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arXiv:1107.1040v1 [cond-mat.str-el] 6 Jul 2011

Quantum Dynamics of a Driven Correlated System, Coupled to Phonons

L. Vidmar,1J. Bonˇ ca,2,1T. Tohyama,3and S. Maekawa4,5

1J. Stefan Institute, SI-1000 Ljubljana, Slovenia

2Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

4The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan

5CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan

Nonequilibrium interplay between charge, spin and lattice degrees of freedom on a square lattice

is studied for a single charge carrier doped in the t–J–Holstein model. In the presence of an uniform

electric field we calculate the qusistationary state. With increasing electron-phonon (e-ph) coupling

the carrier mobility decreases, however, we find increased steady state current due to e-ph coupling

in the regime of negative differential resistance. We explore the distribution of absorbed energy

between the spin and the phonon subsystem. For model parameters as relevant for cuprates, the

majority of the gained energy flows into the spin subsystem.

PACS numbers: 71.27.+a, 72.10.Di, 87.15.ht

Introduction.—One of the outstanding contemporary

challenges in condensed matter physics is to understand

dynamics of interacting quantum systems exposed to

an external perturbation. Advanced pump and probe

techniques with few femtosecond time-resolution and

broad-band THz spectroscopy were developed [1–4] to

drive the system out of equilibrium and measure its

nonequilibrium physical properties. These measurements

were complemented by time-resolved photoemission spec-

troscopy [5, 6], electron cristallography [7] and Raman

scattering [8]. In the systems with competing interac-

tions the most demanding task is to disentangle different

elementary excitations arising at comparable energy-time

scales.An example of such materials are cuprate su-

perconductors, where the interplay between strong cor-

relations and lattice degrees of freedom is responsible

for many unconventional properties.

electron-phonon (e-ph) interaction on ultrafast dynam-

ics was recently investigated, and different mechanisms

were proposed [3, 5, 8, 9].

The influence of

Despite a considerable ongoing effort to understand

nonequilibrium dynamics of quantum systems, a siz-

able gap perseveres between theory and experiments.

A theoretical insight into many-body quantum phenom-

ena far from equilibrium has been obtained, among oth-

ers, through works on nonlinear transport of half-filled

systems in one dimension (1D) [10, 11], infinite dimen-

sions [12], and analysis on correlation induced quench-

ing of Bloch oscillations [13]. Lately, an effort has been

devoted to study response of pump-excited Mott insula-

tors [14], and a problem of complex thermalization pro-

cess of strongly correlated systems was addressed [15].

Still, the nonequilibrium interplay between strong corre-

lations and lattice effects as described within a many-

body quantum model remains for the most part un-

resolved. Various different approaches have been ap-

plied to explain ultrafast dynamics in experimets on

cuprates, however, the majority of data has been con-

sidered in the framework of phenomenological mod-

els [2, 5, 16], d-wave BCS theory [17], and Boltzmann

equations within the LDA approxmation [8] or nonther-

mal electron model [3, 18].

In contrast to the latter approaches, we study a time

evolution of a microscopic model under the influence of

a uniform electric field, containing charge, spin and lat-

tice degrees of freedom, while maintaining the full quan-

tum nature of the model. We address a fundamental,

yet unresolved question concerning the interplay between

strong correlations and e-ph interaction in a driven quan-

tum system far from equilibrium. For this purpose we

investigate time propagation of a single charge carrier

doped into two-dimensional (2D) plane within the t–J–

Holstein model, representing a prototype model for the

description of competing interactions in cuprates.

far, nonequilibrium response of the generalized Hubbard–

Holstein model has been analyzed on a 1D chain [19] and

on 8-site 2D cluster [4]. While qualitatively different be-

havior is expected in 1D systems due to a spin-charge

separation [20], a detailed investigation of 2D systems is

still pending. By generalizing recently developed method

[21, 22], we time evolve the system until it reaches a qua-

sistationary (QS) state with a constant current and as

well as a constant energy flow to the system. To our

knowledge this is the first study of a 2D strongly corre-

lated system (SCS) coupled to phonons where QS condi-

tions are achieved.

So

In this Letter, we explore two important aspects of

nonequilibrium carrier dynamics: (i) We establish the in-

fluence of e-ph coupling on the nonlinear transport prop-

erties of a carrier in SCS. We show that the coupling

to phonons on the one hand decreases carrier mobility,

on the other, it leads to an enhancement of QS current

in the regime of negative differential resistance; (ii) We

compare the energy absorbed by the spin subsystem and

the one absorbed by lattice vibrations. Taking into ac-

count model parameters fitting cuprates we find that the

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spin subsystem absorbs the energy from the electric field

more efficiently than the lattice.

Model and numerical method—We define the time–

dependent t–J–Holstein Hamiltonian as

H = −t0

?

?lj?,σ

?

eiφlj(t)˜ c†

l,σ˜ cj,σ+ H.c.

?

+ J

?

?lj?

Sl· Sj,

+ g

?

j

nj(a†

j+ aj) + ω0

?

j

a†

jaj, (1)

where ˜ cj,σ = cj,σ(1 − nj,−σ) is a projected fermion op-

erator, φlj(t) is time–dependent magnetic flux and ?lj?

denote nearest neighbors.

pled to Einstein phonons with the energy ω0 via e-ph

coupling constant g, where aj is the phonon annihila-

tion operator and nj =?

k0 = (π/2,π/2) is calculated by exact diagonalization

defined over a limited functional space (EDLFS) [23–25].

To construct functions of the Hilbert space we use the

basis generator {|ϕnh

for nh= 0,...,Nh. Here, Hk,˜HJ and Hg represent the

off-diagonal parts of the first, second and third part of

Eq. (1), respectively, and |ϕ0? = ck0|N´ eel? represents a

translationally invariant state of a carrier in the N´ eel

background.We switch on the uniform electric field

F along the diagonal at time t = 0 and perform the

time evolution by iterative Lanczos method [26].

cordingly, we define the charge current along the diago-

nal j(t) and set φlj(t) = −Ft/√2 for (positive) ˆ x– and

ˆ y–direction. We measure F in units of [t0/e0a] and set

t0 = e0 = a = 1. Recently, the EDLFS method was

reported to effectively calculate the QS state of a doped

charge carrier within the 2D t–J model [21] as well as of

the Holstein polaron [22]. The strength of the numerical

method is in construction of the Hilbert space that en-

ables not only an accurate description of the ground state

of the spin–lattice polaron, but it allows for enough extra

spin and phonon excitations to absorb energy, emitted by

the field driven carrier, until the system reaches the QS

state. This enables a proper description of the QS state

without coupling the system to an external thermal bath.

Results.—We focus mostly on weak and moderate val-

ues of e-ph coupling λ = g2/8t0ω0and different regimes

of ω0, while keeping J = 0.3 constant. Results, describ-

ing the real-time propagation of a spin-lattice polaron

are shown in Fig.1(a)-(c) for λ = 0.2 and ω0= 0.5. After

a short transient regime t/tB<

QS state with the steady current j(t) =¯j and the linear

increase of the total energy, satisfying ∆˙E(t) = F¯j (com-

pare Fig. 1(a) and (b)), where ∆E(t) = ?H(t)?−?H(0)?.

The longest time of propagation in the QS regime is lim-

ited due to a finite number of spin and phonon excitations

in the Hilbert space acting as reservoars for the energy

absorption. They are determined roughly by Nh, and

choosing Nh = 10 enables computation of QS quanti-

ties with good accuracy. In this work, we are interested

The charge carrier is cou-

σnj,σ. The ground state at

l?} = [Hk(φlj= 0)+˜HJ+Hg]nh|ϕ0?

Ac-

∼1 the system enters the

FIG. 1: (Color online) (a) ∆E(t) and (b) j(t) for F = 1.6,2.8

and 4.0. We measure time in units of Bloch time tB, where

ωB = F/√2. Dashed lines in (a) represent extrapolation of

linear increase of ∆E(t) in the QS state and dashed lines

in (b) the corresponding QS current¯j.

λ = 0.2 and ω0 = 0.5.Numerical accuracy of the time

propagation is checked by the total-energy-gain sum rule

∆E(t) = ωB?t

Dashed lines represent their extrapolation to the QS state.

(d) Energy distribution ratio in the QS state ¯ η vs. F.

We set J = 0.3,

0j(t′)dt′. (c) ∆Es(t) and ∆Eph(t) for F = 2.8.

only in values of F when the response of the system is

dissipative, i.e.,¯j ?= 0. The time evolution for F ≪ 1 be-

comes adiabatic with¯j = 0, as pointed out in discussion

of Ref. [21].

We next turn to the calculation of energy flow to the

spin and phonon subsystem. In contrast to the nonequi-

librium studies of closed systems at half-filling where the

current response is considerably influenced by the Joule

heating [27], the problem of a single carrier in dissipative

medium enables the investigation of the steady growth of

energy due to carrier propagation in initially undistorted

background. In Fig.1(c) we show ∆EJ(t) and ∆Eω0(t),

i.e., expectation values of the second and the fourth term

of Eq. (1). The energy flows to both subsystems are de-

termined by PJ,ω0(t) = ∆˙EJ,ω0(t), defining the distribu-

tion ratio η(t) = Pω0(t)/PJ(t). In the QS state both,

∆EJ(t) and ∆Eω0(t), reveal a linear time dependence

(see dashed lines in Fig. 1(c)), and therefore η(t) = ¯ η.

In Fig. 1(d) we show ¯ η(F) which exhibits only tiny vari-

ation around the constant value. This result is rather

surprising since F strongly influences the energy flow

into the system, nevertheless, ¯ η remains fairly field-

independent.

This result facilitates the investigation of the efficiency

of the energy absorption through spin and phonon chan-

nel when e-ph coupling and phonon frequencies are var-

ied. With increasing ω0, ¯ η increases at fixed λ as seen

in Fig. 2(a). We are particularly interested in the case

¯ η = 1, i.e., when a propagating carrier deposits equal

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FIG. 2:

J = 0.3. At ω0 = 0.2, ¯ η is calculated for λ ≤ 0.2, while for

higher ω0 a Nh-independent ¯ η is obtained until λ = 0.4. We

parametrize ¯ η = a(ω0)b, where a = 0.12,0.17,0.21,0.29,0.35

and b = 1.64,1.65,1.64,1.53,1.54 with increasing λ. (b) Red

dots denote values of ω0,¯ η=1 for which the condition ¯ η = 1

is fulfilled, and dashed line represents a fit ω0,¯ η=1 = c/√λ,

where c = 1.83. Left and right side of the plot correspond to

the SC and WC regime of e-ph coupling, respectively. WC-

SC crossover was calculated by the EDLFS method [24] and

is in good agreement with other numerical methods [28, 29].

Filled ellipse represents parameters as relevant for cuprates.

(Color online) (a) ¯ η vs.ω0 for different λ and

amount of the gained energy to the spin and the phonon

subsystem alike. For this purpose we calculate ω0,¯ η=1

and plot it vs.λ in Fig. 2(b).

ing ω0,¯ η=1 ∼ 1/√λ is found.

energy flow to phonons dominates over the flow to the

spin subsystem only for large values of ω0> 1. Recent

photo-emission [24, 28] and optical experiments [30, 31]

on cuprates that were interpreted within the t-J-Holstein

model, assigned the realistic λ to be in the interval

[0.2,0.3] and ω0∼ 0.2, as indicated by the filled ellipse

in Fig. 2(b).In the parameter regime as relevant for

cuprates, the majority of the absorbed energy via the

charge carrier driven by the constant electric field flows

in the spin subsystem.

We next explore the influence of e-ph coupling on car-

rier’s QS propagation. In Fig. 3 we plot¯j–F character-

istics calculated in the QS state. Solid line represents

¯j–F characteristics for a plain t-J model at J = 0.3, as

calculated in Ref. [21]. In the latter work, it was shown

that a regime of positive differential resistivity (PDR)

at small F evolves into a negative differential resistivity

(NDR) regime at crossover field F0 ∼ 2.3J. The effect

of increasing λ, displayed in Fig. 3(a) at ω0 = 0.5, is

to extend the region of PDR toward larger F, and to

decrease the carrier mobility µ as shown in the inset.

A similar tendency is observed when increasing ω0while

keeping λ fixed, see Fig. 3(b). The carrier mobility in the

PDR regime calculated from the¯j–F characteristics is in

qualitative agreement with the linear response (LR) the-

ory [31], and allows one to detect the limit of LR regime

occuring below the crossover field F0. The variation of

F0with λ and ω0is shown in the inset of Fig. 3(b).

While a decrease of mobility with larger e-ph coupling

Remarkably, a scal-

In the WC regime the

FIG. 3: (Color online)¯j–F characteristics of the QS state.

(a)¯j–F at ω0 = 0.5 and (b) at λ = 0.2. Solid line shows¯j–F

at λ = 0. Inset of (a): µ =¯j/F (squares) calculated from the

dashed lines in (a), LR mobility µLR = 2πD (triangles) with

charge stiffness D as calculated in Ref. [31]. Maximal current

shows slight dependence on Nh (compare¯j–F at λ = 0 with

Ref. [21]), and contributes to discrepancy between µ and µLR.

Inset of (b): crossover field F0between PDR and NDR regime

vs. ω0 for different λ. Dashed line shows F0 for λ = 0.

in the PDR regime can be intuitively understood due to

increased scattering on phonon excitations, the most in-

triguing result of Fig. 3 represents the phonon-induced

enhancement of¯j in the NDR regime. In general, ap-

pearance of NDR regime for large F is a consequence of

limited degrees of freedom contained in the model that

are available to absorb the excess energy. The lack of

sufficient degrees of freedom impedes the carrier motion

along the field at large F. For instance, a NDR regime

in 2D t-J model is characterized by pronounced trans-

verse oscillations of the carrier that serve to emit the

excess energy (gained by hopping along the filed direc-

tion) to spin excitations [21]. Alternatively, carrier prop-

agation can be for large F interpreted in the basis of

Wannier-Stark (WS) states where phonon assisted hop-

ping between these states leads to nonzero¯j [22, 32]. In

this picture one can explain two phenomena observed for

large F in Fig. 3(a): (i) with increasing F, the overlap be-

tween neighboring WS states mediated by the combined

phonon- and magnon- assisted hopping decreases, hence

¯j decreases with F, (ii) increasing the e-ph interaction

seems to boost the already existing magnon- mediated

overlap between WS states at λ = 0. In contrast to low

F, where additional scattering on phonons at λ > 0 di-

minishes the carrier mobility, at large F the increase of

¯j is due to opening of additional channels for deposit-

ing the excess energy through simutaneous emission of

magnons as well as phonons. This explains the seem-

ingly counterintuitive effect observed in our results, i.e.,

that the current is enhanced due to an increased e-ph in-

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teraction. In a similar fashion, increasing of ω0enhances

the phonon assisted hopping between WS states at fixed

λ and leads to an enhancement of¯j shown in Fig. 3(b).

Discussion and Conclusion.—Our results reveal the

complex interplay between strong correlations and

phonons in the 2D lattice under nonequilibrium condi-

tions. While the ratio of the energy flow to the phonon

relative to the spin subsystem remains field-independent,

it increases with increasing λ as well as ω0. Nevertheless,

when taking into account model parameters relevant for

cuprates, the energy flow into the spin subsystem remains

dominant. This result may signal stronger coupling of

charge carrier to the spin system in comparison to e-ph

coupling. An intuitive physical picture emerges when

considering hopping of the hole in the spin background,

simultaneously coupled to phonons. The hole can hop

a few lattice spacings without exciting a single phonon,

however each hop of the hole through a N´ eel background

unavoidably generates spin excitations.

While our investigations of carrier dynamics under the

external field are limited to modeling driving-induced

low-energy intraband excitations, to mimic the situa-

tion realized in ultrafast experiments, one has to combine

mechanisms emerging due to both intraband as well as

interband transitions. In particular, relaxation of carriers

from high-energy unoccupied bands is most likely to oc-

cur via phonon emission, and recently the phonon occu-

pation rate after the photoexcitation was monitored on a

femtosecond timescale [2]. Still, a theoretical description

of a quantum nonequilibrium dynamics in a microscopic

correlated multiband model remains an open problem.

The influence of phonons on carrier’s QS propagation

shows strong dependence on the particular regime of the

¯j–F characteristics. While increasing e-ph coupling leads

to a decrease of the carrier mobility at small F, the QS

current increases due to increased coupling to phonons

for large F, where the system enters NDR. In several re-

cent theoretical studies of SCS the onset of NDR regime

has been reported at large F [12, 21, 33], and its origin

has been interpreted as inefficient dissipation of the en-

ergy gained by hopping along the field direction [21]. Its

characteristic features depend on the system’s topology

since the carrier deposits its energy to model’s excitations

rather than a thermal bath. Nevertheless, we expect the

phonon enhancement of QS current to be a general in-

trinsic feature of closed SCS in the NDR regime.

L.V. and J.B. acknowledge stimulating discussions

with M. Mierzejewski, P. Prelovˇ sek and V. V. Kabanov.

This work has been support by the Program P1-0044

of the Slovenian Research Agency (ARRS) and REIMEI

project, JAEA, Japan.

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