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Strongly anharmonic collective modes in a coupled electron-phonon-spin problem

Sauri Bhattacharyya1, Sankha Subhra Bakshi1, Saurabh Pradhan2and Pinaki Majumdar1

1Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

2Department of Physics and Astronomy, Uppsala University, 751 05 Uppsala, Sweden

(Dated: December 18, 2019)

We solve for the ﬁnite temperature collective mode dynamics in the Holstein-double exchange problem, using

coupled Langevin equations for the phonon and spin variables. We present results in a strongly anharmonic

regime, close to a polaronic instability. For our parameter choice the system transits from an ‘undistorted’

ferromagnetic metal at low temperature to a structurally distorted paramagnetic insulator at high temperature,

through a short range charge ordered (CO) phase near the ferromagnetic crossover at TFM . The small amplitude

harmonic phonons at low temperature cross over to large amplitude dynamics around 0.5TFM due to thermally

generated short range correlated polarons. The rare thermal “tunneling” of CO domains generates a hitherto

unknown momentum selective spectral weight at very low energy. We compare our results to inelastic neutron

data in the manganites and suggest how the singular low energy features can be probed.

Collective modes play a crucial role in dictating low-energy

spectral and transport properties in correlated electron sys-

tems [1–3]. The most detailed information about them comes

from inelastic neutron scattering (INS) experiments [4–18],

which probe the momentum resolved spectrum of lattice,

magnetic, or density ﬂuctuations. The interpretation of INS

results has depended, traditionally, on schemes like the ran-

dom phase approximation (RPA), for phonons, or the 1/S ex-

pansion for spins. These methods are meant to access low

amplitude ﬂuctuations and are limited to low temperature.

Correlated electron systems often show thermal phase tran-

sitions, or strong short range correlated distortions [1–3],

where the low temperature ‘linearised’ dynamics is no longer

useful in describing the modes. While methods like time

dependent Ginzburg-Landau (TDGL) theory [19, 20] or full

scale ‘molecular dynamics’ (MD) [21, 22] are available in

classical systems, equivalent real time methods are rare in

quantum systems. To address collective mode physics in these

systems a method should handle the strong interactions re-

liably, capture large dynamical ﬂuctuations in real time, be

sensitive to spatial correlations, and access ﬁnite temperature.

In this paper we demonstrate a ‘real space’, real time

method that probes the dynamics of phonons and spins

strongly coupled to an electron system, handles strong inter-

actions non perturbatively, and accesses thermal physics. We

deliberately choose an ‘operating point’ where anharmonic ef-

fects in the phonons - arising from mode coupling, polaron

‘tunneling’, and magnetic ﬂuctuations - are large. This allows

us to demonstrate the uniqueness of the method, and also ad-

dress dynamics in the manganites as a non trivial test case.

The manganites provide a concrete template for thermally

induced anharmonic ﬂuctuations. The itinerant egelectrons in

these materials are strongly coupled to lattice modes via Jahn-

Teller (JT) coupling [2], and to t2gbased core spins via large

Hund’s coupling [2, 23]. The JT coupling favours polaron for-

mation (and electron localisation), while the Hund’s coupling

favours a ferromagnetic metallic (FM-M) state. There are ma-

terials, e.g, La1−xCaxMnO3, with x∼0.3, where thermally

induced polaronic distortions convert a homogeneous low T

FM-M to a structurally distorted polaronic insulator above

magnetic transition at TF M . The structural and transport fea-

tures of this problem [2, 24–30] has seen much analysis [31–

33] over the last two decades while the thermal dynamics re-

mained virtually unaddressed.

Experiments exist on the phonon [6–9] and spin dynamics

[12–18]. INS results reveal that (i) in La1−xSrxMnO3with

x∼0.2−0.3, the transverse acoustic phonons show anoma-

lous softening and broadening [6, 7] on heating through TF M ,

while (ii) the magnons along (100) and (110) directions in

La0.7Ca0.3MnO3[12, 15, 18] and La0.8Sr0.2MnO3[13, 16]

show large linewidths and softening near TFM , which can’t

be explained using a simple spin model.

Our work reveals that for a model that involves thermally

induced metal-insulator transition via polaron formation, the

dynamical features above arise naturally, and, additionally,

are accompanied by rare events relating to polaron tunnel-

ing. This leads to a remarkable low energy signature in the

spectrum. We use a Langevin dynamics approach [34, 35],

new to these problems, a Holstein model, rather than Jahn-

Teller, for the phonons, a large Hund’s coupling to drive fer-

romagnetism, and solve a two dimensional [36] electron prob-

lem. The electron-phonon (EP) coupling is chosen so that the

T= 0 system is just below the polaronic instability. The bare

phonon frequency is Ω. Our main results are the following.

(A). Phonons: The phonon spectrum exhibits the expected

‘RPA’ dispersion at T= 0, with resolution limited lineshapes,

but increasing Tleads to three prominent effects: (a) The

growing phonon distortions, and the associated electron den-

sity, order in a short range pattern with wavevector Q∼

(π, π)and the phonon dispersion for momentum q∼Qsoft-

ens signiﬁcantly. (b) The damping Γqis strongest for q∼Q

and grows rapidly with Tdue to a combination of anharmonic

phonon interaction and magnon-phonon coupling. (c) The

short range correlated structures have their own slow dynam-

ics - involving thermally assisted tunneling - and this gener-

ates visible spectral weight at frequency ωΩfor q∼Q.

(B). Magnons: The low temperature magnons are as ex-

pected in a ferromagnet, with ωs(q)∼Jef f (2 −cos(qxa)−

cos(qya)), where Jeff ∼0.1t, and resolution limited widths.

Temperature brings in two features: (a) the spectrum narrows

arXiv:1912.07810v1 [cond-mat.str-el] 17 Dec 2019

FIG. 1. Phonon and spin snapshots at temperatures T /TF M = 0.01,0.5,1.0,1.5and the corresponding spectral maps. The Tare chosen

to represent low, intermediate, ‘critical’, and high temperature regimes. First row: snapshot of the phonon ﬁeld xi(t), showing the change

from an undistorted low Tstate to a progressively large distortion checkerboard correlated state with temperature. Second row: snapshot of

nearest-neighbour spin correlation: Oi(t) = 1

4Pδ~

Si(t).~

Si+δ(t), showing the evolution from a perfect FM to a disordered state on heating

through TF M . Third row: phonon power spectrum 1

T|X(q, ω)|2for qvarying along Γ−X−K−Γin the Brillouin zone. We see a thermally

induced softening and broadening for modes with q∼(π, π), alongwith an unexpected ω→0feature for T>

∼0.5TF M . Fourth row: magnon

power spectrum 1

T|S(q, ω)|2, starting from a Heisenberg-like spectrum with Jef f ∼0.1t, shows overall softening and large dampings near

(π, 0) and (π, π )on heating up.

and the mean dispersion shows a softening that is roughly lin-

ear in T, and (b) The damping stays small till T∼0.5TF M

and then shows a dramatic increase. Unlike phonons, for

whom the principal weight remains at ω∼Ω, the high tem-

perature magnon lineshape is very broad.

Model and method: We study the Holstein-double ex-

change (HDE) model on a two-dimensional square lattice.

H=−t

σ

X

<ij>

c†

iσcj σ −JHX

i

~

Si.~σi−µX

i

ni

−gX

i

nixi+X

i

(p2

i

2M+1

2Kx2

i)(1)

We study a nearest neighbour model with t= 1 at density n=

0.40.Kand Mare the local stiffness and mass, respectively,

of the optical phonons, and g= 1.40 is the electron-phonon

coupling. We set K= 1. In this paper, we report studies

for Ω = pK/M = 0.1t, which is a reasonable value for

real materials. ~

Si’s are ‘core spins’, assumed to be large and

classical. The chemical potential µis varied to maintain the

electron density at the required value. We work in the Hund’s

coupling limit JH/t 1.

The thermal dynamics of the phonons and spins is solved

using the coupled Langevin equations (see Supplement):

Md2xi

dt =−Dph

dxi

dt −Kxi−∂hHel i

∂xi

+ξi(t)

d~

Si

dt =−~

Si×(∂hHeli

∂~

Si

+~

hi) + Ds~

Si×(~

Si×∂hHeli

∂~

Si

)

Hel =X

ij

(tij −µδij )γ†

iγj−gX

i

nixi

tij /t =q(1 + ~

Si.~

Sj)/2(2)

The phonon equation [35] involves inertia, damping, an effec-

tive force from the electronic energy, and noise. The noise sat-

isﬁes the ﬂuctuation-dissipation theorem (FDT) and is speci-

ﬁed by hξi(t)i= 0,hξi(t)ξj(t0)i= 2DphkBT δij δ(t−t0).

The spin dynamics follows a Landau-Lifshitz-Gilbert-Brown

(LLGB) equation [37]. The ﬁrst term on the right hand side

of the spin equation is the torque and the second term is the

Gilbert damping. The noise ~

hialso satisﬁes FDT but enters

in a ‘multiplicative’ form, crossed with the spin ﬁeld ~

Siitself.

The spin evolution conserves |~

Si|.

There are multiple timescales involved. We set the bare os-

cillation period for the phonons, τph = 2π/Ω, as the unit of

time. The low Trelaxation time for phonons is 2M/Dph ∼

60τph, for Dph = 0.05t. For magnons, the typical time pe-

riod is set by τs= 1/Jeff of the effective Heisenberg model,

roughly 10/t. The low Tmagnon relaxation timescale is

D−1

s∼4τs. For our parameter choice τph ∼6τs. The largest

timescale is for phonon relaxation, at 60τph, and the small-

est is for magnetic oscillations at ∼(1/6)τph. The numeri-

cal scheme has to use time discretisation and overall runtime

keeping these in mind. The choice of Dph and Dsis discussed

in the Supplement.

The evolution equations are numerically integrated using

the Euler-Maruyama [38] scheme for phonons and a Suzuki-

Trotter decomposition based method [39] for spins. The time

step is ∆t= 1.6×10−4τph. We typically ran our simulations

on systems of size 24 ×24 for 107steps, i.e ∼103τph. This

ensured that we had enough time for equilibration and enough

frequency resolution to compute the power spectrum.

Results: We organize the results in three parts: (a) T

dependence of the typical instantaneous phonon and spin

backgrounds and gross spectral features, (b) the phonon and

magnon lineshape at a few momenta, and (c) comparison of

our results with inelastic neutron data on the manganites.

Fig.1 correlates the typical phonon and spin backgrounds,

in the upper two rows, obtained as instantaneous Langevin

conﬁgurations, with the momentum resolved power spectrum

of the phonons and spins in the bottom two rows. The top

row shows phonon conﬁgurations {xi}. We see an undis-

torted state at low T(left panel) gradually forming patches of

checkerboard ordered large distortions on heating up. These

patches proliferate in the critical regime. The changing xi

background leads to a rise in the density correlation function

Sn(q)at q= (π, π), shown in the Supplement. The sec-

ond row shows snapshots of the nearest neighbour summed

overlap Oi(t) = 1

4Pδ~

Si(t).~

Si+δ(t), which indicate a ferro-

magnetic low Tstate, and progressively spin disordered con-

ﬁgurations on heating across TF M . The Tdependence of the

associated FM peak, Ss(0,0), in the structure factor, is shown

in the Supplement. The magnetic disorder aids lattice polaron

formation by suppressing the hopping.

The third row shows 1

T|X(q, ω)|2, where X(q, ω ) =

Pieiq.riRdte−iωtxi(t). Sample behaviour of xi(t)in dif-

ferent Tregimes is shown in the Supplement. The low tem-

perature phonon spectrum (ﬁrst column) is accessible through

a harmonic theory, equivalent to RPA in the quantum context,

with intersite phonon correlations arising via the ‘bare’ elec-

tronic polarisability Π0(q). The phonon dispersion has a form

ω0

ph(q)∼p(K+g2Π0(q))/M and the damping Γph(q)of

these ‘normal modes’ is ∼Dph/M. On heating up, the typi-

cal xiincrease in magnitude and the spectrum displays three

distinct features- (i) increased damping due to anharmonicity

induced coupling between normal modes, (ii) ‘softening’ of

the dispersion near q= (π, π), related to enhanced CO cor-

relations, and (iii) the appearance of spectral weight at low

frequencies, ωΩ! The low-energy feature arise from rare

tunneling of checkerboard correlated patches that lead to large

local ‘switching’ of the xi. The low energy weight reduces

when TTF M where large amplitude oscillations and tun-

neling events can no longer be distinguished. We will discuss

the impact of the magnetic degrees of freedom on the phonons

later in the paper.

The fourth row shows 1

T|~

S(q, ω)|2, where ~

S(q, ω) =

Pieiq.riRdte−iωt ~

Si(t). The low temperature spectrum

corresponds to FM spin waves, with ω0

s(q)∼Jef f (2 −

cos(qxa)−cos(qya)) and can be reproduced using a nearest

neighbour FM Heisenberg model with Jeff ≈0.1t. Increas-

ing Treveals a suppression of the intensity and a slow in-

crease in magnon damping. Beyond ∼0.5TF M the magnon

lines broaden rapidly, notably near (π, 0) and (π, π). Beyond

TF M , most of the modes are diffusive in nature, except near

FIG. 2. Top row: Phonon lineshape for q= (π, π)on a lin-

ear (a) and logarithmic (b) scale. The temperatures chosen are-

T/TF M = 0.01,0.5,1.0,1.5. We observe a ∼15% softening of

mode frequency and a sharp increase in the linewidth with Tin (a).

The accumulation of low-energy weight is emphasized in (b), where

the inset shows a detailed Tdependence. Middle: the same analysis

is repeated for q= (π/2, π/2) in (c) and (d). Similar trends persist

with much reduced extent. Bottom: Theoretically extracted ‘soften-

ing’ ∆¯ω(π,π )(T) = ¯ω(π,π )(0) −¯ω(π,π)(T)and thermal broadening

∆Γ(π,π)(T)=Γ(π,π)(T)−Γ(π,π )(0) in (e) is compared to corre-

sponding quantities for q=qCE from experiments (f). Qualitative

trends are similar.

the zone center. These thermal trends are qualitatively similar

to the Heisenberg model [40, 41] and also to a pure double ex-

change model at the same density. We will discuss the relative

insensitivity of magnons to phonon physics later.

Fig.2 examines phonon lineshapes in detail at two mo-

menta, q= (π, π)and (π/2, π/2). Panel 2(a) focuses on

the ‘high energy’ part of the spectrum, ω∼Ω, at q= (π, π),

while 2(b) uses a logarithmic frequency and amplitude scale

to show the full q= (π, π)data. In 2(a), we see a strik-

ing enhancement of broadening in the high-energy part of the

spectrum on heating from the T= 0.01TF M to T∼TFM .

A reduction of the mean frequency is also observed. To clar-

ify the behaviour at ωΩpanel 2(b) depicts the full spec-

trum in a log-log plot. This reveals the low frequency spec-

tral weight arising from polaron tunneling. The inset shows

the Tdependence of the low frequency weight wlow(q, T ) =

RΩlow

0dω|X(q, ω)|2, normalised by the the full weight. We

use Ωlow = 0.4Ω. The weight wlow is non monotonic in T

with a reasonable maximum value ∼10%. 2(c)-(d) repeats

the same analysis for (π/2, π/2), which is considerably sep-

arated from the ‘CO’ wavevector. The trends are similar to

q= (π, π)but the peak wlow is much smaller, ∼3%.

Figs.2(e) and 2(f) show a comparison of the softening

and damping inferred from our (π, π)phonon lineshape with

that extracted from experimental data at the ‘CE ordering’

wavevector q=qCE . The experimental data is on acoustic

phonons, but its has been argued that the behaviour should be

similar to that of the JT phonons. Both theory and experimen-

tal results are normalised by the respective low Tbandwidth,

and temperatures are scaled by the respective TFM ,∼0.1t

in the model and 305K in experiments. The Tdependence

in panels (e) and (f) share similarities but actual numbers dif-

fer [42] by ∼a factor of 2. We discuss the comparison in

more detail later. To the extent we know, experiments have

not probed the low frequency part of the spectrum.

Figs.3 is focused on magnons, at q= (π, 0) and (π , π).

The lineshapes show that the sharp dispersive feature for

T<

∼0.5TF M and then rapid broadening as T→TFM . The

(π, 0) mode softens much less than the mode at (π, π ). In 3(c)

and 3(d), the detailed temperature dependence of softening,

∆ωq(T), and damping, ∆Γq(T), are shown. The normaliz-

ing energy scale is the low Tmagnon bandwidth (∼0.8tin

our case). We have not been able to ﬁnd systematic temper-

ature dependence data on magnon lineshapes in the mangan-

ites, although a body of results [12, 13, 15, 18] point out low

temperature magnon ‘anomalies’ in these materials.

Discussion: Having presented the results, in what follows

we provide an analysis of the features in Figs.1-3, and point

out where our results match with, differ from, and go beyond

measurements in the manganites.

We broadly observe four phonon regimes- (i) harmonic,

∼0−0.1TF M , (ii) anharmonic, ∼0.1TF M −0.3TF M ,

(iii) polaronic, ∼0.3TF M −1.5TF M , and (iv) large oscilla-

tions, >

∼1.5TF M , in terms of real-time dynamics (see Supple-

ment for xi(t)data). The harmonic to anharmonic crossover

is reﬂected in the Tdependence of Γ(q)due to mode cou-

FIG. 3. Top panel: Magnon lineshapes for q= (π, 0) (a) and

q= (π, π)(b) at the same temperatures. The low Tspectrum is con-

cordant with a nearest neighbour Heisenberg model for Jeff ∼0.1t.

Near TF M , asymmetric and broad lineshapes are seen. Bottom: The-

oretical estimates of ‘mode softening’ and broadening for magnons

(similar to phonons) for (π, 0) (c) and (π, π )(d). We see enhanced

softening in the latter.

pling. In the polaronic regime the distortions increase and

we observe ‘burst like’ events - anticorrelated between near-

est neighbour sites. For T>

∼1.5TF M the oscillations are even

larger but the spatial correlations begin to weaken. Regimes

(iii) and (iv) contain appreciable effect of magnetic disorder,

which results in a peak in Γph(q, T )for T∼TF M for the

present case - in contrast to a pure Holstein model.

The observed magnon spectra are similar to those of

a nearest-neighbour Heisenberg model. The ‘square root’

renormalization of stiffness, and phonon couplings, are both

seemingly irrelevant. There are broadly three magnon

regimes- (i) free, ∼0−0.5TF M , (ii) interacting, ∼0.5TF M −

TF M , and (iii) diffusive, >

∼TF M . The (1 + hSi.Sji)1/2fac-

tor varies by ∼15% from 0−TF M , while hγ†

iγjiis almost

Tindependent. Due to this an essentially phonon insensitive

Heisenberg description arises. The overall picture holds even

in presence of a small JAF , whose main effect is bandwidth

reduction at low T.

Our parameter choice was meant to mimic the physics in

La1−xSrxMnO3and La1−xCaxMnO3for x∼0.2−0.3. Un-

like the real material, the model we use is two dimensional, in-

volves Holstein rather than cooperative JT phonons, and does

not include AF couplings. As Figs.2(e)-(f) demonstrate the

phonon softening at the short range ordering wavevector fol-

lows similar trends in theory and experiment roughly upto

TF M , beyond which they deviate. The fractional softening

near TF M however differs by more than a factor of two. Sim-

ilarly, the thermal component of phonon broadening, Γ, has

very similar Tdependence in Figs.2(e) and 2(f), but the the-

ory value is now smaller by about a factor of 2. What has

not been probed experimentally is the signature of low energy

weight at q∼qCE at T<

∼Tc- in manganites which show

a thermally induced metal-insulator transition. This weight at

∼10% of the bare phonon scale is the key dynamical sig-

nature of short range correlated large amplitude distortions.

Unless ionic disorder pins polarons, this low energy feature

should be visible.

Conclusions: We have presented the ﬁrst results on the

coupled anharmonic dynamics of phonons and spins that

emerges with increasing temperature in the Holstein-double

exchange model. Past the low temperature harmonic win-

dow, we observe the expected nonlinearities attributable to

‘phonon-phonon’ and ‘magnon-magnon’ interactions. Be-

yond this, however, we see a striking ‘two peak’ structure in

the momentum resolved phonon spectrum, involving: (i) low

energy weight at ωΩ, for q∼(π, π), from slow tunnel-

ing of thermally generated spatially correlated polarons, and

(ii) enhanced damping of the high energy, ω∼Ω, feature

due to scattering from magnetic ﬂuctuations. The magnetic

dynamics itself remains mostly insensitive to the phonon ef-

fects and can be described by a Heisenberg model. Our ‘high

energy’ phonon trends compare well with inelastic neutron

scattering in the manganites, although numerical values dif-

fer, and the low energy features should be visible in the low

disorder samples at T>

∼0.5TF M .

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try as JT displacements.

Supplemental Material for “Strongly anharmonic collective modes in a coupled

electron-phonon-spin problem”

Sauri Bhattacharyya1, Sankha Subhra Bakshi1, Saurabh Pradhan2and Pinaki Majumdar1

1Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

2Department of Physics and Astronomy, Uppsala University, 751 05 Uppsala, Sweden

(Dated: December 18, 2019)

We ﬁrst discuss the derivation of the respective Langevin

equations for phonons and spins. Next, the consistency be-

tween the results obtained using the present scheme and exact

diagonalization based Monte Carlo is shown. Then, to hint

at the various thermal regimes in phonons, the distribution of

displacement and density ﬁelds are featured. A comparison

of spectra in the present model with that obtained from a pure

Holstein and a pure double exchange model are depicted after-

wards. This elucidates the role of magnon-phonon coupling.

Finally, real-time trajectories of phonon and spin ﬁelds are ex-

hibited to create an insight into the detailed results.

DERIVING THE LANGEVIN EQUATION

We reproduce the evolution equation below for reference

Md2xi

dt =−Dph

dxi

dt −Kxi−∂hHel i

∂xi

+ξi(t)

d~

Si

dt =−~

Si×(∂hHeli

∂~

Si

+~

hi) + Ds~

Si×(~

Si×∂hHeli

∂~

Si

)

Hel =X

ij

(tij −µδij )γ†

iγj−gX

i

nixi

tij /t =q(1 + ~

Si.~

Sj)/2

In what follows we comment on (i) the origin of the phonon

and spin equations, (ii) the ‘noise’, ξi(t)and ~

hi(t), (iii) the

choice of the damping parameters Dph and Ds, and (iv) the

robustness of the ‘equal time’ properties (at equilibrium) to

variation in the damping parameters.

Phonon equation

The derivation of the phonon equation of motion has been

outlined before [2] in case of the pure Holstein model, which

is in turn motivated by earlier work [3]. To quickly recapit-

ulate, (i) A real time path integral is set up in the Keldysh

formalism in terms of coherent state ﬁelds corresponding to

xiand cioperators. (ii) Integrating out the quadratic elec-

trons we obtain an effective action for the phonon ﬁelds on the

Keldysh contour. (iii) We transform to ‘classical’ and ‘quan-

tum’ variables, the average and difference of phonon ﬁelds on

the forward and backward branches of the Keldysh contour,

and expand the action perturbatively in the ‘quantum’ ﬁelds,

assuming that the phonon timescale, Ω−1, is much larger than

the electronic timescale, t−1(tis the hopping). (iv) Inte-

grating out the ‘quantum’ ﬁelds at Gaussian level leads to a

stochastic equation of motion for the ‘classical’ ﬁelds.

The main difference here with respect to the pure Holstein

model is the (time dependent) modulation of the hopping am-

plitude, tij , due to ﬂuctuation in the orientation of the spins.

The implicit dependence of the ‘force’ term ∂hHel i

∂xi=gni(t)

on the spin conﬁguration is relatively weak.

Spin equation

The spin equation is written based on arguments presented

in Ref.4. The signiﬁcance of the different terms is as follows:

(i) The torque term describes semiclassical dynamics of the

ﬁxed magnitude moments in an effective magnetic ﬁeld. In

our case, this ﬁeld is computed as a gradient of the instan-

taneous electronic energy with respect to the local spin ~

Si.

(ii) The dissipation is proportional to the angular momentum

of the moments, in analogy with the motion of ions. When the

the dynamics is restricted to the sufrace of a unit sphere, our

LLGB equation emerges.

There are alternate derivations starting from a model of lo-

cal moments coupled to phonons [5] and/or conduction elec-

trons [6, 7] and then employing the Keldysh formalism. One

makes similar assumptions there as in the electron-phonon

case, namely that the moments are ‘slow’ compared to other

microscopic degrees of freedom and that the electronic den-

sity of states is gapless and conforms to the Ohmic dissipation

model.

The spin equation differs from that for the double exchange

model only in the implicit dependence of bond kinetic energy

average hγ†

iγjion the phonon ﬁeld. It differs from the Heisen-

berg model, additionally, via the factor 1/q(1 + ~

Si.~

Sj)/2,

which modiﬁes the local ﬁeld acting on the site i.

Specifying the noise

The ‘noise’ which drives the phonon ﬁeld has a correlator

hξi(t)ξj(t0)i=g2Πij (t, t0)

where Πis the Keldysh component of the electronic polariz-

ability, in the background xi(t). On the assumption that we

are at ‘high temperature’, ΩkBT, and that the electronic

density of states is gapless, we obtain the Langevin form used

arXiv:1912.07810v1 [cond-mat.str-el] 17 Dec 2019

FIG. 1. (a)-(b): Comparison of Langevin results with Monte Carlo.

(a) The ferromagnetic feature Ss(0,0) in the magnetic structure fac-

tor and (b) the Fourier transform of the density-density correlation

Snat q= (π, π). There is a very good agreement for the chosen

Dsand Dph values. (c) The dependence of Ss(0,0) on Dsfor three

values Ds= 0.025,0.05,0.1, and (d) of Sn(π, π)on Dph for three

values Dph = 0.1t, 0.5t, 1.0t. In both cases, we see an insensitivity

to dissipation rates.

in the paper. Although we use the high temperature approxi-

mation throughout to simplify the noise correlator, the essen-

tial dynamical features of the phonons are retained even for

kBT<

∼Ωthrough the ‘electronic force’ term.

The spin noise ~

hiis chosen to be Markovian, obeying

the ﬂuctuation-dissipation theorem, for simplicity. In an ac-

tual derivation from a Keldysh action, the noise correlator is

proportional to the Keldysh spin polarizability [ΠS

ij (t, t0)]µν ,

that has a natural frequency scale ∼t. For low frequencies

(ωt) and high temperatures (ωkBT), the present

Markovian form may be used. Once again, we apply this

approximation at all temperatures, assuming that the key dy-

namical features are captured through the ‘electronic torque’

term.

The damping parameters

Microscopically, in the Holstein double exchange model,

the phonons couple to the local density and the moments to the

local conduction electron spin. Hence, the dissipation chan-

nels arise from particle-hole excitations in the charge and spin

sectors. These involve the charge and spin susceptibilities of

the parent electronic system. At low temperature one may ap-

proximate these by their ‘bare’ versions. The dissipation rates

Dsand Dph arise from the imaginary parts of the ‘linear’,

low-frequency part of these functions. Dph carries a prefactor

of g2, while Dsis proportional to t2. An actual calculation

reveals that Ds∼0.1tand Ds∼0.1at low temperatures.

CONSISTENCY BETWEEN LANGEVIN DYNAMICS AND

MONTE CARLO

The ‘static’ physics at equilibrium for the Holstein-double

exchange model can be extracted from an exact diagonali-

sation based Monte Carlo (ED-MC), when we assume the

phonons and spins to be classical. That assumes M→ ∞,

i.e,Ω→0, and the ‘size’ of the spin S→ ∞ (to be ab-

sorbed in the Hund’s coupling). In that case the probability

for a phonon-spin conﬁguration is:

P{x, S} ∝ T r e−β(Hel+Hstiff )

Hel =X

ij

(tij −µδij )γ†

iγj−gX

i

nixi

tij /t =q(1 + ~

Si.~

Sj)/2

Hstiff =1

2KX

i

x2

i

The trace over the electrons can be performed numerically

by diagonalising Hel. This forms the heart of ED based MC

and yields results that do not have the effect of the parameters

M, Dph , Dsthat enter the Langevin equation. It is impor-

tant to verify that the Langevin approach, with its additional

dissipative parameters, yields equilibrium results that are con-

sistent with MC.

In Fig.1, we show the behaviour of the spin structure factor

Ss(q)and the density structure factor Sn(q)at q= (0,0) and

(π, π)respectively.

Ss(q) = 1

N2X

ij

~

Si.~

Sjeiq.(~ri−~rj)

Sn(q) = 1

N2X

ij

hniihnjieiq.(~ri−~rj)

FIG. 2. (a)-(b): The distribution of distortions (P(x)) and density

(P(n)) for four temperatures- T/TF M = 0.01,0.5,1.0,1.5. The

former broadens gradually on heating up. The latter shows a more

interesting evolution from a sharp unimodal proﬁle (T << TF M )

through a positively skewed, long tailed distribution (near TFM ) to

ﬁnally a plateau-like structure beyond TFM . The quick onset of an-

harmonicity, presence of large distortions near TFM , and the merg-

ing of large oscillations with polarons at high T, are all hinted at.

We test out different Dsand Dph values. In the top panel, the

results on these obtained using the two methods (Langevin

dynamics and Monte Carlo annealing) are shown. The match

is remarkable. For both the spin and density cases, the bottom

panel shows an insensitivity of the Langevin inferred structure

factor to the damping rates Dph and Ds.

DISTRIBUTION OF DISPLACEMENT AND DENSITY

In Fig.2, we show the distribution of displacements

(P(x, T )) and electron density P(n, T )in four temperature

regimes. The low temperature state is uniform and has a sharp,

unimodal proﬁle for both of them. The former shows grad-

ual broadening thereafter, not underscoring the thermal po-

laron formation. However, the density distribution broadens

FIG. 3. (a)-(b): Comparison of ‘high energy’ part of phonon line-

shapes at q= (π, π)for different temperature regimes between the

present model (H-DE) and a pure Holstein model (H) at comparable

density. (c)-(d): Same for ‘low energy’ part of the spectra. Similar

qualitative features are seen in both cases, but the effects are stronger

in the former. (e): The extracted softening ∆¯ω(π,π)(T)compared

between the H-DE and Holstein models. The results are broadly

similar. (f): The linewidth ∆Γ(π,π)(T)compared between the H-

DE and Holstein models. In the Holstein model the width saturates

at some temperature, without the peak that one observes in the H-DE

model. The larger H-DE damping and the peak feature in Tdepen-

dence arise due to the magnetic degrees of freedom.

quickly (T/TF M = 0.5), signifying the onset of highly an-

harmonic behaviour. Near TF M , a left-skewed, long-tailed

shape is seen, which depicts thermally induced polarons. Fi-

nally, beyond TFM , a broad, plateau-like structure emerges,

hinting at the merging of distortions of different sizes.

PHONON SPECTRA IN THE HOLSTEIN-DOUBLE

EXCHANGE AND HOLSTEIN MODELS

In Fig.3 we ﬁrst compare lineshapes in four temperature

regimes between the Holstein-double exchange (H-DE) and

the Holstein (H) model at comparable density. (a) and (b)

feature the ‘high energy’ spectra. The basic trends are simi-

lar. However, the extent of softening and broadening close to

TF M in the former is much more accentuated compared to the

Holstein model for the same temperature. The basic reason is

the presence of increasing spin disorder in the former which

effectively reduces the electronic bandwidth and enhances the

FIG. 4. (a)-(b): Comparison of magnon lineshapes at q= (π, 0) for

different temperature regimes between the present model and a pure

double exchange model at comparable density. (c)-(d): Same for

q= (π, π). One observes a qualitative similarity in all regimes, sig-

nifying the subdued effect of magnon-phonon coupling on magnons.

(e)-(f): Magnon spectra at the same wavevectors for a DE model

with small (JAF = 0.2Jeff ) AF coupling. We observe a bandwidth

shortening at low Tand smaller dampings compared to the former.

FIG. 5. Top panel: Real-time trajectories of xi(t)on the central site at four temperatures- T /TF M = 0.01,0.5,1.0,1.5. One observes a

gradual increase in the amplitude of oscilations on heating up. The ‘ﬂip’ moves, involving large (∼O(g/K)) changes in xi, are merged with

large oscillations near TF M . They ﬁrst start showing up near ∼0.7TF M . Bottom: Trajectories of the magnetic overlap function Oi(t)on the

same site. Here, a proliferation of ‘sign changes’ are seen with a large period close to criticality.

effective coupling to phonons. However, we comment that

the quantitative increase in damping is underestimated by just

considering this effect, which hints at a crucial role of spatial

correlations. The middle panel ((c) and (d)) exhibits the ther-

mal evolution of low-frequency weight, which is again sharper

in the H-DE case, due to well-formed polarons and their as-

sociated tunneling. In the bottom panel ((e) and (f)), the soft-

enings and linewidths extracted from the corresponding line-

shapes are shown ((c) and (d)) . We see a more detailed ther-

mal comparison here, which corroborates the above ﬁndings.

The main result is a gradual and substantial rise in phonon

damping near TF M in the present model, which is absent in

the pure Holstein case, where damping saturates by 0.5TF M .

Moreover, the rise in softening is sharper in the pure Holstein

case.

MAGNON SPECTRA IN THE HOLSTEIN-DOUBLE

EXCHANGE AND DOUBLE EXCHANGE MODELS

Fig.4 highlights the comparison of magnon spectrum for

the H-DE model with that of the DE model. The top and mid-

dle panels feature lineshapes at q= (π, 0) and q= (π, π )

respectively. The qualitative features remain the same, which

conﬁrms that the effects of magnon-phonon coupling on the

spin dynamics are weak. Moreover, the double-exchange

model spectra are qualitatively similar to those of a nearest-

neighbour Heisenberg model with Jeff ∼0.1t. This means

that the magnon dynamics is fairly conventional. In the bot-

tom panel ((e) and (f)), we show lineshapes at the same

wavevectors for a DE model with a small (JAF = 0.2Jef f )

AF coupling, which is realistic from the materials point of

view. The main result is a smaller low Tbandwidth and ﬁnite

temperature linewidths limited by this scale. The TF M in this

case is ∼0.08t.

REAL TIME TRAJECTORIES

In Fig.5, we depict the representative real time trajectories

from which we obtain the phonon and magnon power spec-

trum. The top row shows xi(t)while the lower row shows the

magnetic overlap function:

Oi(t) = 1

4X

δ

~

Si(t).~

Si+δ(t)

where δare the four neighbours of site i. We show results in

four temperature regimes- harmonic, anharmonic, polaronic

and high for phonons and low, intermediate, critical and high

for magnons. The chosen site is located at the middle of the

system. The top panel also follows its right neighbour.

In the harmonic regime, the phonon dynamics is saturated

by small oscillations about a homogeneous state. Anharmonic

behaviour sets in quickly (T∼0.2TF M ), where oscillations

of larger amplitude (>10% of mean value) prevail and ef-

fect of mode coupling shows up in Γph(q). Next comes the

polaronic regime (T/TF M = 0.7), where short-range corre-

lated ‘burst’ like events coexist with anharmonic oscillations.

These lead to the remarkable low-frequency weight transfer.

Finally, well beyond TFM , distortions of several scales get

merged together and short-range correlations weaken.

The magnetic overlap Oi(t)is perfect at low T, but shows

a gradual proliferation of ‘sign changing’ moves on heating,

which eventually kill the ferromagnetic order.

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