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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 $T_{FM}$. The small amplitude harmonic phonons at low temperature cross over to large amplitude dynamics around $0.5 T_{FM}$ 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.
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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, La1xCaxMnO3, with x0.3, 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 La1xSrxMnO3with
x0.20.3, 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 . 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 qQsoft-
ens significantly. (b) The damping Γqis strongest for qQ
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 ωfor qQ.
(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 ΓXKΓin the Brillouin zone. We see a thermally
induced softening and broadening for modes with q(π, π), alongwith an unexpected ω0feature 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 T0.5TF M
and then shows a dramatic increase. Unlike phonons, for
whom the principal weight remains at 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
cj σ JHX
i
~
Si.~σiµX
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 KxihHel i
∂xi
+ξi(t)
d~
Si
dt =~
Si×(hHeli
~
Si
+~
hi) + Ds~
Si×(~
Si×hHeli
~
Si
)
Hel =X
ij
(tij µδij )γ
iγjgX
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(tt0).
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/, 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
D1
s4τs. 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= 1.6×104τph. 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.riRdteiωtxi(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, ω! 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.riRdteiωt ~
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, 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 TTFM .
A reduction of the mean frequency is also observed. To clar-
ify the behaviour at ωpanel 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 ) =
Rlow
0|X(q, ω)|2, normalised by the the full weight. We
use low = 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 TTFM . 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,
00.1TF 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 TTF 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, 00.5TF M , (ii) interacting, 0.5TF M
TF M , and (iii) diffusive, >
TF M . The (1 + hSi.Sji)1/2fac-
tor varies by 15% from 0TF 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
La1xSrxMnO3and La1xCaxMnO3for x0.20.3. 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 qqCE 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 ω, for q(π, π), from slow tunnel-
ing of thermally generated spatially correlated polarons, and
(ii) enhanced damping of the high energy, 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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finite size calculation it is difficult to distinguish a merely long
range ‘correlated’ state from a genuinely ordered one.
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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 KxihHel i
∂xi
+ξi(t)
d~
Si
dt =~
Si×(hHeli
~
Si
+~
hi) + Ds~
Si×(~
Si×hHeli
~
Si
)
Hel =X
ij
(tij µδij )γ
iγjgX
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, 1, is much larger than
the electronic timescale, t1(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’, kBT, 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<
through 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 S
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 Ds0.1tand Ds0.1at 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,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 eβ(Hel+Hstiff )
Hel =X
ij
(tij µδij )γ
iγjgX
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 (T0.2TF 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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