The Separation
Author: K.D. Edwards
Date: 2012-11-13 02:10:03+00:00
Keywords: Fantasy; urban-fantasy
Description: A short story about baby Rune and baby Brand. Available to download in the extra content page on the author's website.

Phase separation in the Edwards model S. Ejima,1 S. Sykora,2 K. W. Becker,3 and H. Fehske1

1 Institut f¨
ur Physik, Ernst-Moritz-Arndt-Universit¨
at Greifswald, 17487 Greifswald, Germany 2 Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany 3 Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany (Dated: November 13, 2012)

The nature of charge transport within a correlated background medium can be described by spinless fermions coupled to bosons in the model introduced by Edwards. Combining numerical density matrix renormalization group and analytical projector-based renormalization methods we explore the ground-state phase diagram of the Edwards model in one dimension. Below a critical boson frequency any long-range order disappears and the system becomes metallic. If the charge carriers are coupled to slow quantum bosons the Tomonaga-Luttinger liquid is attractive and finally makes room for a phase separated state, just as in the t-J model. The phase boundary separating the repulsive from the attractive Tomonaga-Luttinger liquid is determined from long-wavelength charge correlations, whereas fermion segregation is indicated by a vanishing inverse compressibility.

On approaching phase separation the photoemission spectra develop strong anomalies.

I.

PROBLEM
band filling n. Thereby we demonstrate that this model indeed captures a number of very interesting phenomena, The Edwards fermion-boson model1,2 constitutes a including e.g. electronic phase separation (PS). To obtain paradigmatic model for the theoretical description of reliable information about the ground-state and spectral quantum transport in solids. Charge transport normally properties of the model in the thermodynamic limit, we takes place in the presence of some background medium. 3
employ numerical pseudosite density-matrix renormaliza-Examples for such a ”background” could be a spin-, tion group (DMRG) and dynamical DMRG (DDMRG) charge- or orbital-ordered lattice, 4,5 but also a sequence techniques (supplemented by a careful finite-size scaling of chemical side groups along the transport path, a de-procedure, for details see Refs. 9, 21–24), in combination formable medium, or even a heat bath might be possible.

with the analytical projective renormalization method In all these cases, the transfer of the charge carriers will (PRM). 25,26
be strongly influenced by fluctuations, which may exist So far the 1D Edwards model has been solved ex-in the background medium. The other way around, the actly by numerical approaches for two cases.22,24,27 The properties of the background are quite often determined first, in no way trivial case concerns just a single particle by the motion of the particle itself.

on an infinite lattice, where—depending on the model Correlations inherent in such a complex interactive sys-parameters—transport appears to be quasi-free, diffu-tem are mimicked by a boson-affected hopping of spinless sive, or boson-assisted. 6,28 When strong correlations de-fermionic particles in the Edwards model. It reads: velop at small λ and large ω0, the background becomes

“stiff” and the spinless particle’s motion resembles those X

X

X
of a hole in an antiferromagnetic spin background, 29,30

H = −tb
f †f (b† + b ) − λ

(b† + b ) + ω
b†b .

j
i
i
j
i
i

0
i i
as known from the t-J model. Interestingly, the Ed-hi,ji
i
i
wards model allows for so-called Trugman paths31 in an (1)

1D setting (Trugman paths usually unwind the string of Every time a fermion f (†) hops, it creates or absorbs a bo-i
misaligned spins a mobile hole leaves behind in two di-son b(†) of energy ω
mensions). The second case is half-filling n = 1/2. Here i

0 at the sites it leaves or enters. Such an excitation or de-excitation corresponds to a local “disa metal-insulator transition has been proven to exist: For
arXiv:1209.1810v2 [cond-mat.str-el] 10 Nov 2012
tortion” of the background. Because of quantum fluctu-small λ and large ω0 the repulsive Tomonaga-Luttinger ations the distortions are able to relax (∝ λ). The physi-liquid (TLL) gives way to a CDW. 7,8 The CDW consti-cally most interesting regimes in this setting are those of tutes a few-boson state that typifies a Mott-Hubbard in-vanishing fermion density and of a half-filled band, rep-sulator rather than a Peierls state (normally established resenting doped Mott insulators, polaronic organics and by the softening of a boson mode).

charge-density-wave (CDW) systems6–9 with possible rel-evance to high-Tc superconductors,10,11 colossal mag-netoresistive manganites, 12–14 carbon nanotubes, 15,16

II.

THEORETICAL RESULTS
graphene17,18 and CDW transition metal complexes, 19,20
respectively. However, the Edwards model also reveals A.

Ground-state properties
fascinating properties over the whole density range.

On these grounds, the main goal of the present work is The situation at finite density n (n 6= 1/2) is much less to pinpoint the ground-state phase diagram of the one-understood. By analogy with the t-J model one might dimensional (1D) Edwards model as a function of the expect that the system is metallic for 0 < n < 1/2,

2
at least if the background is not too stiff. If so, the 1.5

0.008
next question will be, whether the Edwards model might

–1

)/2]

ρ
λ = 0.2
κ

K

0.004
support the pairing of electrons in a certain parameter 2

/L

1
regime (λ, ω
π

0

0.6 0.8

1
ω = 1

0). Apparently a second electron, following (2
ω =

0

0.6
ω
c

0

1

1.5

0
the path of a first one, can take advantage of the back-ω

–1 0.01
ω = 0.8

0

LS

0
κ
ground excitations (bosons) left behind by the first elec-ω = 0.8

0

)[=
tron. This acts like an effective attractive interaction.

ω = 0.7

(L
ω = 1

0
ρ

0

If this attraction completely dominates the kinetic en-K 1
ω = 0.6
ω =

0
ergy the system might even phase-separate into particle-1.4

0

0
enriched and particle-depleted regions.32 Since the Ed-0

0.02

0.04

0

0.02

0.04

1 / L

1 / L
wards model is not particle-hole symmetric, it is a moot point whether a corresponding hole pairing mechanism is Figure 1. Finite-size scaling of the TLL parameter Kρ (left at play also for 1/2 < n < 1.

panel) and inverse compressibility κ−1 (right panel) for the To address these problems, we analyze the charge cor-1D Edwards model with λ = 0.2 at n = 1/8. Insets show relations existing in the ground-state of the 1D Edwards the variation with ω0 of the L → ∞ extrapolated values.

model. Firstly, we calculate the charge structure factor Results are obtained by DMRG for a lattice with L sites and OBC. In all DMRG runs we take into account up to nb =

1 X

E

4 bosonic pseudosites and determine n S
b by the requirement
c(q) =
eiq(j−l) Df †f − n
f †f − n

, (2)

L
j
j
l
l
that the local boson density of the last pseudosite is less than j,l

10−6 for all j. Using selectively nb = 5, we controlled that convergence is truly achieved. Furthermore we keep up to for a system with N fermions, L sites, and open boundary m = 1200 density-matrix eigenstates in the renormalization conditions (OBC). So the particle density is n = N/L
steps to ensure that the discarded weight is smaller than 10−8.

and the momenta q = 2πs/L with integers 0 < s < L.

The TLL charge exponent is proportional to the slope of Sc(q) in the long-wavelength limit q → 0+ (cf. Refs. 33
and 34):

Proceeding in the same manner for different values of n and λ respectively ω0, we can map out the phase diagram S
of the 1D Edwards model. The outcome is displayed in K
c(q)

ρ = π lim

.

(3)

Fig. 2. Let us first consider the case of a not too small q→0
q
boson relaxation (∝ λ), which ensures that the system is Then, K
metallic for large ω
ρ

> 1 (Kρ < 1) characterizes an attrac-0 in the whole density regime. Then, tive (repulsive) TLL for our spinless fermion transport as the upper panel of Fig. 2 shows, depending on n we model (1), and K
find up to three different regimes: For small and large ρ = 1/2 will define a metal-insulator transition point at n = 1/2.35 That the finite-size scaling particle densities a repulsive TLL, an attractive TLL and of S
a PS state appear in sequence as the energy of the bosons c(q) and Kρ works well has been demonstrated for the half-filled band case. 8 Secondly, in order to comprise is lowered. In contrast, around half-filling only the repul-potential PS, we determine the finite-size equivalent of sive TLL exists. At this point we would like to remind the charge compressibility, 36
the readers that at half-filling, for smaller values of λ, (λ < λc), a CDW is formed for ω0 > ω0,c.8 Our unbi-L E

−1
ased DMRG calculations give no evidence for any other κ =

0(N + 2) + E0(N − 2) − 2E0(N )

, (4)

N 2

4
phases. Note that the behavior in the low-density regime is consistent with what is found for the 1D t-J model36,37, with E0(N ) being the ground-state energy for N elec-where the holes correspond to the spinless fermions in the trons on L sites. An infinite compressibility signals the Edwards model. As mentioned before, the phase diagram formation of a PS state.

is not symmetric with respect to n = 1/2. This is because Fig. 1 demonstrates that all this works in practice. Ex-the hopping of a hole (missing electron) to a neighboring emplarily choosing n = 1/8 and λ = 0.2, we show how site creates in the Edwards model a boson at the arrival the TLL exponent Kρ and the inverse compressibility site and not on the departure site as in the motion of κ−1 scale with increasing system size at various ω0. The a single particle. Since this boson can be destroyed im-transition point between a repulsive TLL—obviously re-mediately when the hole makes a further hop, there is alized at ω0 > ω0,c1 when excitations of the background no string effect and a few holes propagate more easily are energetically rather costly—and an attractive TLL at than a few electrons. Hence the background at fixed ω0
smaller ω0, can be read off from the inset [depicting the and λ appears to be less stiff for n . 1 than for n & 0
extrapolations Kρ(L → ∞)] to be ω0,c1(λ = 0.2) ≃ 1.118.

and, as a result, the boundary between the repulsive and Reducing ω0 further, in the attractive TLL phase, a dra-attractive TLL is shifted to larger values of ω0,c1. Con-matic increase in Kρ is observed at ω0,c2 ≃ 0.6. Our in-trariwise, since carriers will be less mobile in a state with verse compressibility data indicate that at this point the charge separation, the boundary between the attractive attraction among the particles becomes so strong that TLL and the PS state ω0,c2 is shifted to smaller values if the system shows PS, i.e., κ−1 = 0 for ω0 < ω0,c2.

one compares the corresponding results at high and low

3
carrier density.

1.5

We next consider a fixed particle density n = 1/8 and λ=0.2
track the phase boundaries in the λ-ω0 plane; see the lower panel of Fig. 2. The repulsive TLL established for large ω0 is strongly renormalized if the ability of the back-1
ground to relax is low. For example, we find Kρ = 0.527

TLL (attractive)

for ω

0

0 = 2 at λ = 0.01.

Lowering ω0 for such a stiff
ω

TLL (attractive)

background, the transition to the PS state happens al-TLL (repulsive)

most instantaneously with a narrow intervening attrac-0.5
tive TLL state; at λ = 0.01 we have ω0,c1 ≃ 0.747 and ω0,c2 ≃ 0.672. If we fix, on the other hand, ω0 = 2 and increase λ, we observe a strong enhancement of K

PS
ρ in

PS
the interval 0 < λ < 0.3 (Kρ = 0.767, 0.879, 0.932 for λ = 0.1, 0.2, 0.3, respectively), followed by a very gradual 00

0.2

0.4

0.6

0.8

1
increase until the transition to the attractive TLL takes n
place at about λc1 ≃ 1. Obviously, the region where the attractive TLL (PS state) constitutes the ground state 2
n=1/8
expands (shrinks) as λ gets larger, which can be traced back to the subtle competition between kinetic energy TLL (repulsive)

and charge segregation effects.

1.5

To gain deeper insight into the different mechanisms at play, we analyze the variation of the local fermion 0ω

[nf = hf †f i] and boson [nb = hb†b i] densities in real 1

TLL (attractive)

j
j
j
j
j j
space. In Fig. 3 we show both density profiles for n = 1/8
and characteristic values of λ and ω0, implementing repulsive TLL, attractive TLL, and PS states.

In the

0.5
metallic regime, the OBC lead to Friedel oscillations in PS
the particle density. These oscillations are especially pro-nounced for the repulsive TLL [panel (a)], and they will 00

0.2

0.4

0.6

0.8

1
be even stronger for ω0 = 2 (not shown), where the num-λ
ber and fluctuations of the bosons is reduced. Note that the Friedel oscillations caused by the OBC will be alge-Figure 2. Ground-state phase diagram of the 1D Edwards braically reduced, if we move for very large systems to-model, showing repulsive TLL (Kρ < 1), attractive TLL
wards the central part of the chain. If we enter the attrac-

(Kρ > 1), and PS (κ−1 = 0) regions. The phase boundaries tive TLL regime by increasing λ at fixed ω
were obtained by DMRG (filled symbols) in the course of a 0 the Friedel
oscillation will be smeared out [panel (b)]. Thereby, the careful finite-size scaling analysis and for the infinite system number of oscillations stays constant, which means that a directly by PRM (open symbols). The upper panel displays the phase diagram as a function of n, varying ω
pairing of electrons does not occur. This is an important 0 at fixed
λ = 0.2; the lower panel gives the phase diagram in the λ-ω
difference from the spinful 1D t-J model, where a recent 0
plane for fixed density n = 1/8.

DMRG study reveals that the number of Friedel oscillations is halved (by increasing J), corresponding to half the number of particles, which indicates pairing. 37 Next simply because the dimension of the Hilbert space in-lowering ω0 to a point where PS sets in, the particle den-creases dramatically. To access the low-ω0 regime and sity oscillations vanish, see panel (c). There, just above reconfirm the DMRG phase boundaries, obtained for the the PS boundary, no evidence is found for the clustering 1D Edwards model at larger values of ω0, we employ of multiple particles in several groups. This agrees with the analytical PRM approach.25 The basic idea of the the findings for the 1D t-J model. 37 Once we are deep PRM is to construct—performing a sequence of discrete inside the PS region, a density distribution results, where unitary transformations and eliminating all transitions a single island of particles at the central part of the sys-with energy larger than a given cutoff energy—an effec-tem appears, leaving a sizable number of almost empty tive non-interacting Hamiltonian ˜

H with renormalized
sites at both ends of the chain [see panel (d)]. Looking parameters (in the limit of vanishing cutoff energy). For at the bosonic degrees of freedom, we see that the strong the metallic state of the Edwards model, in momentum attraction among these particles is mediated by a boson space, it takes the form
cloud covering the electron-rich region. As a result the kinetic fluctuations will be strongly quenched.

˜

X

X

H =

˜
εkf †f +

˜
ω
b + ˜

E ,

(5)

k
q b†
q q

It is all but impossible to comprise by DMRG the large k
k
q
number of bosons and the strong bosonic fluctuations, which appear at still smaller ω0 in the Edwards model, where the renormalization equations for ˜
εk and ˜
ωq have

4

B.

Spectral properties
ω

(a)

b
j 0.2

0=1, λ=0.01
n,

Let us finally discuss the single-particle spectrum of f
jn
the 1D Edwards model. The single particle excitations associated with the injection (+) or emission (−) of an 0
electron with wave vector k,

X

A±(k, ω) =

|hψ± |f ±|ψ
m
k

0i|2 δ(ω ∓ ω±) ,

(6)

ω

(b)

m
b
j 0.2

0=1, λ=0.5
n, f j
can be computed by DDMRG,23,24 where f + = f †, f − =
n
k
k
k
f , |ψ
k

0i is the ground state of a L-site system in the N -
particle sector, and |ψ±i denote the m-th excited states m

0
in the (N ± 1)-particle sectors with excitation energies ω± = E± − E
m
m

0.

0.6

Within PRM we find for the photoemission part ω

(c)

b
j

0=0.5, λ=0.5
n 0.4

,

A−(k, ω) = ˜
α2 ˜
nf δ(ω − ˜
ε
k
k
k)

f
jn

X

0.2

+

˜
β2 (1 + ˜
nb)˜
nf
δ(ω + ˜
ω
k,q
q
k+q
q − ˜
εk+q)

q

0

X

+

˜
γ2k,q ˜nbq ˜nf δ(ω − ˜
ω
k−q
q − ˜
εk−q) ,

(7)

q

1 ω0=0.5, λ=0.01

(d)

b
jn
where ˜
nf (˜
nb) are the fermion (boson) occupation num-q
q

,
bers in momentum space calculated with the renormal-f
j 0.5
n
ized Hamiltonian ˜

H.

The coefficients ˜
α2, ˜
β2 , and
k
k,q

˜
γ2 follow from renormalization equations. 26 Taking the k,q

00

10

20

30
corresponding expression for A+(k, ω), the sum rule 40

50

R ∞
j
dω[A+(k, ω) + A−(k, ω)] = 1 is fulfilled.

∞Figure 4 gives the combined photoemission spectrum, A(k, ω) = A+(k, ω) + A−(k, ω), as obtained by DDMRG
for quasimomenta k. In the repulsive TLL regime (left Figure 3.

Local particle (filled circles) and boson (open squares) densities for the 1D Edwards model with n = 1/8, panel, ω0 = 2) we find a rather coherent signal with calculated for a 48-site system with OBC at various λ and ω
comparable spectral weight for all k values k . π/2. For 0.

larger k, excitations with at least one additional ω0-boson involved become important (recall that both initial N -
particle- and target (N ± 1)-particle states are multibo-been derived in Ref. 26 (for the half-filled band case).

son states with a momentum being the total momentum In order to fix the boundary between metallic and PS
of electrons and bosons). The spectrum in the attractive states, we (i) solve the renormalization equations at a given EF in the TLL phase, (ii) determine the corre-k
sponding fermion density n, and (iii) slightly vary EF
n=1/8
ω
ω

(to get closer to the PS instability), determine n, and re-0=2

0=0.7
peat the whole procedure self-consistently until the func-tional dependence of n on E

)

0

F is established. A vanish-
ω
ing inverse compressibility can then be read off from a

, k
k

(k
sudden jump of n under a tiny variation of EF or, the A
other way around, if a plateau in the EF versus n plot appears. The transition points, determined in the manner described, were inserted in Fig. 2 to complete the ground-state phase diagram in the n-ω
π

0 plane. Whenever

-2

0

2

-5

0

5
points can be compared we find a remarkable agreement ω – E
ω – E

F

F
between our DMRG and PRM data. The deviation of the PRM data at larger particle densities n & 0.75 re-Figure 4. Line-shape of the single-particle spectral function sults from uncertainties in fixing the jump of n under A(k, ω) for the 1D Edwards model with λ = 0.2 at n = 1/8.

E

Results are obtained by DDMRG for an 16-site chain with F variation. Here we should trust in the DMRG phase boundary.

quasimomenta k = πs/(L + 1), using a broadening η = 0.2.

5
k
ical density nc2 for PS. Since in the PRM derivation of A±(k, ω) terms with two (and more) bosonic creation or

)

n=3/10
ω0=0.5
annihilation operators were neglected,26 the photoemis-

,ω
sion spectrum of Fig. 5 does not feature the multiboson (k

0
related signatures found within a DDMRG treatment.

+ A),

III.

CONCLUSIONS

,ω
π/2

(k

To summarize, the combination of analytical (PRM)

– A
and numerical (DMRG) approaches permits the precise determination of the ground-state phase diagram of the π

1D Edwards model in the whole parameter regime. In the low- and high-density regions, the attraction between

-4

0

4

8
the particles mediated by the bosonic degrees of freedom ω − E
representing the background medium might become so F
strong that electronic PS sets in. In the remaining re-Figure 5. PRM [inverse] photoemisson spectrum [A+(k, ω)]

gion the system realizes a TLL or, at n = 1/2, possibly A
even a truly long-range ordered CDW state. Depending

− (k, ω) for the Edwards model with λ = 0.2, n = 3/10. The bold line marks the signal at the Fermi momentum k on the properties of the background medium, the TLL

F = 0.3π.

might be attractive or repulsive. The richness of the phase diagram is remarkable. The model captures im-TLL phase (right panel, ω
portant features of Holstein, t-J, Hubbard, and Falicov-0 = 0.7) shows a sharp absorp-
tion signal in the vicinity of k

Kimball type models.6–9,38 Since the Edwards model is F only. Here the (inverse)

photoemission spectrum for k < k one of the simplest models for studying transport in low-F (k > kF ) exhibits a
few absorption maxima at multiples of the boson energy.

dimensional systems, an inspection of its predictions with Obviously, due to the smaller ω
ultracold fermion/boson quantum gases, 39 as, e.g., car-0, here the dynamics of
the system becomes dominated by bosonic fluctuations.

ried out for the 1D t-J model, 40 would be of great inter-The (inverse) photoemission close to the transition to est.

the PS state is depicted Fig. 5, as calculated by PRM
for ω0 = 0.5. Here the almost dispersionless signal for k ≤ 0.212π (k ≥ 0.788π) is a precursor of the PS instabil-ACKNOWLEDGMENTS
ity. As we noted above, EF does not show any variation with n if n is smaller (larger) than the lower (upper) crit-This work was supported by DFG through SFB 652.

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Table of Contents

- l
- ,
- 2.
- (1)
- 1
- 3