin) f (k i + Uin)].(99)

0 20 40 60 80 100 120 1400.0

0.1

0.2

0.3

0.4

0.5

t

m

FIG. 30. (Color online) The time evolution of m for the interactionramp U = 0 1.25 in the Hubbard model at half filling with = 40,h = 104, and tramp = 10 obtained from the Hartree approximation.

Thus, we obtain

f0(k) = f (k i + Uin) f (k i + Uin). (100)Now we have determined all the components of f k(t), whichare consistent with the equation of motion (71), the self-consistency condition (76), and the initial condition (96)-(98).

If no constant anth exists which satisfies Eq. (95), then nosymmetry breaking occurs. Hence, the existence of a solutionfor anth in Eq. (95) is a prerequisite for dynamical symmetrybreaking. As we see below, this corresponds to the conditionfor a symmetry-broken solution in equilibrium; that is, the dy-namical symmetry breaking occurs if and only if U f exceedsU thc determined by Eq. (79) with the initial temperature.

In Fig. 30, we plot as an example the solution for the in-teraction ramp U = 0 1.25 given by the Hartree approx-imation. If the ramp is performed fast enough compared tothe development of the order parameter, it can be consideredas a quench. The interaction ramp generates an exponentialgrowth of the order parameter, after which it goes through amaximum and returns back to the initial value. The curve ofm(t) in Fig. 30 looks like a soliton. In fact, Eq. (93) allows foran analytical soliton solution,

m(t) =2anth

U f cosh(antht)

, (101)

for the initial condition that m() is infinitesimal. Whenthe initial value is nonzero, the solution corresponds to a trainof solitons as shown in Fig. 30. The period of the soliton traindepends on the initial condition and is hence nonuniversal,while the maximum of m(t), mmax = 2

anth/U f , does not. As

we see below, this mmax exhibits a universal behavior, whichobeys a scaling law different from that for the conventionalGL theory.

The nature of the Hartree solution is quite distinct fromthe DMFT results (Figs. 23 and 27): The Hartree approxi-mation gives a permanently oscillating m, whereas the DMFTsolution indicates that the amplitude oscillation damps, and meventually converges to the thermal value mth. The difference

23

is apparently coming from the lack of scattering processes inthe Hartree approximation. In other words, it is due to the in-tegrability of the Hartree equation. However, there seem toexist common universal features in both results. For example,the universality of mmax in the Hartree approximation some-how survives even after we take account of correlations in thenonequilibrium DMFT. We examine this point later.

Let us first take a closer look at the coefficient anth of thequadratic term in Fnth(m), since it controls the phase tran-sition. anth is implicitly determined by Eq. (95), throughwhich anth can be regarded as a function of U f . Assum-ing that anth = anth(U f ) is a reversible function, we writeU f = U f (anth). We now prove that in the vicinity of anth = 0,U f varies as

U f (anth) = Unthc + canth + O(a1nth); (102)

i.e., U f has a square-root dependence on anth (c is an arbi-trary constant). Unthc limanth0 U f (anth) can be interpretedas the critical interaction strength; i.e., the dynamical symme-try breaking is generated when U f > Unthc .

To identify Unthc , we consider the limit anth 0 in Eq. (95).Substituting anth = 0 in Eq. (95) gives

Unthck

f (k i + Uin) f (k i + Uin)2k

= 1,

(103)

which is equivalent to the static mean-field Eq. (79) if we iden-tify iUin in Eq. (103) with U thc n in Eq. (79) and Unthc inEq. (103) with U thc in Eq. (79). This identification is allowed ifthe particle number is conserved (the Hartree approximationis conserving). Hence, the relation Unthc = U

thc Uc holds

exactly for arbitrary filling within the Hartree approximation.To see the anth dependence of U f , we take the derivative of

Eq. (95) with respect to anth,

dU1fdanth

=k

2k[(2k)2 + anth]2

f0(k). (104)

This leads to

dU fdanth

= 2U2fanth

dU1fdanth

= 2U2f

d D()(2)2

anth

[(2)2 + anth]2

f ( i + Uin) f ( i + Uin)2

. (105)

Let us consider the quantity

(2)2anth

[(2)2 + anth]2. (106)

The integral of this quantity does not depend on anth,

d(2)2

anth

[(2)2 + anth]2=

d(2)2

[(2)2 + 1]2=pi

4, (107)

and

limanth0

(2)2anth

[(2)2 + anth]2= 0 ( , 0), (108)

which means

limanth0

(2)2anth

[(2)2 + anth]2=pi

4(). (109)

By taking the limit anth 0 in Eq. (105), we have

limanth0

dU fdanth

= 2U2c

d D()pi

4()

f ( i + Uin) f ( i + Uin)2

= pi2U2cD(0) f

(i Uin) c, (110)

which is finite as long as f (i Uin) is finite. As a re-sult, one obtains the expansion (102). [Zero temperatureis an exception, since f (i Uin) diverges or vanishes.At half filling there is a logarithmic correction U f (anth) c(ln anth)1, while away from half filling it has a linear de-pendence U f (anth) Uc + canth around anth = 0.]

The result (102) implies

anth (U f Uc)2 (U f Uc), (111)which strikingly contrasts with the behavior of the conven-tional GL free energy Fth(m), having ath (U f Uc)1. Thescaling (111) is natural from the point of view of the powercounting, since anth has the dimension of (energy)2. Puttinganth = a0(U f Uc)2 (a0 is a dimensionless constant), the non-thermal potential becomes

Fnth(m) = 12a0(U f Uc)2m2 +

U2f8m4. (112)

The scaling law (111) is universal, i.e., the exponent doesnot depend on details of the problem (, i, Ui, U f , and otherparameters). It defines a new universality class that character-izes the nonequilibrium dynamical symmetry breaking. Forexample, the maximum of the magnetization curve, mmax, orthe middle point mnth = (mmax + mmin)/2 = mmax/2, scales as

mnth mmax (U f Uc) (113)with

= 1. (114)

By comparing this with the thermal scaling (61), we noticethat mth becomes much bigger than mnth when Unthc = U

thc

(which is the case in the Hartree approximation). That is, inthe vicinity of the critical point the magnitudes of the ther-mal and nonthermal order parameters are very different. Thisleads us to the following scenario. When one goes beyondthe Hartree approximation by including correlation effects, mapproaches mth in the long-time limit. However, if there ex-ists a nonthermal critical point, which may govern the tran-sient order-parameter dynamics, m is trapped for some dura-tion around mnth, which can deviate strongly from the finalvalue (mth).

24

0 20 40 60 80 100 1200.00

0.05

0.10

0.15

0.20

0.25

0.30

t

m

FIG. 31. (Color online) Time evolution of m for interaction rampsU = 1.0 1.05, 1.1, . . . , 1.5 in the Hubbard model at half fillingwith = 40, h = 104, and tramp = 10.

i-1H4Lmmaxmminmth

1.0 1.1 1.2 1.3 1.4 1.50.00

0.05

0.10

0.15

0.20

0.25

0.30

U f

FIG. 32. (Color online) The initial exponential growth rate i andthe maximum (minimum) value of m at the first peak (dip) of theoscillations for the interaction ramps in Fig. 31.

To confirm the validity of this scenario, we solve a nearlyintegrable system, i.e., the case with smaller Ui and U f thanin Fig. 23 or 27, using the nonequilibrium DMFT with thethird-order perturbation of type (I) (beyond the Hartree ap-proximation). In Fig. 31, we plot the time evolution of m forinteraction ramps Ui = 1 1.05, 1.1, . . . , 1.5. As we de-crease Ui and U f , the magnetization curves approach the formof