andreas martin lauchli- quantum magnetism and strongly correlated electrons in low dimensions
TRANSCRIPT
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Diss. ETH No. 14908
Quantum Magnetism and
Strongly Correlated Electronsin Low Dimensions
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of
Doctor of Natural Sciences
presented by
Andreas Martin Lauchli
Dipl. Phys. ETH
born February 8th, 1972
citizen of Remigen (AG)
accepted on the recommendation of
Prof. Dr. T. M. Rice, examiner
Prof. Dr. M. Troyer, co-examiner
Prof. Dr. F. Mila, co-examiner
2002
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Abstract
In this thesis low dimensional strongly correlated electron systems and frustrated
quantum magnets are investigated employing large scale numerical simulations.
The difference between doping of lithium and zinc in undoped two leg spin ladders
is discussed. While zinc dopants induce local moments and suppress the spin gap,
weak Lithium doping is predicted to maintain a stable spin gap. The relevance of
our results to recent Zn and Li impurity experiments in the underdoped cuprates
is discussed. New experiments are proposed.
A novel numerical approach to the understanding of the strong coupling fixed points
of perturbative Renormalization Group (RG) treatments is introduced and its ap-
plicability to different low dimensional models is demonstrated. The method is
a combination of a standard perturbative RG treatment followed by a numerical
analysis of the flow to strong coupling by exact diagonalization methods. For sys-
tems such as the one dimensional g1g2 model or the two leg Hubbard ladder athalf filling, good agreement with existing analytical predictions is found. Future
applications to the two dimensional Hubbard model are outlined.
The phase diagram of a spin ladder with cyclic four spin exchange has been in-
vestigated. The phase diagram is surprisingly rich. In addition to conventional
phases such as the rung singlet phase, the ferromagnetic phase and the dimerized
phase, two more exotic phases with strong chiral correlations are found. One shows
long range order in the staggered scalar chirality, while the other has short range
order in the staggered vector chirality. First results for the square lattice indicate
that a phase with long range order in the staggered vector chirality is stabilized for
strong cyclic four spin exchange. The influence of four spin exchange on the magnon
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dispersion is determined and compared to recent experiments on La2CuO4.
Finally the phase diagram of a generalized Shastry-Sutherland model is reported.
We find two different Neel ordered phases, two short range ordered resonating va-
lence bond phases (with strong dimers or strong plaquette singlets) and along the
standard Shastry-Sutherland line a valence bond crystal phase with long range order
in plaquette singlet correlations, thereby breaking a discrete lattice symmetry.
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Zusammenfassung
In der vorliegenden Doktorarbeit werden stark korrelierte Elektronensysteme und
frustrierte Quantenmagnete in niedrigen Dimensionen mittels Computersimulatio-
nen untersucht.
Der Unterschied zwischen Zink und Lithium-Dotierung in undotierten Spinleitern
wird erlautert. Zink Storstellen induzieren lokale magnetische Momente und un-
terdrucken dadurch die Spinanregungslucke. Im Gegensatz dazu lasst die Dotierung
mittels Lithium die Anregungslucke intakt. Die Auswirkungen unserer Resultate
und die Interpretation von Experimenten mit Zink- und Lithium-Dotierung in den
unterdotierten Kupraten werden diskutiert.
Wir stellen eine neue numerische Methode zur Analyse von Renormierungsgrup-
penflussen hin zu starker Kopplung vor. Wir zeigen die Anwendbarkeit der Methode
auf verschiedene niedrigdimensionale Modelle. Der Zugang besteht aus der Anwen-
dung einer storungstheoretischen Renormierungsgruppe auf das System, welche in
einem zweiten Schritt durch die numerische Diagonalisierung des asymptotischen
Flusses erganzt wird. Fur Systeme wie das eindimensionale g1g2 Modell oder diezwei-Bein Hubbard Leiter bei halber Fullung finden wir sehr gute Ubereinstimmung
mit analytischen Berechnungen. Zukunftige Anwendungen auf das zweidimension-
ale Hubbardmodell werden skizziert.
Das erstaunlich reichhaltige Phasendiagramm einer Spinleiter mit Ringaustausch
wurde bestimmt. Zusatzlich zu konventionellen Phasen wie der Sprossen-Singlet
Phase, der ferromagnetischen Phase und einer dimerisierten Phase finden wir zwei
Regionen mit starken chiralen Korrelationen. Eine davon ist besitzt langreichweit-
ige Ordnung in der alternierenden skalaren Chiralitat. Die andere tragt kurzreich-
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weitige vektor-chirale Korrelationen. Erste Resultate fur das Quadratgitter weisen
darauf hin, dass fur grossen Ringaustausch die letztgenannte Phase ordnet. Wir
berechnen den Einfluss des Ringaustauschtermes auf die Dispersion der Magnonen
im Antiferromagneten auf dem Quadratgitter und vergleichen die Resultate mit
La2CuO4 Experimenten.
Zuletzt behandeln wir das Phasendiagram eines verallgemeinerten Shastry-Sutherland
Modells. Wir charakterisieren zwei Neel geordnete Phasen, zwei Resonating Valence
Bond Phasen mit starken Dimer- oder Plaketten-Singlets und einen Valence Bond
Kristall mit langreichweitiger Ordnung in den Plakett-Singlet Korrelationen fur das
normale Shastry-Sutherland Modell.
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Acknowledgements
First of all I would like to thank my advisor Prof. Maurice Rice for the opportunity
to work with him on a number of exciting and challenging physical questions. I
learned a lot about condensed matter physics and the way to look at strongly
correlated electrons.
My thanks also go to Prof. Matthias Troyer for teaching me all the fine details
about powerful algorithms in strongly correlated systems, about C++, generic pro-
gramming and supercomputers.
I am very grateful to Prof. Frederic Mila for accepting to be one of my coreferees
and for many stimulating discussions throughout my PhD time.
Some of the projects have been done in collaboration with other people. I found
it very interesting to work in a collaboration where different approaches to the
same problem meet. I would therefore like to thank Andreas Honecker, Carsten
Honerkamp, Didier Poilblanc, Guido Schmid, Manfred Sigrist, Stefan Wessel and
Steven White for their valuable contributions.
What would life be at the institute without all my colleagues: Malek, Hanspeter,
Jerome, Prakash, Stefan, Mathias, Igor, Samuel, Guido, Arno, Simon, Paolo, Mar-
tin, Fabien and Synge. I thank you all for many stimulating coffee breaks including
discussions about physics and the rest of life, and not to forget the outstanding
Toggeli1 games.
Im especially grateful to my parents. They always provided me strong support
throughout my Studienjahre. Thank you very much.
Finally I would like to thank Johanna. She knows why.
1swiss german: tabletop soccer
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Contents
1 Introduction 1
1.1 The cuprate high Tc superconductors . . . . . . . . . . . . . . . . . 1
1.2 Frustrated quantum magnets . . . . . . . . . . . . . . . . . . . . . . 4
2 Lithium induced charge and spin excitations in a spin ladder 7
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Binding energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Magnon-Lithium bound state . . . . . . . . . . . . . . . . . . . . . 14
2.5 Local density of states . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Numerical analysis of Renormalization Group flows 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Mesh in k-space . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 The coupling function . . . . . . . . . . . . . . . . . . . . . 25
3.2.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Test case : a one-dimensional problem . . . . . . . . . . . . . . . . 28
3.4 The two-leg Hubbard ladder at half filling . . . . . . . . . . . . . . 31
3.4.1 Repulsive U - The D-Mott phase . . . . . . . . . . . . . . . 35
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3.4.2 Zoo of insulating phases . . . . . . . . . . . . . . . . . . . . 36
3.5 The Two-patch model . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Quantum magnets with cyclic four spin exchange 45
4.1 Phase diagram of a two leg ladder with cyclic four-spin interactions 46
4.1.1 Rung singlet phase . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 Staggered dimer phase . . . . . . . . . . . . . . . . . . . . . 50
4.1.3 Scalar chirality phase . . . . . . . . . . . . . . . . . . . . . . 51
4.1.4 Dominant vector chirality region . . . . . . . . . . . . . . . . 52
4.1.5 Dominant collinear spin region . . . . . . . . . . . . . . . . . 54
4.1.6 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . 54
4.1.7 Phase transitions and universality classes . . . . . . . . . . . 54
4.1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Square lattice with cyclic four spin exchange . . . . . . . . . . . . . 57
4.3 Magnon dispersion of La2CuO4 . . . . . . . . . . . . . . . . . . . . 66
5 Phase diagram of the quadrumerized Shastry-Sutherland model 71
5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Boson operator approach . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1 Dimer-boson approach . . . . . . . . . . . . . . . . . . . . . 74
5.2.2 Quadrumer-boson approach . . . . . . . . . . . . . . . . . . 75
5.3 Exact Diagonalization studies . . . . . . . . . . . . . . . . . . . . . 78
5.4 Shastry-Sutherland model . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 Numerical techniques 89
6.1 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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6.1.2 Basis construction, Symmetries . . . . . . . . . . . . . . . . 90
6.1.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1.4 Lanczos Algorithm . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.6 Implementation Details . . . . . . . . . . . . . . . . . . . . . 99
6.2 Density Matrix Renormalization Group . . . . . . . . . . . . . . . . 101
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Chapter 1
Introduction
1.1 The cuprate high Tc superconductors
The study of strongly correlated electron systems is one of the most active fields in
condensed matter physics. Since the seminal discovery of high Tc superconductivity
in the cuprates in 1986 [1] steady progress has been made in the understanding of
these strongly interacting systems. But still a consistent theory is lacking. It is
generally believed that the strong Coulomb repulsion inside the two dimensional
CuO2 planes plays an important role.
d-wave
Superconductivity
Fermi Liquid
Non-Fermi Liquid
Pseudogap
NeelOrder
TemperatureT
hole concentration x
Figure 1.1: Schematic phase diagram of the hole doped cuprate high Tc superconductors.
The phase diagram of the hole doped cuprates has the schematic form shown in
Fig. 1.1. The undoped system is a good example of a Mott insulator (i.e. an
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insulating state induced by correlation effects, not by band structure) and exhibits
antiferromagnetic long range order. This Neel order is rapidly destroyed by the
doping of holes and a strange metallic state emerges which goes superconducting
for low enough temperatures. Upon further doping the critical temperature Tc
of the superconducting phase raises as high as 133 K for certain mercury based
compounds. Doping beyond optimal doping reduces Tc again and the material
turns into a conventional metal (Fermi liquid). Superconductivity is not the only
unconventional phenomenon in this phase diagram. Another puzzle is the presence
of a pseudogap in the single particle spectral function within the underdoped region
of the phase diagram. An early high Tc paradigm stated that the key to the solution
of the high Tc puzzle lies in the understanding of the strange normal state properties
of the cuprates. This seems to be true still today.
The broad range of phenomena present in the phase diagram illustrate the difficulty
of developing a consistent theory of the cuprate materials. Parts of the phase
diagram are well understood in their respective framework, but these cease to be
valid for other regions.
Soon after the discovery of the high Tc superconductors it was realized that analyt-
ical treatments alone will not immediately solve the puzzle. The reason for this are
the strong correlations and fluctuations present in the CuO2 planes which render
the usual mean field approaches unreliable. Also perturbative schemes are difficult
to put to work as there is no well defined limit about which to expand. Therefore
numerical simulations became very important tools in this field and algorithms such
as Quantum Monte Carlo, Exact Diagonalization, Density Matrix renormalization
Group and series expansions helped improve our understanding that the basic mod-
els such as the single band Hubbard model or its descendant, the tJ model capturethe essential physics of the cuprates. In this thesis we will mainly use Exact Diago-
nalization (ED) and the density matrix renormalization group (DMRG) algorithm.
These algorithms are presented in chapter 6.
In recent years it has been realized that the presence of impurities in these strongly
correlated materials can actually provide new insights in the properties of the host
system. Beautiful scanning tunnelling microscope (STM) experiments [3] on super-
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conducting BSCCO1 samples were able to measure the local density of states around
individual Zn impurites on an atomic scale. The local density of states exhibited
a d-wave like pattern. A different series of experiments using Nuclear Magnetic
Resonance (NMR) techniques analyzed the effect of Zinc and Lithium impurities
in YBCO [19] in the metallic and the superconducting state. These nonmagnetic
impurities both induce a magnetic moment close to the impurity site. This mag-
netic moment is finally Kondo screened at low temperatures. Our work presented
in chapter 2 was motivated by these fascinating results and we discuss the behavior
of Zinc and Lithium impurities in an undoped Resonating Valence Bond (RVB)
system, the two leg spin ladder.
180 K
95 K
120 K
cT =85 K
Figure 1.2: Evolution of the Fermi surface as a function of temperature in an underdoped
sample. Schematized ARPES results (taken from [7])
As pointed out before the nature of the pseudogap phase is one of the hotly debated
topics today. In this region of the phase diagram the single particle spectral function
develops a gap in certain regions of the Fermi surface despite the fact that the
system is not yet superconducting. Very nice photoemission experiments [6, 4, 5]
revealed a successive destruction of the Fermi surface in underdoped BSCCO as the
temperature is lowered. This is illustrated in Fig. 1.2. For high temperatures the
Fermi surface is intact. As the temperature is lowered extending regions close to
1BSCCO stands for the compound Bi2Sr2CaCu2O8x, YBCO stands for YBa2Cu3O6+x
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to the (, 0) and (0, ) points develop a gap. Finally below the superconducting
transition at 85 K only the d-wave nodal quasiparticles are left. Various theoretical
approaches to this phenomenon have emerged in the meantime. Here we briefly
focus on the results of weak coupling Renormalization Group (RG) calculations for
the twodimensional t
t Hubbard model [38, 37, 40]. These studies suggest a weak
coupling instability towards a state which develops a gap in the aforementioned
regions of the Brillouin zone. Interestingly this state shares much of the physics
of the two leg Hubbard ladder. In order to characterize these RG results more
precisely, we propose and discuss a novel numerical approach for the analysis of the
flow to strong coupling in chapter 3.
1.2 Frustrated quantum magnets
Frustrated (quantum) magnets form another class of strongly interacting electron
systems. In these systems all the elementary interactions between spins cannot
be satisfied simultaneously. Therefore they are called frustrated. The inherent
competition induces strong fluctuations. These in turn can induce unconventional
phases in which no simple magnetic structure such as ferromagnetism or Neel order
is stabilized.
An exciting example with classical spins is the so called spin ice [8]. For example
the Ho2Ti2O7 compound is a magnetic system on the pyrochlore structure. The
Ising anisotropy constrains the spins to point either in or out of the elementary
tetrahedron. The competition between exchange and dipolar interactions dictates a
local structure with two spins pointing in a two spins pointing out. This rule, which
is very similar to the ice rules proposed by Pauling, leads to a highly degenerate
groundstate with a finite entropy per spin at T = 0. This has been confirmed
experimentally.
When turning to fully quantum spins (S = 1/2) a few models with unconventional
behavior are known. For reviews see [9]. The antiferromagnetic Heisenberg model
on the Kagome lattice for example does not seem to order and might also have a
groundstate with extensive entropy. Numerical calculations report a finite triplet
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gap and a large number of singlets below the triplet gap. In the multiple spin
exchange models on the triangular lattice a variety of different phases has been
reported, including a long sought spin liquid without any local order parameter.
The discovery of significant cyclic four spin exchange in the La2CuO4 [54] compound
sparked our interest in the physics of higher order spin interactions. We determine
numerically the phase diagram of a spin ladder and of the square lattice Heisenberg
model with additional four spin interactions. Our results reveal several unexpected
phases and also underline the fact that the physics of multiple spin exchange models
on square geometries is rather different compared to the triangular lattice.
Finally we calculate the phase diagram of a generalized Shastry Sutherland model.
This model has attracted a lot of interest, especially because it is realized in the
magnetic structure of the SrCu2(BO3)2 compound. A open problem was the identi-
fication of the phase between the established Neel order in one limit and the exact
dimer product state in the other limit. We present evidence for a valence bond
crystal phase with plaquette singlets in an intermediate range of parameters.
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Chapter 2
Lithium induced charge and spin
excitations in a spin ladder
In this chapter we investigate properties of strongly correlated systems upon impu-
rity doping. In recent years beautiful experiments demonstrated that the doping of
impurities into strongly correlated systems leads to interesting phenomena which
reveal a lot about the properties of the host material itself. Important advances in
experimental techniques such as NMR and STM lead to these interesting experi-
mental results.
We concentrate on the impurity doping of the high-Tc cuprate superconductors
which is an effective tool to explore the low temperature physics of these strongly
correlated systems.
The similarities or differences observed upon doping non-magnetic zinc (Zn) and
lithium (Li) ions must find their explanations in the nature of the host and in
the peculiarities of each dopant. In the antiferromagnetic (AF) phase of La2CuO4,
Li [11] is far more effective at suppressing AF order than Zn [12], although both enter
the same planar Cu(2) site. Li introduces a hole (due to the difference in formal
valence of Cu2+ and Li+) which is tightly bound since the alloy series La2LixCu1xO4
remains insulating for all 0 < x < 0.5. The rapid destruction of AF order is
attributed to the effect of the bound hole1. On the other hand Zn (which has the
1The detailed mechanism is however still not understood. The skyrmion topological defect
scenario proposed in Phys. Rev. Lett. 77, 3021 (1996) has failed to be reproduced in our extensive
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same valence as Cu) does not destroy AF order up to the percolation threshold.
This has been evidenced in experimental [14] and numerical studies [15].
Strikingly, non-magnetic Zn and Li ions behave similarly in conducting (hole-doped)
YBa2Cu3O6+x (YBCO) by inducing local magnetic moments [16] which sit predom-
inantly on the four nearest neighbor (NN) Cu. They both exhibit static [17] and
dynamic [18] susceptibilities reminiscent of a Kondo-like behavior with a very low
effective temperature in the underdoped samples (TK 2.8 K) [19]. Previous cal-culations using a vacant site model for a Zn-dopant found that it acted as a strong
scattering center for holes with even a bound state (hole-Zn bound state) which
could be the source of an effective Kondo coupling [20].
Undoped spin ladders [21] offer an ideal system to investigate doping in a spin liq-
uid or resonating valence bond [22] (RVB) state with short range AF correlations
and a spin gap, which can help the understanding of their two dimensional (2D)
analogs. Zn doping into the (undoped) spin-1/2 Heisenberg two-leg ladder com-
pound SrCu2O3 leads to local moments which form an AF ordered state [23] at
low temperature. Local moments and a rapid suppression of the spin gap were ob-
tained theoretically in a Heisenberg ladder using the vacant-site model [24, 25] for
Zn doping (without additional holes). Further simulations [26] led to an effective
model with coupled local moments with an interaction which decays rapidly with
separation.
Doping Li into a two-leg spin ladder is an open problem both experimentally and
theoretically. Novel physics can be expected due to the additional (with respect to
Zn) hole when Li+ replaces Cu2+. In the following, we use a vacant-site model for
Li+ and show that Li+, unlike Zn2+, does not introduce low-energy spin excitations
but forms a dopant-magnon bound state (BS) just below the spin gap of the undoped
ladder. It follows that, unlike Zn-doped ladders, Li-doped ladders will keep a robust
spin liquid character at low temperature.
calculations.
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(a)
(b)
5
4 4221
53 3
Figure 2.1: Schematic representations of a Li-doped spin ladder. The cross denotes the dopant
site (Li+ ion). The circle stands for the injected mobile hole and the thick lines sketch the attractive
hole potential.(a) Pictorial representation of the groundstate where spins are paired in singlets
(shown shaded). Sites are labelled for convenience. (b) Sketch of the lowest triplet excitation:
dopant-magnon bound state (discussion in the text).
2.1 The model
For dilute concentrations, a single dopant as in Fig. 2.1 suffices. We model a Li+
dopant by an vacant site with a hard-core repulsion for holes and an attractivenearest neighbor (NN) Coulomb potential due to its negative charge with respect
to Cu2+. The Hamiltonian reads:
H = J
(Si Sj 14
ninj) (2.1)
t
,
(ci,cj, + h.c.) V
lI
(1 nlI ) ,
using standard notations and the primed sum is restricted to the NN bonds < ij >notconnected to the dopant. The sum over lI runs over dopant NN sites. Note, for
simplicity, we restrict ourselves to the case of a magnetically isotropic ladder i.e.
with equal rung and leg couplings, J. We use Exact Diagonalisations (ED) of small
periodic ladders (up to 2 12) supplemented by Density Matrix RenormalisationGroup (DMRG) calculations on larger open systems (up to 2 128). In open
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ladders, the dopant is placed in the center.
2.2 Binding energies
First, we investigate the localization of the injected hole versus the strength of the
Coulomb potential. Following Ref. [20], we define the hole-dopant binding energy
as,
S=01 h,1 do p = E0(1h, 1i) + E0(0h, 0i) (2.2)
(E0(1h, 0i) + E0(0h, 1i)) ,
where E0(nh, mi) is the groundstate (GS) energy with n = 0 or 1 (m = 0 or 1)
holes (dopants). It is negative when the hole and Li
+
-ion form a stable BS. SinceLi-doping removes two spins we expect a magnetically inert groundstate i.e. a
singlet (S = 0). As seen in Fig. 2.2(a) a stable bound state is found for almost
all couplings, even when V = 0, but the binding strength increases considerably
with V. Note, the magnitude of the binding energy is slightly larger than in a
2D planar geometry [20]. Fig. 2.1(a) shows schematically how the absence of local
moment can be understood from the RVB nature of the host (all remaining spins
are paired in singlets). Although a single Li-dopant binds the injected hole, caution
is required at finite concentration and the possibility of other decay channels has
to be considered, e.g. 2 holes from 2 dopants recombine into an itinerant hole pair.
This is ruled out since the dopant-hole binding energy is always larger in absolute
value than half of the hole-pair binding energy (see Fig. 2.2(a)) even when V = 0.
Note that the other decay channel consisting of a single dopant trapping two holes
can also be rejected on energetic grounds since the two hole-dopant binding energy
S=02 h, 2 dop
defined as E0(2h, 1i) + E0(0h, 1i)
2E0(1h, 1i) was always found to be
positive. At low concentrations we therefore find decoupled dopant bound states
each with one hole. Since the spatial extent of an isolated BS is small ( = 2 to 4
rungs even when V = 0), the system remains insulating up to large doping.
The binding energies related to the two decay processes in the triplet channel that
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0 0.5 1
J/t1
.2
0.8
0.4
0.0
Binding
en
ergies/t
V/t=0
V/t=0.5
V/t=1
2holes
/2
0 0.5 1
J/t0
.8
0.6
0.4
0.2
0.0
0 0.5 1
J/t0
.1
0.075
0.05
0.025
0
0.02
5
(a)(b) (c)
Figure 2.2: Various binding energies to the dopant (see text for definitions) vs J/t obtained
by ED on a 211 ladder. V denotes the attractive NN potential. (a) BE of the single hole inthe singlet GS. (for comparison, half of the hole-pair binding energy 2holes is also shown). (b)
Binding energy of the hole to the dopant-magnon complex (see equation 2.3) in the lowest triplet
state. (c) Binding energy of the magnon to the hole-dopant complex (see equation 2.3) in the
lowest triplet state.
are plotted in Figs. 2.2(b),(c) are defined as
S=11 h,1 do p = E1(1h, 1i) + E0(0h, 0i) (E0(1h, 0i) + E0(0h, 1i))and as
S 0S = E1(1h, 1i) + E0(0h, 0i) (E0(1h, 1i) + E1(0h, 0i)), (2.3)
where E0 (E1) is the lowest energy in the singlet (triplet) sector.
2.3 Magnetic properties
We now turn to the magnetic properties of the Li-doped spin ladder. We compute
the dynamical spin structure factor S(q, ) for an undoped ladder (Fig. 2.3(a)),
a Zn-doped ladder (Fig. 2.3(b)) and a Li-doped ladder described by Eq. (2.1)
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S(q,
)[a
.u.
]
0 1 2
/J0 1 2
/J0 1 2
/J
q=(0,)
q=(/5,)
q=(2/5,)
q=(3/5,)
q=(4/5,)
q=(,)
x 1/3
x 1/10x 1/10
x 1/3
x 1/20
x 1/6
x 1/6
x 1/3
x 1/3
x 1/3
(a) (b) (c)
Figure 2.3: Spin structure factors S(q, ) calculated on 2 10 ladders. The different curvescorrespond to decreasing q, from q = (, ) (bottom) to q = (0, ) (top). For clarity, reducing
scaling factors (as indicated) are applied on some curves. (a) Undoped periodic ladder; (b) Spin
ladder doped with a single Zn (full line) or two Zn dopants separated by the maximum distance
on the same leg (dashed blue line); The arrow marks the spectral weight which is generated inside
the spin gap. (c) Ladder doped with a single Li dopant with t = 2J and V /t = 0.5. The peak
originating from the bound state mentioned in text is marked with an arrow
(Fig. 2.3(c)). The dynamical structure factor is defined as:
S(q, ) =
n
|n|Szq|0|2 ( n), (2.4)
where these sum runs over all eigenstates n with energy n. The RVB picture, in
which spins are paired up into short range singlets, gives a qualitative understanding
of the magnetic properties. In the undoped ladder, a triplet excitation (magnon)
is well described by exciting a rung singlet into a triplet. Fig. 2.3(a) shows the
single magnon dispersion with a minimum at q = (, ) and = 0S, the spin
gap [27] of the undoped ladder. Introducing a Zn atom on a rung releases a free
spin-1/2 which leads to zero-energy spin fluctuations, predominantly at q = (, )
(Fig. 2.3(b)) and the undoped ladder magnon survives. Two Zn-dopants behave as
two S=1/2 moments with a weak effective exchange interaction, Jeff, which decays
rapidly with separation, in agreement with Ref. [24]. A small spectral weight at
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q = (, ) and low energy ( Jeff) appears below the undoped spin gap results.Li-doping (Fig. 2.3(c)) is drastically different with no weight at small energy. Since
a Li+ dopant has a bound hole, there is no free spin but a new type of excitation
appears just below the unperturbed spin gap - a bound state of a magnon with the
hole/Li+ as naively depicted in Fig. 2.1(b). Its binding energy defined as the energy
difference with respect to the free magnon energy 0S remains in general quite small
(in absolute value) as seen directly in Fig. 2.3(c) (and quantitatively in Fig. 2.2(c)).
Therefore a drastic reduction of the spin gap does not occur in this case. We checked
this conclusion by extending the DMRG calculations of Fig 2.4 to two Li dopants
separated by 64 sites for the case V /t = 1 (Fig. 2.5. In this case the lowest triplet
excitation is a magnon-hole BS strongly localized near one dopant quite different to
the case of two Zn dopants discussed above. The spin susceptibility should remain
activated with only a small reduction in the activation energy in the presence of
Li-dopants unlike the Curie term introduced by the Zn-dopants.
28 32 36
rung
0
0.2
0.4
0 16 32 48 64
rung
0
0.05
0.1
V/t=1.0
V/t=1.3
V/t=2.0
(a) (b)
Figure 2.4: Hole rung density (a) and Sz rung density (b) along the ladder direction in the
lowest energy triplet state calculated by DMRG for J/t = 0.5. The rung density is defined as the
algebraic sum of the densities (if any) on the two sites of a given rung. Different values of V /t (as
indicated) are shown.
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0 50 100
rung
0
0.05
0.1
0.15
Sz
(x)on
rung
0 50 1000
0.1
0.2
0.3
0.4
nh
(x)on
rung
J/t =0.5 , V/t=1
Spin Gap (two dopants) : 0.2412 tSpin Gap (one dopant): 0.2413 tSpin Gap (undoped): 0.2512 t
Figure 2.5: Hole rung density (upper panel) and Sz rung density (lower panel) along the ladder
direction in the lowest energy triplet state for a 2 128 ladder with two Li dopants, calculated byDMRG for J/t = 0.5 and V /t = 1. Each Li dopant confines one hole. The magnon is bound to
one of the Li-hole complexes. The values of the spin gap indicate that there is no drastic reduction
for two Li dopants contrary to the Zn case.
2.4 Magnon-Lithium bound state
The physical origin of this BS is of particular interest. It can be attributed to the
gain in hole kinetic energy associated with the spin triplet. DMRG calculations
give different spatial extents of the charge and spin perturbations. While the hole
is localized on the scale of a few rungs (Fig. 2.4(a)) the rung magnetization can
extend to large distances (Fig. 2.4(b)). This is consistent with our finding that the
binding energies of a hole to a dopant-magnon complex and that of a magnon to a
hole-dopant complex are quite different (Figs. 2.2(b) and 2.2(c)). In fact, increasing
the attraction V binds the hole more strongly and limits the ability of the hole toreduce its kinetic energy by moving in the spin polarized background of the magnon,
hence reducing the binding of the magnon to the dopant-hole complex [28]. Directly
from the binding energy (Fig. 2.2(c)), we can conclude that above a critical value
of V (typically VC/t 2 for J/t = 0.5.) the magnon escapes from the hole-dopantcomplex, as can be seen also in Fig. 2.4(b), where the rung Sz-density for V /t = 2
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indicates an unbound magnon.
2.5 Local density of states
-6
-4
-2
0
/t
-6
-4
-2
0
2
/t
impurity
site
12
2
3 3
4 4
5 5
Figure 2.6: Local DOS around a Li dopant obtained by ED of a 2 9 ladder with J/t = 0.5 andV/t = 0.5. Each panel corresponds to a site in the vicinity of the dopant (site labels correspond
to those of Fig. 2.1(a)). Occupied (empty) electronic states are shaded (left blank).
We calculate also the local density of state (LDOS) near the Li-dopant. The local
density of states is defined as follows:
Ni,i() =
n
|n|ci,|0|2 ( (n 0)), (2.5)
Results are shown in Fig. 2.6 for the spatially resolved DOS spectra. The LDOS can
be measured directly in scanning tunneling microscope (STM) experiments. The
> 0 ( < 0) spectra give the weights of the neutral (charged) target S = 1/2
states accessed by removing the hole (adding an extra hole) to the singlet GS. The
large peak at small positive energy on site # 1 (i.e. on the same rung as the dopant)
corresponds to a local moment (S=1/2)-dopant resonance. Other resonances are
seen at higher energies with larger spatial ranges i.e. 2 or 3 sites on the leg opposite
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to the dopant. Similarly, < 0 resonances are seen when adding an extra hole on
the same leg as the dopant, e.g on sites # 3 and 5. The lowest resonance energy
(in absolute value) is obtained when the second hole is added on sites # 3 next to
the dopant. This might indicate the possibility of a bound state of two holes close
to one impurity2. Note the local > 0 ( < 0) integrated weight provides directly
the local hole (electron) density in the GS. Hence the bound hole is located mainly
on the leg opposite to the dopant and extends roughly over three sites.
2.6 Conclusions
Our theory can be directly tested, if Li can be substituted for Cu in the ladder
compound, SrCu2O3. The extra bound hole around a Li dopant should ensure that
a free local S=1/2 moment is not created, unlike Zn doping. Hence SrCu2xLixO3
should not order antiferromagnetically at low temperature, unlike SrCu2xZnxO3
(Ref. [23]). Further the nature of the magnon-dopant bound state, the charge
distribution and local DOS could also be examined experimentally in these systems.
However, our analysis raises interesting questions regarding the close similarity be-
tween Li and Zn substitution in superconducting YBCO samples [19]. In particular
if we interpret the spin gap phase in underdoped YBa2Cu3O6.6 as a doped d-RVB
phase, then there should be a close similarity to the behavior of the doped lad-
der. However Bobroff et al. [16, 17] report a free S=1/2 moment (which is Kondo
screened only at very low temperatures) for both Zn and Li doping of the un-
derdoped samples. A possible way to reconcile this apparent contradiction is to
postulate that Li+ does not bind a hole in YBa2Cu3xLixO6.6, unlike the case of
La2Cu1xLixO4. This could occur if the mobile O2-ions in the chains were repelled
from the Li+-ions in the planes. A test of this hypothesis can be made by doping
Li+ and Zn2+ in YBa2Cu4O8 which as a stoichiometric compound has no mobile
O-ions. Our analysis then predicts free S=1/2 moments only for Zn-doping and not
for Li-doping in this case.
2In that case we would probably need to include the hole-hole repulsion into the problem as
well
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In conclusion, our analysis predicts a clear distinction between the magnetic prop-
erties of the two non-magnetic ions, Zn2+ and Li+ when doped into spin liquids due
to the binding of a hole in the latter case. Experiments to test these predictions
are proposed.
The content of this chapter has been published in
Phys. Rev. Lett. 88, 257201 (2002)
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Chapter 3
Numerical analysis of
Renormalization Group flows
This chapter is devoted to the presentation and analysis of a new numerical ap-
proach to correlated fermion systems. It is a combination of a weak-coupling renor-
malization group (RG) treatment of the fermionic system followed by a numerical
analysis of the asymptotic flow by an exact diagonalization (ED) scheme. It is
intended to give insights into the strong coupling state starting from the weak
coupling limit.
The outline of this chapter is as follows: In the first section we introduce the nu-
merical scheme in detail. The method is then illustrated with an application to a
one-dimensional system, the Luther-Emery liquid. Next we investigate the weak
coupling phase diagram of the two-leg Hubbard ladder at half-filling. Bosonization
studies revealed that the Hubbard ladder at half-filling can accommodate a large
variety of ordered and quantum disordered phases. All of them are insulating. We
show that the numerical scheme is able to characterize these phases in accord with
the Bosonization treatments. In particular we confirm the simultaneous enhance-
ment of several correlation functions. Finally results for the two-dimensional
two-patch model, a simplified model relevant for the 2D t t Hubbard model arepresented.
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3.1 Introduction
Renormalization group approaches to physical problems are powerful conceptual
and calculational tools. Initially developed for problems in particle physics, the
method was successfully extended to statistical and condensed matter physics. In
the context of strongly interacting electrons a milestone was set by the solution of
the Kondo problem by Wilson [33]. In the spirit of Wilsons ideas the RG method was
subsequently applied to one dimensional conductors. (see Refs. [29, 30, 31, 32, 34]
for reviews.)
In the framework of strongly correlated electrons the Renormalization Group is
often implemented in k-space. One starts with a theory at an initial cutoff1 0 with
bare (initial) couplings gi(0). In the next step one lowers the running cutoff
and integrates out the fermions in the narrow shell between 0 and . The mode
elimination leads to changes in the couplings and may also generate couplings which
were not present at the initial stage. The interest lies in the behavior of the various
coupling constants and susceptibilities as one lowers the cutoff to zero energy,
i.e. to the Fermi surface. This information is contained in the Renormalization
group equations. Depending on the physics of the system we can scale to zero
energy without encountering any singularity, which signifies that the bare system is
attracted to a weak coupling fixed point; or we find a divergency at a finite cutoff
c. This indicates an instability of the initial theory towards a strong coupling
fixed point. In one dimensional models both situations are known to occur (c.f.
section 3.3). For the t t Hubbard model in two dimensions however the couplingsgenerically flow to strong coupling.
In one dimension the perturbative RG approach is on solid grounds. In general
one encounters logarithmically divergent zero incoming momentum particle-particle
and q=2kF momentum transfer particle-hole diagrams. These can be treated consis-
tently in a one loop approach. For flows to weak coupling the RG approach remains
in its domain of validity. For flows to strong coupling the coupling constants leave
the perturbative regime and other methods are needed for a reliable analysis. Due
1The precise definition of the cutoff depends on the chosen approach. It could be a bandwidth
cutoff, a temperature cutoff, etc.
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to the special structure of the low energy excitations in one dimension we can resort
to the bosonization approach, where the fermionic operators are mapped to bosonic
ones. The resulting bosonic theory is weakly interacting and a semiclassical anal-
ysis yields an accurate description of the low energy physics. A different source of
understanding are the few one dimensional models which are exactly solvable. So
for example the standard Hubbard model for general filling and U/t and the tJmodel at J/t = 2, both solvable by Bethe-Ansatz; the Tomonaga-Luttinger liquid
and the Luther-Emery liquid, both solvable by Bosonization.
The discovery of the high-Tc superconductors in 1986 has sparked a tremendous
amount of research on two dimensional strongly correlated electron models, since
it is believed that superconductivity emerges mainly due to electron-electron inter-
actions, not due phonons as in standard BCS theory. Weak coupling approaches
played an important role from the start: By concentrating on the regions around the
van Hove singularities the two patch approach for the Hubbard model was derived
[49, 50]. In recent years improved RG schemes with a higher k-space resolution (N-
patch schemes) have been developed by several groups [35, 36, 37]. These studies
reported various instabilities but all agreed on the fact that the system scales to
strong coupling. In contrast to one dimension one can not apply the bosonization
mapping in two dimension to the cases of our interest. Exact solutions for non-
trivial models are also lacking. Therefore there is a need for an unbiased method
to complement the RG analysis.
3.2 The method
Our approach is built on a numerical investigation of the Hamiltonian which results
from the asymptotic couplings in the RG procedure. The scheme we developed
consists of three steps:
1. A weak coupling Renormalization Group scheme in kspace is implemented.We calculate the flow of the couplings g(k1, k2, k3) as a function of by
integrating the RG equations. The ratios of the couplings close to the crit-
ical scale are determined. In some cases analytical results for the ratios are
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available.
2. We map the coupling function gc(k1, k2, k3) to a Hamiltonian on a finite mesh
of k-points.
3. The discretized problem is diagonalized exactly with a numerical algorithm.
In practice a Lanczos type algorithm is implemented. This limits the maxi-
mum number of k-space orbitals to about 20. Energies and correlation func-
tions are calculated, enabling us to determine the energy gap structure and
different order parameter susceptibilities directly in a fermionic language.
In the following we discuss some details of the setup of the mesh in k-space and the
mapping of the asymptotic couplings to the k-space Hamiltonian.
3.2.1 Mesh in k-space
The first ingredient of an numerical implementation is the mesh of k-points in
reciprocal space. The mesh consists of a number of patches Np which corresponds to
the number of Fermi points in the case of a one dimensional system, or to a N-patch
approximation in 2D systems. In each of the patches we distribute Nppk k-points
with a fixed momentum assigned. This gives a total number ofk-points Nk=NpNppk .
The distribution of the k-points in a 1D setting with two Fermi points is illustrated
in Fig. 3.1. The k-points are chosen below a bandwidth cutoff , distributed in
a uniform way throughout the allowed region. The momenta of the individual k-
points should respect certain relations: when total incoming momentum zero pairing
instabilities could arise we should have pairs with momenta k and k present, andthe common 2kF instabilities should be respected by the presence of momenta k and
k2kF. These requirements are satisfied with the uniform spacing described above.
In the 1D situation illustrated in Fig. 3.1 the degeneracy of the noninteracting
system with Ne = Nk depends on the number of orbitals per patch Nppk . For N
ppk
even (odd) the noninteracting groundstate is not degenerate (is degenerate). This
will sometimes have an influence on the finite size behavior of the gaps for example.
Note that due to the initial RG procedure on the couplings and the bandwidth
cutoff, our approach is not a simple exact diagonalization of a Fourier transformed
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F-k
EF
kF
k
k
Figure 3.1: Schematic representation of the discretized k-space around the Fermi surface in a
1D geometry with two patches.
problem, but actually a simulation of a system of effective length L = 2/k
Nk.
Our calculations are carried out with Ne=Nk for the groundstate sector. The Fermi
energy EF then lies in the middle of the bandwidth . Gaps are calculated with
respect to that state. The actual filling of the parent systems is encoded in the
asymptotic interactions, e.g. by the presence or absence of umklapp processes,
differences in Fermi velocities, etc.
ky
=
E
-kF A
kF A
-kF B
kF B
FE
Fk
y= 0
Figure 3.2: Schematic representation of the discretized k-space around the four Fermi points in
a ladder geometry.
The mesh of k-points for the two leg Hubbard ladder is analogous to the 1D case,
we just have four patches instead of two (Fig. 3.2). Now umklapp processes are
included as well. This is ensured by the following relation on the longitudinal Fermi
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momenta: |kF,A + kF,B | = . The important wavevector at half filling therefore is(, ).
The situation is different for the 2D two patch case, as illustrated in Fig. 3.3. Due
to the restricted number of available orbitals in the numerical calculation together
with the physical requirements on the momenta very few discretization schemes are
possible. We have chosen the two arrangements in Fig. 3.3 for our calculations.
The noninteracting groundstate has a closed shell structure and all desired k-point
relations are satisfied. The mesh is however not a homogeneous refinement and
could therefore pose some difficulties in the finite size scaling process. Qualitative
results should nevertheless be possible.
Figure 3.3: Discretization of the k-space in the two patch model. The left panel is the choice for8 (resp. 12 with points on the Fermi arcs) k-points. The right panel for 16 (resp. 20) k-points. The
grey points show optional points on the Fermi arcs which could be included in future calculations.
The empty points represent folded ( = ) existing k-points.
In our program code we exploit the conservation ofSztot, the number of particles Ne
and the conservation of momentum to reduce the size of the Hilbert space before
the diagonalization. This gives us the additional advantage to resolve energies as
a function of total momentum. The reduction factor for a subsector with fixedmomentum can be important: e.g. for a two leg Hubbard ladder at half filling
with Nk = 16, Ne = 16, Sztot = 0 and total momentum (0, 0) we reduce the size
of the Hilbert space from (C168 )2 = 165636900 down to 3370670. In comparison
to standard real space exact diagonalizations where the number of different total
momenta is equal to the number of orbitals, our special Hilbert space structure
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allows for much more total momentum sectors. In the one dimensional situation
in Fig. 3.1 with Ne=18 our allowed momenta are clustered with a certain width
around specific momenta ranging from 18kF (all particles on the left branch) to+18kF (all particles on the right branch) in steps of 2kF. (transfer of one particlefrom the left to the right branch).
3.2.2 The coupling function
The Hamiltonian which acts on the mesh of k-points is generically of the following
SU(2) invariant form:
H =k,
(k) ck,ck,
+ 12
k1,k2,k3
V(k1, k2, k3),
ck3,ck4,ck2,ck1,. (3.1)
Where (k) denotes the kinetic energy, is a global coupling constant, the total
volume (usually Nk), V(k1, k2, k3) is the discretized coupling function and k4 =k1 + k2 k3 (modulo umklapp) by momentum conservation. The fact that it onlycontains four-fermion terms can be justified by RG arguments2.
The functional dependence of the coupling function V(k1, k2, k3) is basically deter-
mined by the RG couplings close to the critical scale:
V(k1, k2, k3) = g[c] (Patch(k1), Patch(k2), Patch(k3)) ; (3.2)
Here g[c] denotes the ratios of the diverging couplings and Patch(k) assigns a
patch index to every k-point. If we discretize the coupling function this way we will
generate O(N3k ) different scattering processes. As this number can become quite
large even for small Nk we would like to minimize the number of coupling processes3.
One possible way to reduce the number of couplings is to retain only those processeswhich exactly satisfy the momentum relations of the suspected instabilities. In a
certain sense this amounts to solving an effective reduced Hamiltonian similar to
2The processes with higher order fermionic interactions have a different scaling dimension and
are therefore irrelevant.3For a two leg Hubbard ladder at half filling with Nk = 16 we have 1200 different couplings
in the D-Mott phase.
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k1,
k2,
k ,
k ,4
3
Figure 3.4: The elementary scattering vertex V(k1, k2, k3). Two incoming particles with mo-menta and spin (k1, ) and (k2,
) are scattered to (k3, ) and (k4, ). Momentum is conserved
by k4 = k1 + k2 k3 (modulo umklapp).
mean-field theories, e.g. the reduced BCS Hamiltonian only scatters pairs with
total momentum exactly zero. In a 1D setting we would then only keep processes
with a) k2 = k1 b) k3 = k1 2kF and c) k4 = k1 2kF after the reduction. Thislimits the number of processes to O(N2k ). We have noted in the application of the
present scheme that the finite size behavior of gaps and structure factors depends tosome extent on the discretization of the couplings. For the one dimensional system
in section 3.3 the reduced number of couplings gave results in good agreement
with the theoretical expectations. For the two leg ladder case, where we have an
insulating, fully gapped state without quasi-long range order, we had to include
all the couplings in order to obtain stable, finite gaps. With the restricted set of
couplings the spin and the two particle gap would scale to zero. The single particle
gap however was stable even for the reduced set. These observations might reflectthe fact that the 1D system has a single dominant correlation function, whereas the
two leg ladder has a spin liquid groundstate with several equally dominant short
range correlations.
3.2.3 Observables
As in the well-known real space ED calculations we can measure finite size expec-
tation values of almost any observable. In our approach we are mainly interested
in energy gaps and structure factors related to several types of orders. The energy
gaps we determine are defined as follows:
Spin Gap:
(S=1)(Nk) = E0(Nk, Ne, 1) E0(Nk, Ne, 0) (3.3)
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Single particle gap:
1p(Nk) =1
2
E0(Nk, Ne + 1,
1
2) + E0(Nk, Ne 1, 1
2)
E0(Nk, Ne, 0) (3.4)
Two particle gap:2p(Nk) =
1
2(E0(Nk, Ne + 2, 0) + E0(Nk, Ne 2, 0))
E0(Nk, Ne, 0) (3.5)
where E0(Nk, Ne, Sz) denotes the groundstate energy of the discretized system for a
fixed number of k-points Nk, fixed number of particles Ne and total magnetization
Sz. The energies have to be measured in the appropriate total momentum sector,
but the momentum of the gap is model dependant.
The next observables are the structure factors associated to different order param-
eters. The particle-hole (p-h) instabilities with momentum q and form-factor fA(k)
are defined as follows:
Spin singlet channel :
OACDW =
1
Nk k, fA(k) ck,ck+q, (3.6)
Spin triplet channel (z-component) :
OASDW = 1Nk
k,
1
2fA(k) c
k,ck+q, (3.7)
whereas the particle-particle (p-p) instabilities are defined as:
OASC =
1
Nk k fA(k) c
k,
c
k,
(3.8)
In the singlet channel the particle-hole order parameters correspond to standard
charge density wave (CDW) instabilities for fA(k) 1 (s-wave). The triplet ana-logue corresponds to a spin density wave (SDW). The higher angular momentum
analogues will be discussed in more detail in section 3.4.
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In order to measure the structure factors in the ED approach one calculates the
groundstate |0 and subsequently applies the appropriate operator on the ground-state:
SX = 0|OXOX |0 = |OX |0|2 (3.9)It is also possible to calculate dynamical response functions (e.g. single particle
spectral functions or dynamical spin structure factors) with the continued fraction
method in the present scheme.
3.3 Test case : a one-dimensional problem
Let us illustrate our approach on a simple one-dimensional problem: the 1D Fermi-
gas with two Fermi points. The relevant couplings are labeled g1 to g4 (illustrated
in Fig. 3.5). The g1 processes denote backscattering processes. The g2 and g4
processes are of the forward scattering type. Finally the g3 processes are so called
umklapp processes which violate momentum conservation in general, but are allowed
at special fillings, e.g. at half filling. In the following we consider a system at a
g1
g3
g2
-k k FF
-k kF F-k k
k-k F F
FF
g4
Figure 3.5: The g-ology of the 1D spinful Fermi gas with two Fermi points. g1 denotes backscat-
tering processes, g2 forward scattering, g3 umklapp scattering and g4 chiral forward scattering
processes.
generic filling (away from half filling) and we therefore neglect the g3 coupling. For
simplicity we also discard the g4 processes. Their effect is more on a quantitative
level, they renormalize velocities but are not expected to change the overall phase
diagram.
The phase diagram of the so called g1g2 model has been the subject of many studiesin the 1970s. Reviews can be found in [29, 34]. The one-loop renormalization group
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g1=2g2
g2
g1
sSC CDW
SDWpSC
Figure 3.6: Phase diagram of the g1g2 model. The phases are characterized by the leadingalgebraic correlation function. The two phases for g1 > 0 are gapless. The two regions with g1 < 0
flow to strong coupling and develop a spin gap.
equations for the g1g2 model have been derived as follows:
g1 = 1
g21
g2 = 12
g21, (3.10)
with g = dg/dl and l = ln(/0) +. These equations have a solution in
closed form:
g1(l) =g1(0)
1 + g1(0)l
g1(l) 2g2(l) = const. (3.11)
The flow of the couplings thus depends on the sign of the initial coupling g1(0).
If g1(0) > 0 (repulsive backscattering) we scale to a weak coupling fixed point:
the Luttinger liquid [g1() = 0, g2() = g2(0) 12g1(0)]. If however g1(0) < 0
(attractive backscattering) the weak coupling fixed point is unstable and we flow
to strong coupling [g1, g2 ]. Using bosonization it has been shown that thestrong coupling fixed point develops a spin gap. The weak coupling fixed point
has no gaps. The leading correlation functions have been determined; the resulting
phase diagram is shown in Fig. 3.6.
As an example we calculate the various gaps in the lower left region of the phase
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0 0.5 1 1.5 2
0
1
2
3
4
5
6
G
ap/
spin gap
0 0.5 1 1.5 2
0
1
2
3
4
5
6
g1=-2, g
2=-2
two particle gap
nk=8
Nk=10
N=12
0 0.5 1 1.5 2
0
1
2
3
4
5
6
single particle gap
Figure 3.7: Gaps as a function of and system size in the Luther-Emery part (dominant
superconductivity correlations and a finite spin and one particle gap) of the g1g2 phase diagram.
diagram, i.e g1 < 0, g2 < g1/2. The dominant correlation function is s-wave sin-
glet pairing. For this state we expect a finite spin and single particle gap but no
two particle gap in analogy to a superconducting state. Our numerical results are
shown in Fig. 3.7. The different curves in each panel represent different numbers of
k-points. The horizontal axis denotes the interaction strength . This parameter
allows us to tune between the noninteracting limit and the fully interacting limit,
where kinetic energy plays almost no role anymore. The evolution of the gaps as a
function of indicates also where the finite size effects due to the discretization be-
come unimportant. Note that the Nk=10 system has zero gaps due to a degenerate
groundstate at =0. The finite size behavior of the gap curves strongly suggests a
finite spin and single particle gap, while the two particle gap scales to zero. At the
present stage we do not attempt to measure the gaps quantitatively, but we merely
determine the qualitative gap signature. We have also calculated the gaps in the
other regions of the phase diagram. It turned out that the results for g1 > 0 are
less regular than for g1 < 0. This could be related to the fact that for the latter
case a large spin gap develops.
We calculated also the structure factors corresponding to charge density wave
(CDW), spin density wave (SDW), singlet superconductivity (sSC) and triplet su-
perconductivity (pSC). The results in Fig. 3.8 display dominant correlation func-
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CDW SDW sSC pSC0
2
4
6
8
10
Structure
Factors
[a.u.]
g1=-2, g
2=-2
CDW SDW sSC pSC0
2
4
6
8
Structure
Factors
[a.u.]
g1=2, g2=-2
Nk=8
Nk=10
Nk=12
Nk=14
CDW SDW sSC pSC0
2
4
6
8
10g
1=-2, g
2=2
CDW SDW sSC pSC0
2
4
6
8g1=2, g2=2
Figure 3.8: Structure factors of different orderparameters for the g1g2 model at four differentpoints in the phase diagram. The phases in the two upper panels scale to weak coupling in the
RG process (g1 > 0), while the phases in the lower panels scale to strong coupling region (g1 < 0).
The dominant correlation function agrees with phase diagram in Fig. 3.6. The finite size behavior
is much more regular in the spin gapped strong coupling phases.
tions consistent with the phase diagram in Fig. 3.6. The finite size behavior is very
regular for the two phases with g1 < 0. For the phases with g1 > 0 we detect a
systematic difference between the systems with and without orbitals at the Fermi
energy.
We conclude from the application of our scheme to the 1D g1g2 model that theresults are consistent with the analytical results. Especially in the phases where
the RG flow diverges to strong coupling we get good agreement.
3.4 The two-leg Hubbard ladder at half filling
The two leg Hubbard ladder is an interesting system on the path from one-dimensional
to two-dimensional physics. At half filling the system is insulating (finite charge
gap, incompressible) for any U > 0. In the strong coupling limit (U t) the modelmaps to a two leg spin ladder (also an insulator) with antiferromagnetic exchange
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interaction (J t2/U). This system is a nice realization of a resonating valencebond (RVB) spin liquid state in the spirit of Andersons early proposal [22]. It
shows exponentially decaying spin correlations and an excitation gap to singlet and
triplet modes. The strong coupling limit is well understood, in particular in the
limit where the coupling on the rung is strongest. The groundstate is then well
approximated by a rung singlet product state. We call such a state with a charge
gap and short range spin correlations an Insulating Spin Liquid (ISL).
Our present interest in the two leg ladder is twofold. First we want to test our
numerical scheme on a different system where reliable results are available. As we
will see the variety of potential phases arising in the weak coupling phase diagram
is fascinating. The fact the we will encounter phases without power-law correlation
functions (only short range order) will mark a difference to the 1D model discussed
before. The second motivation comes from recent results of N-patch RG calcula-
tions of the 2D tt Hubbard model. [37]. The properties of the flow to strongcoupling in the region between antiferromagnetism and d-wave superconductivity
were reminiscent of the two leg Hubbard ladder and a scenario based on the forma-
tion of an ISL in parts of the Brillouin zone was put forward [37, 40]. Our aim is to
characterize the relevant phases of the ladder model within the present numerical
scheme in order to compare to the more complex 2D models later on.
In the following we focus again on the limit of weak coupling (U t). This limithas been discussed in detail in a series of publications [42, 40, 43, 44, 45]. Here
we just give a brief outline of the weak coupling RG results before we compare our
numerical results to the analytical predictions.
Let us first discuss the noninteracting starting point. The band structure of the
nearest neighbor tight binding model on the two leg ladder reads:
(kx, ky) = t cos(ky) 2t cos(kx), ky {0, } (3.12)
where t (t) denotes the hopping amplitude on the legs (rungs). The dispersion is
plotted in Fig 3.9. At half filling and for t/t < 2 both bands are partially filled.
There are four Fermi points present. The Fermi velocities on the two bands are
equal at half filling and the Fermi wavevectors kAF, kBF add up to .
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-1 -0.5 0 0.5 1
kx/
-4
-2
0
2
4
(k
x,
ky
)/t
ky=
ky=0
-kF
Ak
F
A
kF
B-k F
B
Figure 3.9: Noninteracting dispersion of a two leg Hubbard ladder with t/t = 1 at half filling.
kAF + kBF add up to . The Fermi velocities are equal on all four branches (valid for t/t < 2).
The relevant processes4 in the g-ology of the two leg Hubbard ladder at half filling
are sketched in Fig. 3.10 (notation according to [39, 40]). The g3 couplings denote
umklapp processes, i.e processes where the total incoming and the total outgoing
momenta differ by a reciprocal lattice vector. We discard the completely chiral g4
processes in our analysis because they are not important on a qualitative level, i.e.
their main effect is to renormalize velocities.
The one loop RG equations at half filling have the following form [42, 39]:
g1x = g21x + g2yg1p + g
23x g3xg3c
g2x =1
2
g21x + g
22y + g
21p g23c
g3x = 2g1xg3x g1xg3c g2xg3x g2eg3x g2yg3eg3c = 2g1eg3c + g1pg3e g1eg3x g2yg3e g2eg3c g2xg3cg2y = g1xg1p + g2yg2x g2yg2e g3eg3xg1p = 2g1eg1p + g1xg2y
g1eg2y + g1pg2x
g1pg2e + g3eg3c
g3eg3x
g1e = g21e + g
21p g2yg1p + g23c g3cg3x
g2e =1
2
g21e g23e g23x g22y
4Here we call a process relevant if is logarithmically divergent in second order perturbation
theory. The solution of the RG equation will show wether the process is also relevant in the RG
sense.
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g1x
g2x
g3e
g1e
g2e
g3c
g1p
g2y
g3x
Figure 3.10: Coupling constants labelling the relevant processes connecting the four Fermi points
at half filling. The g3s are umklapp processes. The couplings g1x, g2x, g1p and g2y denote cooper
processes, couplings g3e, g2e, g2y and g3x are SDW type processes and finally g3e, g1e, g3c and g1p
are CDW type processes
g3e = 2g1pg3c g1eg3e g2yg3c g1pg3x g2yg3x 2g2eg3e, (3.13)
where g = dg/d and we decrease from an initial cutoff 0.
For general initial couplings gi(0) these RG equations are too complicated for an
analytical solution and we need to perform a numerical integration. It is possible
however to obtain solutions in closed form for special values of the initial couplings.
The ansatz
gi() =gi(0)
log(/c)(3.14)
is a solution of 3.13, provided that all gis and gis are replaced by gi(0) and the
gi(0) solve the resulting algebraic system of equations. c denotes the critical scale
where all the couplings diverge. It depends on the initial couplings and scale. Note
that these solutions dictate the initial couplings at the scale 0. Surprisingly the
numerical integration generally yields divergencies which are well captured by this
ansatz. In particular the couplings diverge with fixed ratios. It is the task of the
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numerical integration to decide for which initial couplings we flow to which special
solution of the above form.
3.4.1 Repulsive U - The D-Mott phase
10-12
10 -9
10 -6
-20
2
4
6
g
/t
/ t
Figure 3.11: Flow of the nine coupling constants of the two leg Hubbard ladder at half filling for
purely repulsive initial couplings. The flow diverges at a critical scale c 0.5 1012 t. The dark,
dashed couplings are the d-wave pairing type, the dark, solid couplings the AF couplings and thelight, dashed couplings are d-density wave couplings diverging more slowly than the others. The
initial couplings at 0 = 0.5 t were 0.1 t. (Plot taken from [40])
In Fig. 3.11 we plot the flow of the couplings for purely repulsive initial couplings
[gi(0) = U] as would be the case for a simple repulsive Hubbard model. Seven out
of nine couplings diverge at a critical scale c. The remaining two couplings diverge
not as fast as the others. The ratios of the diverging couplings for this particular
flow are as follows:
g1x g1e g1p g2x g2e g2y g3e g3c g3x
-1 0 1 -1/2 1/2 1 1 0 1
The fixed ray was characterized using Bosonization in [42]. The resulting phase has
only short range order, no power-law correlations. Nevertheless some correlation
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functions are enhanced, while others are suppressed with respect to the noninteract-
ing limit. In the case at hand enhanced spin density wave correlations and d-wave
superconducting correlations are predicted. The staggered flux correlations (dCDW
or d-density wave) are expected to be enhanced as well. All gaps in this phase have
been predicted to be same, based on a dynamically generated SO(8) symmetry. The
phase was termed D-Mott in [42].
We have again diagonalized the corresponding fermionic Hamiltonian and deter-
mined the gap structure. Our results are show in the uppermost panel of Fig. 3.12.
Interestingly the spin and the two particle gap are identical, while the single particle
gap is slightly larger. On the level of our expected accuracy we consider them to
be the same. The spin and the two particle gap show small finite size corrections,
but are clearly consistent with a finite gap as Nk
.
We then calculated the s-wave and d-wave component of the charge and spin density
wave and the pairing correlations. The resulting structure factors are shown in the
upper left plot in Fig. 3.14. We indeed find enhanced sSDW, dCDW and dSC
short range correlations as expected. The size dependence of the structure factors
is significantly reduced compared to the g1g2 model where the correlations werequasi long ranged.
3.4.2 Zoo of insulating phases
The general weak coupling phase diagram of the two leg Hubbard ladder at half
filling has attracted much interest in recent years. Lin, Balents and Fisher [42]
reported new phases in addition to the D-Mott phase discussed before: an ordered
charge density wave state (CDW), an ordered state with staggered circulating or-
bital currents (dCDW) 5 and a s-wave analog to the D-Mott phase (S-Mott). The
dCDW phase is also known as a staggered flux phase or orbital antiferromag-
net and can be viewed as a charge density wave state with angular momentum
l = 2 (d-density wave). It is also intensively discussed as a candidate ground-
state in the pseudogap region of the underdoped cuprates. In two recent preprints
5This phase was misinterpreted in the original paper, but later identified correctly in [43]
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[44, 45] the zoo of insulating phases at half filling has been expanded by other den-
sity wave states with non-zero angular momentum [pCDW ( l = 1) and fCDW
( l = 3)] and two quantum disordered superconductors (S-Mott and D-Mott),
which presumably differ from the unprimed states in topology.
For the first four phases the asymptotic coupling ratios have been determined in
[42] and we recast them in our notation in table 3.1. It was noticed by Lin, Balents
and Fisher that for a large set of initial couplings the flow is attracted to a manifold
with enhanced dynamical symmetry. In particular the four fixed rays in table 3.1
correspond each to an integrable SO(8) Gross-Neveu model. This allows a detailed
characterization of the phases.
Phase g1x g1e g1p g2x g2e g2y g3e g3c g3x
D-Mott -1 0 1 -1/2 1/2 1 1 0 1S-Mott -1 0 -1 -1/2 1/2 -1 -1 0 1
sCDW 0 -1 -1 0 0 0 -1 -1 0
dCDW 0 -1 1 0 0 0 1 -1 0
Table 3.1: Ratios of the coupling constants for the four dominant fixed rays on the
SO(5) manifold.
We map the asymptotic couplings to a discrete lattice in k-space with 8, 12 and
16 k-points. This corresponds to 2,3 and 4 k-points per patch. The noninteracting
( = 0) groundstate is nondegenerate for the systems with 8 and 16 k-points. We
calculate the gap structure in each phase by plotting the evolution of the three
gaps as a function of . The parameter allows us to follow the gaps between
the noninteracting limit and the limit where kinetic energy is not so important
anymore. The results are shown in Fig. 3.12. A common feature is that for each
phase the triplet gap and the two particle gap are equal. The two disordered and
the two ordered states have exactly the same energy gaps respectively. The gaps in
the first two phases (D-Mott and S-Mott) are small, but remain finite for Nk .This can easily be inferred in our plots. According to the SO(8) symmetry of the
asymptotic Hamiltonian all three gaps in these two phases should be of equal size.
That is however not verified exactly in our calculations, as the single particle gap is
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slightly larger than the other gaps. The reason for this could be attributed to our
discretization scheme. It might not respect all of the the nonlocal SO(8) symmetry
elements of the Hamiltonian.
The ordered phases have very clear-cut gaps with small finite size corrections. The
spin gap and two particle charge gap are roughly twice as large as the single parti-
cle gap. In addition we find that the groundstate is twofold degenerate: the lowest
singlets at momentum (0, 0) and (, ) have exactly the same energy. Such de-
generacies are expected for a spontaneous translation symmetry breaking scenario
(Z2 symmetry breaking). In other situations the two states are very close in en-
ergy because of the finite tunneling amplitude in finite systems but are not exactly
degenerate like in the present case.
0 1 20
2
4
6
CDW
0 1 20
2
4
6
dCDW
0 1 20
1
2
3
S-Mott
0 1 20
1
2
3
D-Mott
Nk=8
Nk=12
Nk=16
Spin Gap /
0 1 2
Interaction Strength
0 1 2
0 1 2
0 1 2
Two Particle Gap /
0 1 2
0 1 2
0 1 2
0 1 2
Single Particle Gap /
Figure 3.12: Finite size scaling of the three relevant gaps in the four phases on the SO(5)
manifold discussed by Lin, Balents and Fisher. The finite size scaling of the gaps in the quantum
disordered phases (D-Mott and S-Mott) is consistent with small but finite gaps in the limit Nk . The two ordered phases display very clear gaps with only small finite size corrections. Contraryto the first two phases the ordered phases have a twofold degenerate groundstate, i.e. a singlet at
momentum (, ) is degenerate with the lowest state at (0, 0). This is another evidence for long
range order.
Let us now investigate the different correlation functions. We first discuss the
particle-hole instabilities with momentum q=(, ) and different formfactors fA(k).
The values of the formfactors fA(k) only depend on the patch index (see Fig. 3.13).
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The s-wave channel corresponds to the well-known charge density wave (CDW)
and spin density wave (SDW) order parameters. The higher angular momentum
analogues are also well known by now. The d-density wave or staggered flux
instability corresponds to the CDW operator with d-wave formfactor. The p-wave
CDW order is a spin-Peierls type instability with an alternation in the kinetic
energy on the leg bonds and a phase shift between the legs6. Finally the fCDW
state has been identified as a directed current phase with currents flowing across
the diagonals of a plaquette [44, 45]. Each of these channels also has a triplet
counterpart. D-wave SDW correlations for example are found in the dominant
vector chirality region on the two leg spin ladder with cyclic four spin exchange
(discussed in section 4.1 of this thesis). They can be understood as staggered spin
currents. In the particle-particle channel singlet pairing correlations are measured.
This is the case for s-wave and d-wave formfactors. The p-wave and f-wave cases
would correspond to triplet cooper pairs.
1 1
11
1 1
-1 -1 -1
-11
1
-1
-1
1
1
f-wavep-waved-waves-wave
Figure 3.13: Formfactors fA(k) for the two leg Hubbard ladder.
Our results are summarized in Fig. 3.14 for each phase. We measured the structure
factor: i.e diverging quantities ( Nk) signal true long range order, while saturatedbehavior indicates short range order. We discuss the results in each of the phases
in the following:
D-Mott - The D-Mott phase discussed before has enhanced SDW, dCDW(d-density wave) and dSC response.
S-Mott - The S-Mott phase was described as a disordered s-wave supercon-ductor in earlier work. Our results agree with this, the s-wave Cooper response
6This state is also known as bond order wave or bond charge density wave
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is clearly enhanced. We detect equally enhanced response in the sCDW and
the dSDW channels, which has not been reported in earlier studies.
dCDW - The true long range order in this phase is reflected in an almostperfect scaling of the staggered flux structure factor with system size Nk. No
other order parameter is enhanced.
CDW - The same statements as for the dCDW phase are valid. There is justan interchange in the formfactors from d-wave to s-wave.
sCDW dCDW sSDW dSDW sSC dSC0
5
10
15
20
S
tructure
Factors
[a.u.]
dCDW / Staggered Flux
sCDW dCDW sSDW dSDW sSC dSC0
1
2
3
4
Structure
Factors
[a.u.]
DMott
Nk
=8
Nk=12
Nk=16
sCDW dCDW sSDW dSDW sSC dSC0
5
10
15
20CDW
sCDW dCDW sSDW dSDW sSC dSC0
1
2
3
4SMott
Figure 3.14: Finite size scaling of the different structure factors in the four dominant phases
on the SO(5) manifold discussed by Lin, Balents and Fisher. The finite size scaling behavior in
the quantum disordered phases (D-Mott and S-Mott), where they are slowly approaching a finite
value, is clearly different compared to the true ordered phases [Staggered Flux (dCDW) and charge
density wave (CDW)], where they are proportional to the system size, thereby signalling long range
order. The structure factors are normalized to the values in the noninteracting groundstate. is
set to 2.
Additional phases at half filling
Recent work [44, 45] reanalyzed the weak coupling phase diagram of the Hubbard
ladder by bosonization and reported four additional phases as possible groundstates.
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Phase g1x g1e g1p g2x g2e g2y g3e g3c g3x
D-Mott -1 0 1 -1/2 1/2 1 -1 0 -1
S-Mott -1 0 -1 -1/2 1/2 -1 1 0 -1
pCDW 0 -1 -1 0 0 0 1 1 0
fCDW 0 -1 1 0 0 0 -1 1 0
Table 3.2: Ratios of the coupling constants for the four new phases discussed by
Wu et al.[44] and Tsuchiizu et al.[45].
The postulated phases encompass the pCDW and fCDW ordered phases and the
two disordered phases S-Mott and D-Mott. We have checked within our approach
that by using the asymptotic couplings in table 3.2 one obtains states with the
correct signals in the correlation function. The pCDW and the fCDW are long
range ordered phases as the sCDW and dCDW phases discussed before. The only
difference between these ordered phases is the formfactor. The gaps behave exactly
the same way. The primed Mott-states are slightly more subtle to characterize.
They are disordered phases and all correlation functions decay exponentiall