Two superconducting phases in the d=3 Hubbard model

Abstract

The phase diagram of the d=3 Hubbard model is calculated as a
function of temperature and electron density 〈ni〉,
in the full range of densities between 0 and 2 electrons per site,
using renormalization-group theory. An antiferromagnetic phase
occurs at lower temperatures, at and near the half-filling density
of 〈ni〉= 1. The antiferromagnetic phase is
unstable to hole or electron doping of at most 15%, yielding to two
distinct“τ" phases: for large coupling U/t, one such phase
occurs between 30–35% hole or electron doping, and for small to
intermediate coupling U/t another such phase occurs between
10–18% doping. Both τ phases are distinguished by non-zero
hole or electron hopping expectation values at all length scales.
Under further doping, the τ phases yield to hole- or
electron-rich disordered phases. We have calculated the specific
heat over the entire phase diagram. The low-temperature specific
heat of the weak-coupling τ phase shows an exponential decay,
indicating a gap in the excitation spectrum, and a cusp singularity
at the phase boundary. The strong-coupling τ phase, on the
other hand, has a critical exponent α≈-1, and an
additional peak in the specific heat above the transition
temperature possibly indicating pair formation. In the limit of
large Coulomb repulsion, the phase diagram of the tJ model is
recovered.

Full text

Page 1

arXiv:cond-mat/0503226v2 [cond-mat.str-el] 15 Mar 2005
Two Superconducting Phases in the d = 3 Hubbard Model:
Phase Diagram and Specific Heat from Renormalization-Group Calculations
Michael Hinczewski and A. Nihat Berker
Department of Physics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey,
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A., and
Feza G¨ ursey Research Institute, T¨UBITAK - Bosphorus University, C ¸engelk¨ oy 81220, Istanbul, Turkey
The phase diagram of the d = 3 Hubbard model is calculated as a function of temperature
and electron density ?ni?, in the full range of densities between 0 and 2 electrons per site, using
renormalization-group theory. An antiferromagnetic phase occurs at lower temperatures, at and
near the half-filling density of ?ni? = 1. The antiferromagnetic phase is unstable to hole or electron
doping of at most 15%, yielding to two distinct ”τ” phases: for large coupling U/t, one such phase
occurs between 30-35% hole or electron doping, and for small to intermediate coupling U/t another
such phase occurs between 10-18% doping. Both τ phases are distinguished by non-zero hole or
electron hopping expectation values at all length scales. Under further doping, the τ phases yield
to hole- or electron-rich disordered phases. We have calculated the specific heat over the entire
phase diagram. The low-temperature specific heat of the weak-coupling τ phase shows a BCS-
type exponential decay, indicating a gap in the excitation spectrum, and a cusp singularity at the
phase boundary. The strong-coupling τ phase, on the other hand, has characteristics of BEC-type
superconductivity, including a critical exponent α ≈ −1, and an additional peak in the specific heat
above the transition temperature indicating pair formation. In the limit of large Coulomb repulsion,
the phase diagram of the tJ model is recovered.
PACS numbers: 74.72.-h, 71.10.Fd, 05.30.Fk, 74.25.Dw
I.INTRODUCTION
The Hubbard model [1] is the simplest realistic (in that
it retains particulate dynamics) model of electronic con-
duction systems. This model should constitute a fair
description for many real solid-state physics systems and
a starting-point description for those systems with added
complexities such as quenched randomness, frustration,
and/or spatial anisotropy. The first query that comes
to mind, in the study of either experimental or model
systems, is on the phase diagram, as a function of phys-
ical parameters such as temperature and density. Nev-
ertheless, until recently [2], no estimates, let alone (be it
approximate) solutions, were ventured on the phase dia-
gram of the Hubbard model at dimensions greater than
d = 1, at temperatures greater than T = 0, and densities
away from half-filling.
The first approach to a phase diagram problem, in the
past before the advent of renormalization-group theory
[3], had been through a mean-field approximation. How-
ever, such method is not useful for the Hubbard model,
since, where the characteristic phenomena occur away
from half-filling, the off-diagonal term in the Hamiltonian
plays a determining role, as we shall see below. There is
no ready way to deal with such a dominant quantum
mechanical effect using mean-field theory. On the other
hand, renormalization-group theory, which some time
ago has excelled over mean-field theory in phase diagram
studies, is effective. Previous renormalization-group cal-
culations have concentrated on studying the Hubbard
model in lower dimensions, at zero temperature, or at
half-filling: The zero-temperature (ground-state) prop-
erties were successfully obtained in d = 1,2,3.[4, 5] In
d = 1 at half-filling, the thermodynamic properties were
accurately calculated for finite temperatures.[6] In cases
where comparison is possible due to the availability of
exact results in d = 1, the renormalization-group results
have proven to be very accurate, coming to within about
1% of the exact results.[5, 6] In d = 2 at half filling,
it was found that no phase transition occurs as a func-
tion of temperature.[7, 8] This result was later extended
to other fillings in d = 2 [9] and confirmed by quantum
Monte Carlo calculations [10]. In d = 3 at half filling, an
antiferromagnetic phase transition as a function of tem-
perature was obtained.[8] One calculation done in d = 3
at finite temperature and arbitrary chemical potential
[9] did not obtain the ”τ” phase reported below and in
Ref.[2].
The physics of the Hubbard model in the limit of large
Coulomb repulsion is believed to be described by the
tJ model [11, 12]. Application of renormalization-group
theory to the entire density range of the tJ model at
finite temperatures in d = 3 has yielded [13, 14], be-
tween 30-40% vacancies from ?ni? = 1, a novel (dubbed
”τ”) phase in which the electron hopping strength in the
Hamiltonian renormalizes to infinity under repeated scale
changes, while the system remains partially filled. The
calculated topology of the phase diagram, including near
the τ phase a first-order phase transition that is very nar-
row (less than 2% jump in the electron density) and an
antiferromagnetic phase that is unstable to at most 10%
vacancies from ?ni? = 1, is indeed reminiscent of exper-
imental phase diagram determinations with lanthanide
oxides [15].
While the studies above [4, 5, 6, 7, 8, 9, 13, 14] have
used position-space renormalization-group approaches,
there has recently been a revival of interest in Wilson
perturbative renormalization-group methods applied to

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2
correlated fermion problems. These methods have long
been known to be successful for one-dimensional sys-
tems [16, 17] and, in the last few years, for the d = 2
Hubbard model, they have yielded antiferromagnetic in-
stabilities near half-filling and superconducting instabil-
ities at smaller densities [18, 19, 20, 21, 22, 23]. Because
of the perturbative nature of these treatments, their pre-
dictions are strictly valid only in the case of weak cou-
pling. The position-space renormalization-group method
presented in this paper appears to work over the entire
range of coupling strengths, as seen below, and yields
definite phase diagrams and thermodynamic functions.
In fact, our approach makes an interesting predic-
tion for the evolution of the Hubbard phase diagram as
coupling is increased. We find two distinct τ phases,
one occurring at small to intermediate coupling and the
other, inclusive of the tJ model τ phase, occurring at
strong coupling. From an analysis of their specific heat
behaviors, we find that the two τ phases respectively
have characteristic properties of a weakly-coupled BCS-
type and a strongly-coupled BEC-type superconducting
phase. Since high-Tcmaterials share aspects of both lim-
its, and are thought to lie in some intermediate coupling
range [24], our prediction for the Hubbard phase diagram
may be directly relevant to the physics of high-Tcsuper-
conductors.
II.THE HUBBARD MODEL
The Hubbard model is defined by the Hamiltonian
−βH = − t
?
?
?ij?,σ
?
ni↑ni↓+ µ0
c†
iσcjσ+ c†
jσciσ
?
ni,
(1)
− U0
i
?
i
with β = 1/kT, describing electron conduction on a
d-dimensional hypercubic lattice. Here c†
spectively are creation and annihilation operators, obey-
ing anticommutation rules, for an electron with spin
σ = ↑ or ↓ at the site i of the lattice; niσ = c†
and ni= ni↑+ ni↓are electron number operators. Each
lattice site can accommodate up to two electrons with
opposite spins. The index ?ij? denotes summation over
all nearest-neighbor pairs of sites. The three terms of
this Hamiltonian respectively incorporate kinetic energy
(parametrized by the electron hopping strength t), on-
site Coulomb repulsion (with coefficient U0 > 0), and
chemical potential µ0. It is convenient for our purposes
to rearrange Eq.(1) into an equivalent Hamiltonian by
iσand ciσ re-
iσciσ
grouping into a single lattice summation:
−βH =
?
−U (ni↑ni↓+ nj↑nj↓) + µ(ni+ nj)}
?
?ij?
?
−t
?
σ
?
c†
iσcjσ+ c†
jσciσ
?
(2)
≡
?ij?
{−βH(i,j)} .
The interaction constants are trivially related by U =
U0/2d,µ = µ0/2d, and we have hereby exhibited the
individual-pair Hamiltonian −βH(i,j).
III.RENORMALIZATION-GROUP
TRANSFORMATION
A. Exact Formulation in d = 1
For d = 1 (with lattice sites i = 1,2,3,...), the Hub-
bard Hamiltonian in Eq.(2) takes the form
−βH =
?
i
{−βH(i,i + 1)},(3)
for which an exact renormalization-group transformation
can be formulated. In terms of matrix elements, this
exact transformation is [13]
?u1u3u5···|e−β′H′|v1v3v5···? =
?
w2,w4,w6,...
?u1w2u3w4u5w6···|e−βH|v1w2v3w4v5w6···?,
(4)
where ui, vi, and wiare state variables for lattice site i.
These variables range over the set {◦, ↑, ↓, ?}, by which
we represent the no electron, a single electron with spin
up, a single electron with spin down, and doubly occu-
pied states. Here and below, the quantities referring to
the renormalized (rescaled) system are denoted with a
prime. The transformation in Eq.(4) eliminates half of
the degrees of freedom in the system, while exactly pre-
serving the partition function (Z′= Z). However, the
transformation cannot be readily implemented, due to
the non-commutativity of the operators in the Hamilto-
nian.

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3
B.Approximation in d = 1
The renormalization-group transformation formulated
in Sec.IIIA is implemented approximately, as follows:
Trevene−βH=Trevene
?
?even
i{−βH(i,i+1)}
=Trevene
i
{−βH(i−1,i)−βH(i,i+1)}
≃
even
?
even
?
?even
i
Trie{−βH(i−1,i)−βH(i,i+1)}
=
i
e−β′H′(i−1,i+1)
≃e
i
{−β′H′(i−1,i+1)}= e−β′H′.
(5)
In the two approximate steps, marked by ≃ in Eq.(5), we
ignore the non-commutation of operators separated be-
yond three consecutive sites of the unrenormalized sys-
tem. Since each of these two steps involves the same
approximation but in opposite directions, some mutual
compensation can be expected.
approximation at predicting finite-temperature behavior
has been verified in earlier studies of quantum spin sys-
tems [25, 26].
The algebraic content of the renormalization-group
mapping can be extracted from Eq.(5) as
The success of this
e−β′H′(i,k)= Trje−βH(i,j)−βH(j,k),(6)
where i,j,k are three consecutive sites of the unrenormal-
ized system. The operators −β′H′(i,k) and −βH(i,j)−
βH(j,k) act on the space of two-site and three-site states
respectively, so that, in terms of matrix elements,
?uivk|e−β′H′(i,k)|¯ ui¯ vk? =
?
where ui,wj,vk, ¯ ui, ¯ vkare single-site state variables.
Eq.(7) indicates the contraction of a 64 × 64 matrix on
the right into a 16×16 matrix on the left. This is greatly
simplified by the use of two- and three-site basis states
that block-diagonalize respectively the left and right sides
of Eq.(7). These basis states are the eigenstates of to-
tal particle number, total spin magnitude, total spin z-
component, and parity. We denote the set of 16 two-site
eigenstates by {|φp?} and the set of 64 three-site eigen-
states by {|ψq?}, and list them in Tables I and II. Eq.(7)
is rewritten as
wj
?uiwjvk|e−βH(i,j)−βH(j,k)|¯ uiwj¯ vk? , (7)
?φp|e−β′H′(i,k)|φ¯ p? =
?
¯ v,w
u,v,¯ u,
?
q,¯ q
?φp|uivk??uiwjvk|ψq??ψq|e−βH(i,j)−βH(j,k)|ψ¯ q?·
?ψ¯ q|¯ uiwj¯ vk??¯ ui¯ vk|φ¯ p? . (8)
In the above equation, with the eigenstates shown in Ta-
bles I and II, the largest block in ?φp|e−β′H′(i,k)|φ¯ p? is
2×2 and the largest block in ?ψq|e−βH(i,j)−βH(j,k)|ψ¯ q? is
4×4. (In previous work [2], some matrix elements in these
blocks were incorrectly derived). Eq.(8) yields eleven in-
dependent elements for the matrix ?φp|e−β′H′(i,k)|φ¯ p? of
the renormalized system. These we label γp, as shown in
Appendix A. The values of the γpin terms of the matrix
elements of the unrenormalized system, dictated by the
right-hand side of Eq.(8), are also given in Appendix A.
n p
0 +
1 + 1/2 1/2 |φ2? =
1 − 1/2 1/2 |φ4? =
2 +0
2 −
0
s
0
ms
0
Two-site basis states
|φ1? = | ◦ ◦?
1
√2{| ↑ ◦? + |◦ ↑?}
1
√2{| ↑ ◦? − |◦ ↑?}
|φ6? =
|φ7? =
|φ8? =
|φ9? = | ↑↑?
|φ10? =
1
√2{| ?↑? + | ↑??}
1
√2{| ?↑? − | ↑??}
|φ16? = | ???
0
0
1
√2{| ? ◦? + |◦ ??}
1
√2{| ? ◦? − |◦ ??},
1
√2{| ↑↓? − | ↓↑?}
2 +
2 +
3 + 1/2 1/2 |φ12? =
3 − 1/2 1/2 |φ14? =
4 +0
1
1
1
0
1
√2{| ↑↓? + | ↓↑?}
0
TABLE I: The two-site basis states used in the derivation
of the recursion relations, in Eq. (8). In these basis states,
e−β′H′(i,k)is diagonal, with the exception of a 2 × 2 block
involving |φ6? and |φ8?. The corresponding particle number
(n), parity (p), total spin (s), and total spin z-component
(ms) quantum numbers are also given. The states |φ3?, |φ5?,
|φ11?, |φ13?, |φ15? are obtained by spin reversal from |φ2?,
|φ4?, |φ9?, |φ12?, |φ14?, respectively.
C.Hamiltonian Closed Form under
the Renormalization-Group Transformation
Since eleven interaction strengths can be indepen-
dently fixed by the eleven γp, the Hamiltonian −β′H′
which is embodied in Appendix A has a more general
form than that of the Hubbard Hamiltonian in Eq.(2).
This generalized form of the pair Hamiltonian is
−βH(i,j) =
−
?
σ
[t0hi−σhj −σ+ t1(hi−σnj −σ+ ni−σhj −σ)
+t2ni−σnj −σ]
?
− U (ni↑ni↓+ nj↑nj↓) + µ(ni+ nj) + J?Si·?Sj
+ V2ninj+ V3(ni↑ni↓nj+ ninj↑nj↓)
+ V4ni↑ni↓nj↑nj↓ + G ,
?
c†
iσcjσ+ c†
jσciσ
?
?
− tx
c†
i↑cj↑c†
i↓cj↓+ c†
j↑ci↑c†
j↓ci↓
(9)
where hiσ ≡ 1 − niσ is the hole (vacancy) operator
and?Si =?
σ,¯ σc†
iσ? sσ¯ σci¯ σ, with ? sσ¯ σ the vector of Pauli
spin matrices, is the spin operator at site i. In general,

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4
n p
0 +
1 + 1/2 1/2
1 − 1/2 1/2
2 +0
s
0
ms
0
Three-site basis states
|ψ1? = | ◦ ◦◦?
|ψ2? = |◦ ↑ ◦?, |ψ3? =
|ψ6? =
|ψ8? =1
|ψ9? = |◦ ? ◦?, |ψ10? =
|ψ11? =1
|ψ12? =
|ψ13? =
|ψ14? = | ↑ ◦ ↑?, |ψ15? =
|ψ16? =1
|ψ17? =
|ψ20? =
|ψ21? =1
|ψ23? =
|ψ24? =
|ψ25? =
|ψ26? =
|ψ31? =
|ψ32? =
|ψ33? =
|ψ34? =
|ψ39? = | ↑↑↑?
|ψ40? =
|ψ43? = | ? ◦ ??, |ψ44? =
|ψ45? =1
|ψ46? =1
|ψ47? =
|ψ48? =
|ψ49? = | ↑?↑?, |ψ50? =
|ψ51? =1
|ψ52? =
|ψ55? =
|ψ56? =1
|ψ58? = | ?↑??, |ψ59? =
|ψ62? =
|ψ64? = | ????
1
√2{| ↑ ◦◦? + | ◦ ◦ ↑?}
√2{| ↑ ◦◦? − | ◦ ◦ ↑?}
2{| ↑↓ ◦? − | ↓↑ ◦? − |◦ ↑↓? + |◦ ↓↑?},
1
√2{| ? ◦◦? + | ◦ ◦ ??}
2{| ↑↓ ◦? − | ↓↑ ◦? + |◦ ↑↓? − |◦ ↓↑?},
1
√2{| ↑ ◦ ↓? − | ↓ ◦ ↑?},
1
√2{| ? ◦◦? − | ◦ ◦ ??}
1
√2{| ↑↑ ◦? + |◦ ↑↑?}
2{| ↑↓ ◦? + | ↓↑ ◦? + |◦ ↑↓? + |◦ ↓↑?},
1
√2{| ↑ ◦ ↓? + | ↓ ◦ ↑?}
1
√2{| ↑↑ ◦? − |◦ ↑↑?}
2{| ↑↓ ◦? + | ↓↑ ◦? − |◦ ↑↓? − |◦ ↓↑?}
1
√6{2| ↑↓↑? − | ↑↑↓? − | ↓↑↑?},
1
√2{| ↑? ◦? + |◦ ?↑?},
1
√2{| ↑ ◦ ?? + | ? ◦ ↑?},
1
√2{| ?↑ ◦? + |◦ ↑??}
1
√2{| ↑↑↓? − | ↓↑↑?},
1
√2{| ↑? ◦? − |◦ ?↑?},
1
√2{| ↑ ◦ ?? − | ? ◦ ↑?},
1
√2{| ?↑ ◦? − |◦ ↑??}
1
0
2 −
00
2 +
2 +
1
1
1
0
2 −
2 −
3 + 1/2 1/2
1
1
1
0
3 − 1/2 1/2
3 + 3/2 3/2
3 + 3/2 1/2
4 +0
1
√3{| ↑↓↑? + | ↑↑↓? + | ↓↑↑?}
1
√2{| ?? ◦? + |◦ ???},
2{| ↑↓?? − | ↓↑?? − | ?↑↓? + | ?↓↑?}
2{| ↓↑?? − | ↑↓?? − | ?↑↓? + | ?↓↑?},
1
√2{| ↑?↓? − | ↓?↑?},
1
√2{| ?? ◦? − |◦ ???}
1
√2{| ↑↑?? + | ?↑↑?}
2{| ↑↓?? + | ↓↑?? + | ?↑↓? + | ?↓↑?},
1
√2{| ↑?↓? + | ↓?↑?}
1
√2{| ↑↑?? − | ?↑↑?}
2{| ↑↓?? + | ↓↑?? − | ?↑↓? − | ?↓↑?}
1
√2{| ↑??? + | ??↑?}
1
√2{| ??↑? − | ↑???}
0
4 −
00
4 +
4 +
1
1
1
0
4 −
4 −
5 + 1/2 1/2
5 − 1/2 1/2
6 +0
1
1
1
0
0
TABLE II: The three-site basis states used in the deriva-
tion of the recursion relations, in Eq. (8). In these basis
states, e−βH(i,j)−βH(j,k)is block-diagonal, with the largest
blocks being 4 × 4 (see Table IV). The corresponding parti-
cle number (n), parity (p), total spin (s), and total spin z-
component (ms) quantum numbers are also given. The states
|φ4−5?, |φ7?, |φ18−19?, |φ22?, |φ27−30?, |φ35−38?, |φ41−42?,
|φ53−54?, |φ57?, |φ60−61?, |φ63? are obtained by spin reversal
from |φ2−3?, |φ6?, |φ14−15?, |φ20?, |φ23−26?, |φ31−34?, |φ39−40?,
|φ49−50?, |φ55?, |φ58−59?, |φ62?, respectively.
the Hubbard Hamiltonian, after one renormalization-
group transformation, maps onto this generalized Hamil-
tonian, which has a form that stays closed under further
renormalization-group transformations.
The kinetic energy part of the Hamiltonian in Eq.(9)
distinguishes the four types of nearest-neighbor hop-
ping events: i) vacancy hopping (the t0 term): a
vacancy (hole) hopping against a background of single-
electron occupancy (half-filling); ii) pair breaking or
pair making (the t1term): doubly occupied and com-
pletely unoccupied nearest-neighbor sites reverting to
half-filling, or the reverse process; iii) pair hopping
(the t2 term):a pair hopping against a background
of half-filling; iv) vacancy - pair interchange (the
tx term): doubly occupied and completely unoccupied
nearest-neighbor sites exchanging positions.
The generalized Hamiltonian of Eq.(9) reduces to the
Hubbard Hamiltonian of Eq.(2) for t0= t1= t2= t and
tx= J = V2= V3= V4= G = 0 . The renormalization-
group flows occur in the 10-dimensional interaction space
of the generalized Hamiltonian; the 3-dimensional inter-
action space of the Hubbard Hamiltonian contains the
initial conditions of the renormalization-group flows.
The matrix elements of the renormalized pair Hamil-
tonian −β′H′(i,k) are given in Table III in terms of the
renormalized interaction constants. Table III allows us to
solve for the renormalized interaction constants in terms
of the γpgiven in Appendix A:
t′
0=1
2lnγ4
γ2,t′
1= u
γ0
γ8− γ6,
t′
2=1
2lnγ12
γ14,t′
x=1
2(u − v + lnγ7),
U′=1
2
?
u − v + lnγ2
2γ2
γ2
1γ7
4
?
,µ′=1
2lnγ2γ4
γ2
1
,
J′= −u − v + lnγ9,V′
2=1
4lnγ4
1γ3
γ4
9
2γ4
4
+1
4(u + v),
V′
3=1
2lnγ3
2γ3
γ2
4γ12γ14
1γ7γ3
9
− v,V′
4= lnγ1γ7γ3
γ2
9γ16
12γ2
2γ2
4γ2
14
+ 2v,
G′= lnγ1,(10)
where
v =1
2ln?γ6γ8− γ2
γ8− γ6
0
?,
?γ8+ γ6
u =
?
completes
(γ8− γ6)2+ 4γ2
0
cosh−1
2ev
?
.
Thisthe
transformation,
determinationofour
flowsrenormalization-group
in the ten-dimensional interaction space (t0,t1,t2,tx,
U,µ,J,V2,V3,V4) are to be analyzed. (G is an additive
constant not influencing the flows of the 10 other
interaction constants.However, for expectation value
calculations, its derivatives must be included in Eq.(13).)
whose
D.
d = 1 Renormalization-Group Transformation
Thetransformationdescribedaboveisthere-
moval (decimation) of every other site in a lin-
ear array.This decimation produces the mapping

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5
−β′H′(i, k)
φ1
φ1
G′
φ2
φ4
φ7
φ9
φ10
φ2
−t′
µ′+G′
0+
0
φ4
t′
0+
µ′+G′
φ7
t′
2µ′+G′
x−U′+
φ9
0
2µ′+
1
4J′+
V′
2+ G′
φ10
2µ′+
1
4J′+
V′
2+ G′
−β′H′(i,k)
φ6
φ8
φ6
φ8
2t′
−t′
x− U′+ 2µ′+ G′
2t′
1
1
2µ′−3
φ14
4J′+ V′
2+ G′
−β′H′(i, k)
φ12
φ12
φ16
t′
2− U′+ 3µ′
+2V′
2+V′
3+G′
φ14
−t′
+2V′
2− U′+ 3µ′
2+V′
3+G′
0
φ16
0
−2U′+4µ′+4V′
+4V′
2
3+ V′
4+ G′
TABLE III: Block-diagonal matrix of the renormalized two-
site Hamiltonian −β′H′(i,k). The Hamiltonian being invari-
ant under spin-reversal, the spin-flipped matrix elements are
not shown.
of a Hamiltonian with interaction constants K
(t0,t1,t2,tx,U,µ,J,V2,V3,V4,G) onto another Hamilto-
nian with interaction constants
=
K′= R(K).(11)
The function R is calculated as follows:
(1) The matrix elements of −βH(i,j) − βH(j,k) are
determined in the three-site basis {ψq} given in Table II.
In this basis, this matrix is block-diagonal as shown in
Table IV, with the largest block being 4 × 4.
(2) The above block-diagonal matrix is exponentiated,
yielding the matrix elements ?ψq|e−βH(i,j)−βH(j,k)|ψ¯ q?
which enter on the right-hand side of Eq.(8). This in
turn yields the eleven γp(as given in Appendix A).
(3) Using Eqs.(10), the interaction constants of the
renormalized Hamiltonian −β′H′(i,k), namely (t′
t′
The initial conditions, for the iterated renormalization-
grouptransformations
renormalization-group flow, are the interaction constants
of the Hubbard Hamiltonian, K0= (t0= t, t1= t, t2=
t, tx= 0, U, µ, J = 0, V2= 0, V3= 0, V4= 0, G = 0).
0,t′
1,t′
2,
x,U′,µ′,J′,V′
2,V′
3, V′
4,G′) are found.
thatconstitutethe
E.
d > 1 Renormalization-Group Transformation
The Migdal-Kadanoff approximation procedure [27,
28] (which has been remarkably effective in problems
as diverse as lower-critical dimensions for different types
ψ1
0ψ1
ψ2
ψ3
ψ2
ψ3
2µ−√2t0
µ−√2t0
ψ10
−√2tx
−U + 2µ
√2t1
0
ψ6
µψ6
ψ9
ψ11
ψ12
ψ9
ψ10
ψ11
ψ12
−2U + 4µ
−√2tx
2t1
0
2t1
√2t1
4J + V2
−√2t0
ψ14
0
0
3µ −3
−√2t0
2µ
ψ8
ψ13
√2t1
−U + 2µ
ψ17
−√2t0
2µ
ψ8
ψ13
3µ −3
4J + V2
√2t1
ψ15
−√2t0
4J + V2
ψ14
ψ15
2µ
−√2t0
ψ20
3µ +
4J + V2
3µ +1
ψ16
ψ16
ψ17
3µ +1
4J + V2
−√2t0
ψ20
1
ψ21
3µ +
4J + V2
ψ21
1
ψ24
ψ25
ψ26
ψ31
ψ24
ψ25
ψ26
ψ31
−2U + 5µ +
2V2+ V3
−tx
t2
t1
−tx
t2
−t0
t1
t1
0
−U + 3µ
−t0
t1
−U +4µ+
2V2+ V3
04µ + 2V2
ψ23
ψ32
−√3t1
−2U + 5µ +
2V2+ V3
−tx
t2
ψ33
−√3t1
−tx
−U + 3µ
t0
ψ34
ψ23
4µ − J + 2V2
−√3t1
−√3t1
0
0
ψ32
t2
ψ33
ψ34
t0
−U + 4µ +
2V2+ V3
ψ39
ψ39
4µ +1
2J + 2V2
ψ40
ψ40
4µ +1
2J + 2V2
ψ43
ψ44
−√2tx
−3U + 6µ +
4V2+4V3+V4
−√2t1
0
ψ46
ψ47
ψ43
−2U + 4µ
−√2tx
−2t1
0
−2t1
−√2t1
−U + 5µ −
4J + 3V2+ V3
−√2t2
0
ψ44
0
ψ46
3
−√2t2
−2U + 6µ +
4V2+ 2V3
ψ47
ψ45
ψ48
−√2t1
ψ45
ψ48
−U + 5µ −3
4J + 3V2+ V3
−√2t1
ψ49
−3U + 6µ + 4V2+ 4V3+ V4
ψ50
√2t2
−U + 5µ +1
ψ52
√2t2
−2U + 6µ + 4V2+ 2V3
ψ56
−U + 5µ +
1
4J + 3V2+ V3
ψ49
ψ50
−2U + 6µ + 4V2+ 2V3
√2t2
4J + 3V2+ V3
ψ51
ψ51
ψ52
−U + 5µ +1
4J + 3V2+ V3
√2t2
ψ55
ψ55
−U + 5µ +
4J + 3V2+ V3
1
ψ56
ψ58
ψ59
√2t2
ψ58
ψ59
−2U + 6µ + 4V2+ 2V3
√2t2
−3U + 7µ + 6V2+ 5V3+ V4
ψ62
ψ62
−3U + 7µ + 6V2+
5V3+ V4
ψ64
ψ64
−4U + 8µ + 8V2+
8V3+ 2V4
TABLE IV: Diagonal matrix blocks of the unrenormalized
three-site Hamiltonian −βH(i,j) − βH(j,k). The Hamilto-
nian being invariant under spin-reversal, the spin-flipped ma-
trix elements are not shown. The additive constant contribu-
tion 2G, occurring at the diagonal terms, is also not shown.

Page 6

6
of phase transitions; first- and second-order phase tran-
sitions in q-state Potts models; algebraic order in the
d = 2 XY model; random-field, random-bond, spin-glass
systems; quenched-disorder-induced criticality; etc.) is
used to construct the renormalization-group transforma-
tion for d > 1. In the d-dimensional hypercubic lattice,
a subset of the nearest-neighbor interactions are ignored,
so that a hypercubic lattice (still d-dimensional) is left
behind, in which each lattice point is connected by two
consecutive nearest-neighbor segments of the original lat-
tice. The decimation described above can then be applied
to the site connecting these two segments of the original
lattice, yielding the renormalized nearest-neighbor cou-
plings between the lattice points of the new hypercubic
lattice. To compensate for the nearest-neighbor interac-
tions that are ignored, the couplings are multiplied by a
factor of bd−1after decimation, b = 2 being the length
rescaling factor. Thus, the renormalization-group trans-
formation of Eq.(11) in the previous section generalizes,
for d > 1, to
K′= bd−1R(K).(12)
F.Supporting Results
New global phase diagrams obtained by approximate
renormalization-group transformations are supported by
the correct rendition of all of the special cases of the
system solved. The Hamiltonian in Eq.(9), which is the
system presently solved by approximate recursion rela-
tions, reduces in various limits to the Ising, quantum
XY, and quantum Heisenberg spin systems. Our recur-
sion relations correctly yield the lower critical-dimensions
dl of the Ising (dl = 1), quantum XY (dl = 2), and
quantum Heisenberg (dl = 2) spin systems.
quantum XY spin system in d = 2, this approximation
yields the algebraically ordered Kosterlitz-Thouless low-
temperature phase.[25, 26] For the quantum Heisenberg
spin system in d = 3, our recursion relations yield low-
temperature antiferromagnetically (for J < 0) and ferro-
magnetically (for J > 0) ordered phases, each separated
by a second-order transition from the high-temperature
disordered phase. The antiferromagnetic transition tem-
perature is thus found to be 1.22 times [13] the ferromag-
netic transition temperature, a purely quantum mechan-
ical effect, and to be compared with the value of 1.13
from series expansion [29, 30]. Furthermore, as purely
off-diagonal quantum effects, the hopping-induced anti-
ferromagnetism of the d = 3 Hubbard model is recov-
ered and the scaling of the antiferromagnetic transition
temperature is obtained with an excellent quantitative
agreement, as discussed in Sec.V at Eq.(16) and shown
in Fig.3. In fact, the scaling of the antiferromagnetic
transition at strong-coupling (Fig. 3), as well as the re-
sults quoted above, and the disappearance of the transi-
tion at zero coupling (Fig. 4), indicate the validity of our
approximation across the entire strong-to-weak coupling
For the
range. Finally, the Blume-Emery-Griffiths model is con-
tained in the Hamiltonian of Eq.(9) and its global phase
diagram [31] is obtained from our recursion relations. All
of these results strongly support the validity of the global
calculation here.
IV.RENORMALIZATION-GROUP ANALYSIS:
GLOBAL PHASE DIAGRAM AND
OPERATOR EXPECTATION VALUES
From the recursion equations determined in the pre-
ceding section, flows are generated for initial values
of t, U, and µ in the Hubbard Hamiltonian.
renormalization-group transformation, which constitutes
each step of the flow, is effected numerically. Particular
attention has to be given to the multiplication of small
amplitudes with large exponentials, which can occur in
the right-hand side of Eq.(8) when interaction constants
become large, causing the computational difficulties en-
countered in previous work [2].
Each completely stable fixed point, namely sink of the
renormalization-group flows, corresponds to a thermo-
dynamic phase, and the global phase diagram is found
by identifying the basin of attraction for every sink.[31]
The expectation values for the operators occurring in the
Hamiltonian are obtained from the conjugate recursion
relations, [32]
The
nβ= b−dn′
αTαβ,(13)
with summation over the repeated index α implicit. The
recursion matrix is
Tαβ=∂K′
α
∂Kβ
,(14)
where Kα is an interaction strength, namely a compo-
nent in the interaction strength vector K defined before
Eq.(11); nαis the expectation value of the operator that
occurs in the Hamiltonian with coefficient Kα. Eq.(13) is
iterated along a trajectory until a phase sink limit. The
left eigenvector of Tαβwith eigenvalue bdgives the expec-
tation values at the phase sink, thereby completing the
calculation of the expectation values of the initial point
of the trajectory.
The observed phase sinks in the calculations for the
d = 3 Hubbard model — the details of which are shown in
Table V — have a property in common: at the sink limit,
t1renormalizes toward zero. In the limit t1→ 0, analytic
expressions are derived to first order in t1for the matrix
elements ?ψq|e−βH(i,j)−βH(j,k)|ψ¯ q? on the right-hand side
of Eq.(8). This yields, in the neighborhood of each phase
sink, analytic renormalization-groupequations. The ana-
lytic equations provide a useful check on the accuracy of
the numerical calculations, and lead to closed-form ex-
pressions for limiting values of interaction strengths or
ratios of limiting values of interaction strengths.

Page 7

7
Flows that start at the boundaries between phases have
their own fixed points, distinguished from phase sinks by
having at least one unstable direction.
ing down onto the boundary and from there following
a flow to the neighborhood of the unstable fixed point,
a Newton-Raphson procedure is used to exactly locate
this unstable fixed point. Analysis at these fixed points
determines the phase transition properties. The expecta-
tion values calculated, as described above, at the phase
boundaries allow us to redraw the phase diagram using
expectation values nαon the axes as well as t, U, and µ.
The Hamiltonian of Eq.(9) is covariant under particle-
hole symmetry (c†
iσ→ ciσ), which in Hamiltonian space
takes the form of a mapping¯K = S(K). The function S
is given by
After narrow-
¯ t0= −t2, ¯ t1= −t1, ¯ t2= −t0, ¯ tx= tx,¯J = J,
¯U = U − 2V3− V4, ¯ µ = −µ + U − 2V2− 3V3− V4,
¯ V2= V2+ 2V3+ V4, ¯V3= −V3− V4,¯V4= V4.(15)
The subspace that is invariant under S corresponds to
systems that are invariant under particle-hole exchange,
and therefore are at half-filling: ?ni? = 1 = ?hi?. From
Eq.(15), this subspace occurs at t0= −t2, t1= 0, 2µ =
U − 2V2− V3, 2V3 = −V4. For the original Hubbard
Hamiltonian, all points with µ0/U0 = 1/2 are mapped
onto this subspace after the first renormalization-group
step.
The Hubbard phase diagrams are plotted in the
next section, for fixed U0/t, in terms of 1/t (a tem-
perature variable) versus µ0/U0 or ?ni?.
renormalization-group transformation is also covariant
under particle-hole symmetry, the phase diagrams are
duly symmetric about µ0/U0= 1/2 or ?ni? = 1.
Since our
V. GLOBAL PHASE DIAGRAM FOR d = 3
For d = 3 and a range of couplings U0/t = 5 to 20,
Figs. 1 show Hubbard phase diagrams in terms of tem-
perature (1/t) versus chemical potential (µ0/U0). The
corresponding phase diagrams in temperature (1/t) ver-
sus electron density ?ni? are in Fig. 2. The values of the
interaction constants for each observed phase sink are
listed in Table V. The expectation values for each phase
sink, also listed in Table V, allow us to identify the phases
as follows:
Hole-rich disordered (hD) phase: The electron
density ?ni? is zero at the sink and, concomitantly, the
electron densities ?ni? calculated inside this phase are
low.
Near-half-filled disordered (nHD) phase: The
basin of attraction of nHD occurs at µ0/U0?= 1/2. The
electron density ?ni? is 1 at the sink and, concomitantly,
the electron densities ?ni? calculated inside this phase
are closer to half-filling. Half-filled disordered (HD)
phase: The sink is for the disordered phase at perfect
half-filling, µ0/U0= 1/2 and ?ni? = 1.
Electron-rich disordered (eD) phase: The elec-
tron density ?ni? is 2 at the sink and, concomitantly, the
electron densities ?ni? calculated inside this phase are
high.
Antiferromagnetic (AF) phase: The electron den-
sity ?ni? is 1 at the sink and, concomitantly, the elec-
tron densities ?ni? calculated inside this phase are closer
to half-filling.The expectation value for the nearest-
neighbor spin-spin correlation is ??Si·?Sj? =1
Note that the latter two spins are, on the original cu-
bic lattice, distant spins on the same sublattice; from
this, antiferromagnetism, ??Si·?Sj? < 0 when the spins
are on different sublattices of the original cubic lattice,
is calculationally obtained throughout this phase. Since
there is no explicit antiferromagnetic coupling in the ini-
tial Hubbard Hamiltonian, the antiferromagnetic phase
is completely a quantum mechanical effect resulting from
the kinetic energy term. In fact, at half-filling, second-
order perturbation theory in t, valid for small t/U0, must
yield an effective antiferromagnetic coupling proportional
to t2/U0. Thus, for small t/U0, t2/U0should equal the
same constant at all antiferromagnetic phase transitions
at half-filling (Recall that all of our coupling constants
are dimensionless, incorporating the inverse temperature
factor 1/kT). Equivalently, t/U0should be linear in 1/t
at all antiferromagnetic phase transitions at half-filling,
for small t/U0and therefore for small 1/t (low tempera-
ture):
4at the sink.
1/t ∼ t/U0.(16)
This is indeed rendered by our calculation, as seen in
Fig. 3. For higher values of 1/t, Eq.(16) is not appli-
cable, since second-order perturbation theory does not
hold, and indeed our calculated curve in Fig. 3 deviates
from linearity. On the other hand, the approximation
in our recursion relation is even more justified, since the
commutation relations that are ignored involve terms of
order t2.
The antiferromagnetic transition temperature as a
function of coupling U0/t is also shown in Fig. 4, together
with calculated values from other approximation schemes
for the d = 3 Hubbard model. We see that our results for
intermediate coupling are comparable to those of quan-
tum Monte Carlo studies [33, 34]. As expected, our tran-
sition temperature vanishes in the limit U0/t → 0, since
there are no phase transitions for the non-interacting sys-
tem. Thus, our approximation behaves correctly both at
strong coupling (previous paragraph) and at weak cou-
pling.
τHb and τtJ phases: For large values of U0/t, the
novel phase found in the tJ model [13] (which we call
τtJ) also occurs in the Hubbard model. In addition, we
find a closely related phase (τHb), unique to the Hubbard
model, at smaller U0/t. The two phases are characterized
by very similar properties: the hopping strengths t0, t2,
and tx renormalize to ±∞, and the phase sinks have a

Page 8

8
HeL U0êt = 5
tHb
tHb
AF hD
nHD
eD
nHD HD
HdL U0êt = 7.5
tHb
ttJ
tHb
ttJ
AFhD
nHD
eD
nHD
HD
HcL U0êt = 10
tHb
hD
ttJ
tHb
eD
ttJ
AFhD
nHD
eD
nHD
HD
HbL U0êt = 15
tHb
ttJ
tHb
ttJ
AFhD
nHD
AF
eD
nHD
AF
HD
HaL U0êt = 20
tHb
ttJ
tHb
ttJ
AFhD
nHD
AF
eD
nHD
AF
HD
e r u t a r e
p
m
e
T
1êt
0.45
Chemicalpotential m0êU0
-0.7
-0.3 0.10.50.91.31.7
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.45
0.60
FIG. 1: d = 3 Hubbard model phase diagrams in temperature versus chemical potential. The hole-rich disordered (hD),
near-half-filled disordered (nHD), half-filled disordered (HD), electron-rich disordered (eD), antiferromagnetic (AF), τHb, and
τtJ phases are seen. The full curves are second-order phase boundaries, while the dotted curves are first-order boundaries. The
dashed curves are not phase transitions, but disorder lines between the near-half-filled disordered and the hole-rich or electron-
rich disordered phases. The progression (a) U0/t = 20 through (e) U0/t = 5 shows the changing phase diagram topology from
strong to intermediate coupling. The τtJ phase, which is prominent at strong coupling, disappears entirely for U0/t ? 6, and
the τHb phase is prominent for intermediate couplings.

Page 9

9
HeL U0êt = 5
tHb
tHb
AFhD
nHD
eD
nHD
HD
HdL U0êt = 7.5
tHb
ttJ
tHb
ttJ
AFhD
nHD
eD
nHD
HD
HcL U0êt = 10
tHb
hD
ttJ
AF
hD
nHD
HD
tHb
eD
ttJ
eD
nHD
HbL U0êt = 15
AF
tHb
tHb
ttJ
AF
AF
hD
nHD
AF
tHb
tHb
ttJ
AF
eD
nHD
HD
HaL U0êt = 20
tHb
tHb
ttJ
ttJ
AF
AF
hD
nHD
tHb
tHb
ttJ
AF
eD
nHD
HD
e r u t a r e
p
m
e
T
1êt
0.45
Electron density Xni\
0.50.6 0.7 0.8 0.91.0 1.1 1.21.31.41.5
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.00
0.15
0.30
0.45
0.00
0.15
0.30
0.45
0.60
FIG. 2: d = 3 Hubbard model phase diagrams in temperature versus electron density. The full curves are second-order phase
boundaries. The coexistence boundaries of first-order transitions are drawn with dotted curves, with the unmarked areas
inside corresponding to coexistence regions of the two phases at either side. The dashed curves are not phase transitions, but
disorder lines between the near-half-filled disordered and the hole-rich or electron-rich disordered phases. Noteworthy is the
narrowness of the first-order transitions, with jumps in the electron density of the order of a few percent (i.e., the width of the
coexistence region). The antiferromagnetic phase is unstable to about 8-15% hole (or electron) doping away from half-filling.
In the intermediate U0/t regime, the τHbphase appears for about 10-18% hole (or electron) doping. At larger U0/t, the τtJ
phase dominates, and exists between 30-35% hole (or electron) doping.

Page 10

10
0.00 0.050.10
têU0
0.150.20
0.0
0.2
0.4
0.6
0.8
e r u t a r e
p
m
e
T
1êt
FIG. 3: The data points are the calculated antiferromag-
netic transition temperatures at half-filling. The linear rela-
tion that is expected for strong coupling at low temperatures
(Sec.V) is obtained.
048 1216 20
U0êt
0.00
0.15
0.30
0.45
0.60
0.75
0.90
e r u t a r e
p
m
e
T
1êt
FIG. 4: Comparison of the antiferromagnetic transition tem-
peratures at half-filling for the d = 3 Hubbard model cal-
culated from various approaches: the renormalization-group
method of the present paper (solid line); QMC [33] (dia-
monds); QMC [34] (triangles); DMFT [35] (squares); RPA
weak-coupling expansion [36] (dot-dashed line); and two
approximations for the strong-coupling behavior—the high-
temperature expansion of the Heisenberg model, 1/t =
3.83t/U0 [34] (dotted line) and Weiss mean-field theory, 1/t =
6t/U0 (dashed line).
non-zero vacancy hopping expectation value
??
for µ0/U0< 1/2, and a non-zero pair hopping expecta-
tion value
σ
hi−σhj−σ(c†
iσcjσ+ c†
jσciσ)
?
=
?
−2/3
0.663972
(τtJ)
(τHb)
,
(17)
??
σ
ni−σnj−σ(c†
iσcjσ+ c†
jσciσ)
?
=
?
2/3
−0.663972
(τtJ)
(τHb)
,
(18)
for µ0/U0 > 1/2.
occurrence of hopping, the electron densities at the sinks
have values different from 0 (empty), 1 (half filled), or
2 (doubly occupied): ?ni? = 2/3, 4/3 and ?ninj? = 1/3,
5/3 for the τtJphase, and ?ni? = 0.668014, 1.331986 and
In both cases, as expected for the
?ninj? = 0.336028, 1.663972 for the τHbphase. (At the
sinks of non-τ phases, the electron density is, on the other
hand, 0, 1, or 2.) There are also small spin correlations
in the phase sink limits, ??Si·?Sj? = −1/4 for τtJ, and
??Si·?Sj? = 0.0840069 for τHb, which yield, throughout
these phases, small antiferromagnetic correlations in the
original system.
The boundaries in Fig. 1 are controlled by fourteen
unstable fixed points, given in Table VI. For smaller val-
ues of U0/t, the topology of the phase diagram is that
of Fig. 1(e), where the AF/HD, AF/nHD, AF/τHb, and
hD/τHb boundaries are respectively controlled by the
second-order fixed points C∗
ter three boundaries intersect at the multicritical point
B2, controlled by the fixed point B∗
hD/nHD boundary just above this intersection is second-
order, controlled by the fixed point C∗
multicritical point B1, controlled by the fixed point B∗
The high-temperature section of the hD/nHD boundary
is a disorder line, controlled by the null fixed point N*,
i.e., there is no phase transition above B1.
1, C∗
2, C∗
3, and C∗
4. The lat-
2. A segment of the
5, ending at the
1.
As U0/t is increased, the phase diagram topology be-
comes more complex. For U0/t ? 6, the τtJ phase ap-
pears, its boundary with hD controlled by the second-
order fixed point C∗
boundary between the hD and nHD phases become first-
order (fixed point F∗
the τtJ phase; their boundaries with hD are also first-
order (fixed point F∗
order boundaries with other phase boundaries are con-
trolled by the additional multicritical points B∗
and by the critical endpoint L∗[31].
6. Portions of the lower-temperature
1), and islands of AF appear above
2). The intersections of these first-
3and B∗
4,
As the coupling U0/t varies, a most interesting aspect
of the changing phase diagram topology is the relative
sizes of the τhBand τtJphases. The τhBphase is largest
at intermediate values of U0/t, and gradually decreases in
size as we move into the strong-coupling regime, break-
ing up into narrow slivers until at large values of U0/t
only tiny remnants of it are left in the phase diagram.
The τtJ phase appears at intermediate values of U0/t,
grows in size as U0/t is increased, and occupies a promi-
nent place in the diagram next to the AF phase in the
strong-coupling regime. As discussed in Section VII, this
is precisely what we expect, since the Hubbard phase
diagram should approximately reproduce the tJ model
results [13] in the large U0/t limit.
Phase diagrams in terms of temperature versus elec-
tron density ?ni? are shown in Fig. 2. It is seen that the
antiferromagnetic phase is unstable to at most 15% hole
(or electron) doping at low temperatures. The τHband τtJ
phases exist at different doping values, with τHbappear-
ing for approximately 10-18% doping, directly adjacent
to the AF phase, and τtJin the 30-35% doping range.
The narrowness of the first-order transitions, with jumps
in the electron density of the order of a few percent, is
noteworthy.

Page 11

11
Phase sinkInteraction constants
Additional
properties
t0
0
t1
0
t2
0
tx
0
U
∞
µJ
0
V2
0
V3
0
V4
0 hole-rich
disordered hD
near-half-filled
disordered nHD
half-filled
disordered HD
electron-rich
disordered eD
antiferro-
magnetic AF
−∞
µ/U = const.
∞
≈ 0.62U
∞
=1
2U
∞
µ/U = const.
∞
≈ 0.57U
2 ln30−2 ln3∞∞0−∞∞-∞
U−2µ−2V2−V3=0
2V3+V4=0
≈ 0.24U
0
≈ −0.47U
0
≈ 0.69U
0
≈ −1.38U
0000∞0
U−2µ=0
0000∞0000
−∞0∞∞∞∞ −∞∞ −∞
U−2µ−2V2−V3→0
2V3+V4→0
t2−t0→0
U−2µ−2V2−V3=0
2V3+V4=0
≈ −0.29U
≈ 0.29U
≈ 0.14U
≈ 0.29U
≈ −0.071UV3/U → 0V4/U → 0
(µ0/U0?= 1/2)
antiferro-
magnetic AF
0000∞∞
2U
∞−∞
≈ −1
∞
2J
−∞
≈ −J
≈1
J/U → 0
4J
≈1
(µ0/U0= 1/2)
τHb
(µ0/U0< 1/2)
τHb
(µ0/U0> 1/2)
τtJ
(µ0/U0< 1/2)
τtJ
(µ0/U0> 1/2)
t2−t0=0
t0+µ+1
−∞
≈ −1
−∞
≈ −1
∞
≈ 0.13U
∞
≈ 0.42U
0∞
2U
∞
≈1
−∞
≈ −1.46U
−∞
≈ −0.038U
−∞
≈ −1
−∞
≈ −1
∞
≈ 0.52U
∞
≈ 0.15U
∞∞
4U
∞
≈ U
−∞
∞
2U
∞
≈1
−∞
≈ −0.87U
−∞
≈ −0.25U
−∞
≈ −1
−∞
≈ −1
−∞
≈ −0.50U
−∞
≈ −0.86U
−∞∞
4J+V2
≈−4.35
4U
≈1
2U
≈1
≈1
8U
V3/U → 0
∞
V3/U → 0
−∞
≈ −1.13U
∞
≈ 0.39U
V4/U → 0
∞
V4/U → 0
−∞
≈ −0.21U
−∞
≈ −0.060U
0∞
−t2+U−µ+1
− V2−V3≈−4.35
4J
2U
4U
2U
2U
8U
0∞
≈ −0.022U
∞
≈ 1.62U
0∞
Phase sinkExpectation values
?ni↑ni↓?
0
0
0
1
0


0
1
3
?T0?
0
0
0
0
0
?T1?
0
0
0
0
0
?T2?
0
0
0
0
0
?Tx?
0
0
0
0
0
?ni?
0
1
1
2
1
0.668
014
1.331
986
?
3
??Si·?Sj?
0
0
0
0
1
4
?ninj?
0
1
1
4
1


?ni↑ni↓nj? ?ni↑ni↓nj↑nj↓?
0
0
0
2
0
hD
nHD
HD
eD
AF
0
0
0
1
0
τHb
?0.663
0
972
0
?
0
−0.66
3972
?
3
0
?
0
0.331
986
?



0.0840
069



0.336
028
1.663
972
?
3
?
0
0.331
986
?
3
0
τtJ
?
−2
0
3
0
0
2
0
2
3
4
−1
4
1
3
5
0
1
0
TABLE V: Interaction constants and expectation values at the phase sink fixed points.
µ0/U0<
>
hi −σnj −σ)(c†
For τHband τtJ, the values for
iσcjσ+c†
i↑cj↑c†
1
2are given. The hopping expectation values ?Tα? are: ?T0? =?
σ?hi−σhj−σ(c†
jσciσ)?, ?T1? =?
σ?(ni −σhj −σ+
iσcjσ+ c†
jσciσ)?, ?T2? =?
σ?ni −σnj −σ(c†
iσcjσ+ c†
jσciσ)?, ?Tx? = ?c†
i↓cj↓+ c†
j↑ci↑c†
j↓ci↓?.
VI.SPECIFIC HEAT RESULTS
From the calculated expectation values of the oper-
ators occurring in the Hamiltonian [Eq.(2)], we have
obtained the dimensionless internal energy per bond
?βH(i,j)?. Recall that dimensionless coupling constants
are exhibited in the Hubbard Hamiltonian of Eq.(2), e.g.,
t =
˜t
kBT,
(19)
where˜t is a constant that does not depend on tempera-
ture. The specific heat is calculated with
C =∂?H(i,j)?
∂T
= kB
?
∂
∂t−1?c†
∂
∂t−1?ni↑ni↓+ nj↑nj↓?
iσcjσ+ c†
jσciσ?
+U
t
?
.(20)
The partial derivatives are taken at fixed U0/t and at
fixed density ?ni?.
In Fig. 5 we plot γ = C/T for U0/t = 15 at several
different electron densities.
diagram is shown in Fig. 2(b)). At half-filling, ?ni? =
1.00, we observe a broad peak near the HD/AF transition
temperature, which we can attribute to the onset of spin
order. As we dope the system with holes, this peak gets
sharper, becoming most pronounced near ?ni? = 0.68,
directly above the transition temperature between the hD
and τtJphases. In fact, the C/T curve shows a multipeak
structure near the transition, a general characteristic of
the phase diagram region just above the τtJphase. At
electron density ?ni? = 0.60, no longer in the τtJrange,
the peak decreases in size and broadens out again.
The distinct nature of the τtJand τHbphases becomes
clear when we look at the low temperature specific heat.
In Fig. 6 we plot the coefficient γ = C/T as a function
of electron density for U0/t = 15 and at low temperature
1/t = 0.085. In the limit as T → 0, γ is a measure of
the linear contribution to the specific heat due to quasi-
particle excitations. Near half-filling, γ is close to zero,
increases to a small level with sufficient hole doping, falls
(The corresponding phase

Page 12

12
BasinTypeInteraction constants
Additional
properties
Relevant
eigenvalue
exponents
(y)
t0
t1
0
t2
tx
∞
U
∞
µJ
0
V2
∞
V3
−∞
V4
−∞F∗
1
portion of
hD/nHD
boundary
AF/hD
boundary
AF/HD
boundary
AF/nHD
boundary
AF/τHb
boundary
hD/τHb
boundary
portion of
hD/nHD
boundary
hD/τtJ
boundary
1st
order
2 ln3−2ln3−∞
2µ+V2≈−0.396
U−2µ−2V2−V3→0
3
F∗
2
1st
order
2nd
order
2nd
order
2nd
order
2nd
order
2nd
order
−∞0∞∞∞−∞∞∞ −∞∞
U−2µ−2V2−V3→0
8µ+J+4V2≈−0.658
U−2µ−2V2−V3=0
2V3+V4=0
3
C∗
1
0000∞∞
2U
∞
1.376−0.06500.130−0.2600.715
≈1
C∗
2
−0.554 00.554∞∞1.376−∞∞−∞
U−2µ−2V2−V3→0
2V3+V4→0
t0+µ+1
4J+V2≈−0.739
0.715
C∗
3
−∞0∞−∞∞∞∞−∞∞−∞1.68
C∗
4
−∞0∞−∞∞−∞∞∞ −∞∞
t0+µ+1
4J+V2≈−5.178
8µ+J+4V2≈6.617
1.42
C∗
5
−1.610 0−2ln3−1.594 ∞0.52300.0108−0.569−∞
≈ −3U
1.56
C∗
6
2nd
order
2.9590 −29.5859.629∞1.016−13.692−8.332−18.259−∞1.01
N∗
portion of
hD/nHD
boundary
F∗
basins meet
C∗
basins meet
C∗
C∗
5basins meet
F∗
basins meet
F∗
basins meet
null00−2ln30∞0004ln
√3
2
−∞
≈ −3U
2
L∗
1, F∗
2, C∗
2
critical
endpoint
multi-
critical
multi-
critical
multi-
critical
multi-
critical
−0.554 00.554∞∞−∞1.376∞−∞−∞
U−2µ−2V2−V3→0
8µ+J+4V2≈−0.798
3
0.715155
1.73
0.22
1.15
0.27
2.56
0.96
2.68
1.90
B∗
15, N∗
−1.236 0−2ln3−1.005 ∞0.22100.127−0.652−∞
≈ −3U
−∞
≈ −3U
−∞
≈ −3U
−∞
B∗
22, C∗
3, C∗
4,−2.156 0−1.555−2.708 ∞ 1.559 0.321−0.762 0.201
B∗
31, N∗
00−2ln30∞ −0.68101.089−0.438
3U+V4≈3.044
B∗
42, C∗
3, C∗
4
−∞0∞−∞ ∞−∞ ∞∞−∞
t0+µ+1
4J+V2→0
8µ+J+4V2→0
TABLE VI: Unstable fixed points. The fixed points of the µ0/U0 ≤ 1/2 half space are given here.
to near zero again in the τHbphase, and dramatically in-
creases only after the system makes a narrow first-order
transition to the hole-rich disordered phase. The steady
rise of γ in the hD phase with further hole doping is con-
sistent with a Fermi liquid interpretation of this phase.
The increase in γ is interrupted by the τtJinterval, where
the curve makes a sharp oscillation, but continues in the
hD region on the other side.
We see that the τtJphase has non-zero γ at low tem-
peratures, while the τHbphase does not. In Figs. 7(a)
and (b) we contrast the two τ phases directly, comparing
representative C/T curves for τtJand τHbtransitions. We
observe that in the τHbphase the low-temperature spe-
cific heat exhibits an exponential form characteristic of
a gap in the quasiparticle spectrum. Specific heat data
points for temperatures 1/t < 0.2, shown in the top right
inset of Fig. 7(b), were found to fit a theoretical curve of
the same form as in the T → 0 limit of a weakly-coupled,
BCS-type superconductor,
C
kB
=
A
T3/2exp
?
−∆
T
?
,(21)
with a best-fit coefficient A = 1.02 ± 0.06 and a zero-
temperature gap ∆ = 1.01±0.01, where t−1is used as the
temperature variable. In contrast, the τtJphase clearly
has a gapless spectrum, as we see in the C/T curve of
Fig. 7(a). As mentioned earlier, we also clearly see mul-
tiple peaks in the specific heat just above the hD/τtJ
transition temperature.
The τtJand τHbphases have similar properties at
the phase sink, most notably a non-zero hopping am-
plitude, and thus are both good candidates for super-
conductivity. Since the two phases are dominant in dif-
ferent U0/t regimes, their contrasting specific heat char-
acteristics can potentially be understood as the differ-
ence between strongly-coupled and weakly-coupled su-
perconducting phases. For the strongly-coupled, BEC-
like case, pairing occurs above Tc, and these tightly
bound bosonic pairs condense at the transition temper-
ature. The double-peak structure in the specific heat
above the τtJphase is a possible indicator of such pair
formation. Additionally, we expect that a BEC-like su-
perconducting transition in three dimensions should have
a specific heat critical exponent α = −1 [24]. Analysis
of the C∗
6fixed point, governing the hD/τtJboundary,
yields the result α = −0.97. The presence of low-lying
excitations in a Bose gas is also consistent with the fact
that we do not see a gap in the low-temperature specific

Page 13

13
0.0 0.10.2 0.30.4 0.5 0.6
Temperature 1êt
0
1
2
3
4
g = CêT
0.60
0.68
0.73
0.84
0.93
1.00
FIG. 5: The specific heat coefficient γ = C/T as a function of
temperature for U0/t = 15, at several different electron densi-
ties ?ni? indicated in the legend. For this temperature range
the densities ?ni? = 1.00 and 0.93 lie inside the antiferromag-
netic (AF) phase, 0.84 inside the τHbphase, 0.73 and 0.60
inside the hole-rich disordered (hD) phase, and 0.68 inside
the τtJphase. Here and in the following figures, γ is shown
in units of k2
stant in Eq. (19).
B/˜t, where˜t is the temperature-independent con-
0.50.60.70.8 0.91.0
Electron density Xni\
0.0
0.1
0.2
0.3
0.4
0.5
g = CêT
hD
ttJ
tHb
AF
AF
hD nHD
FIG. 6: The specific heat coefficient γ = C/T for U0/t = 15
at the low temperature of 1/t = 0.085, as a function of elec-
tron density ?ni?. The corresponding phases are indicated
near the top of the figure, with second-order phase bound-
aries marked by thin vertical lines. The interval between the
vertical dashed lines corresponds to the first-order phase tran-
sition.
heat of the τtJphase.
Turning now to the τHbphase, we already noted that
its specific heat can be closely fitted at low tempera-
tures to a BCS-like exponential curve, which is exactly
what we would expect for a weakly-coupled supercon-
ducting phase. Analysis of the C∗
ling the AF/τHbboundary, yields a specific heat coeffi-
cient α = −0.27. This translates into a finite cusp at the
transition temperature, as shown in the top left inset of
Fig. 7(b). For weak and intermediate couplings the su-
perconducting transition is expected to belong to the uni-
versality class of the d = 3 XY model, with α = −0.013
3fixed point, control-
tHb
AF nHD
HbL
0.31150.3200
1.1
1.2
0.00.1
0.0
0.2
tHb
AF
HaL
ttJ
hDnHD
0.0
0.1
0.2 0.30.40.50.6
Temperature 1êt
0.0
1.0
2.0
3.0
4.0
0.0
1.0
2.0
3.0
4.0
5.0
g = CêT
FIG. 7: The specific heat coefficient γ = C/T as a function
of temperature for two different electron densities and values
of U0/t: (a) ?ni? = 0.68, U0/t = 20; (b) ?ni? = 0.875, U0/t =
7.5. Phases are indicated near the top of the figures, with
second-order phase boundaries marked by thin vertical lines.
The interval between the vertical dashed lines corresponds to
a first-order phase transition. In diagram (b) the top left in-
set shows a close-up of the cusp in γ at the AF/τHbtransition
temperature. The data points in the top right inset are calcu-
lated γ values for temperatures 1/t < 0.2, fitted to a BCS-like
exponential curve of the form C/kBT =
best-fit parameters A = 1.02 ± 0.06 and ∆ = 1.01 ± 0.01,
where t−1is used as the temperature variable.
A
T5/2exp?−∆
T
?, with
[37] (examples of transitions in this class include the su-
perfluid transition of4He, the superconducting transi-
tion in certain high-Tcmaterials like Y-123, and also in
conventional superconductors, though for the latter the
critical region is too narrow to be observed experimen-
tally) [24]. Our calculated α is closer to the d = 3 XY
than to the BEC value, supporting the weak-coupling
interpretation of the τHbphase.
VII.THE tJ LIMIT OF THE HUBBARD MODEL
In the strong-coupling limit U0≫ t, second-order per-
turbation theory in t/U0applied to the Hubbard model
leads to the following Hamiltonian (known as the tJ
model) [11, 12, 13, 14, 38, 39, 40, 41, 42],
HtJ= −t
?
?ij?,σ
P
?
c†
iσcjσ+ c†
jσciσ
?
P
+ J
?
?ij?
(?Si·?Sj−1
4ninj),(22)

Page 14

14
HbL
ttJ
AF
AF
AF
AF
hD
0.7
hD nHD
HaL
ttJ
AF
hD
nHD
AF
AF
e r u t a r e
p
m
e
T
1êt
Chemicalpotential m0êU0
-0.10
-0.050.00 0.050.100.15 0.20
0.5 0.60.80.9 1.0
Electron density Xni\
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
FIG. 8:
Coulomb repulsion U0/t = 50 in temperature versus (a) chem-
ical potential, (b) electron density ?ni?. The full curves are
second-order phase boundaries, while the dotted curves indi-
cate first-order boundaries. The dashed lines are not phase
transitions, but disorder lines between the near-half-filled dis-
ordered and hole-rich disordered phases.
d = 3 Hubbard model phase diagram for large
0.00.1 0.20.30.4
Temperature 1êt
0
2
4
6
8
g = CêT
ttJ
nHD AF
nHD
nHD
AF
hD
nHD
FIG. 9: The specific heat coefficient γ = C/T as a function
of temperature for U0/t = 50 and ?ni? = 0.67. Phases are
indicated near the top of the figure, with second-order phase
boundaries marked by thin vertical lines. The dashed lines
are not phase transitions, but disorder lines between the near-
half-filled disordered and hole-rich disordered phases.
0.000.250.500.75 1.00
Electron density Xni\
0.00
0.25
0.50
0.75
1.00
1.25
c i t e
n i
K
y
g r e
n
e
XK\
U0êt = 20
50
1001000 HdottedL
tJ model HblackL
FIG. 10: The kinetic energy per bond ?K? = −?
1/t = 0.2 for Coulomb repulsions U0/t = 20, 50, 100, and
1000 (indicated by numbers next to each curve). The solid
curve at the bottom is the result calculated using the tJ model
renormalization-group equations [13, 14] at the same temper-
ature, with the corresponding J/t = 0.004.
σ?c†
iσcjσ+
c†
jσciσ? as a function of electron density ?ni? at temperature
where J = 4t2/U and P is a projection operator prohibit-
ing double occupation of a lattice site. In addition to the
terms shown above, the perturbation theory generates
a three-site term of the form?
it does not radically alter the physics of the tJ model.
(Our current results, directly from the strong-coupling
limit of the actual Hubbard model, confirm this assump-
tion.) We thus expect that our Hubbard model approach
in the limit of large U0/t should give results qualita-
tively similar to those found for the tJ model in earlier
renormalization-group studies [13, 14]. The phases of the
tJ model found in these studies are identical to those of
the Hubbard model, except that there is no τHbphase.
Fig. 8 shows the Hubbard model phase diagram in
terms of temperature versus chemical potential and tem-
perature versus electron density for U0/t = 50. At this
large coupling, we do indeed observe a phase diagram
very similar to that found in the earlier study of the tJ
model [13, 14]. In particular, the τtJphase is surrounded
by AF islands, and directly above τtJwe get a lamellar
structure of alternating AF, nHD, and hD phases. The
AF phase near half-filling is unstable to only about 5%
hole doping. This phase diagram can be seen as an evo-
lution from the U0/t = 20 result of Figs. 1(a) and 2(a),
with the τhB entirely disappearing at U0/t = 50 except
for infinitesimal slivers. The multiple peaks in the spe-
cific heat above the τtJtransition persist in the strong-
coupling limit, as seen in Fig. 9, which plots the specific
heat coefficient γ for U/t0= 50 at ?ni? = 0.67. The peak
structure here is more complex than in Fig. 7(a), due to
the above-mentioned lamellar phases.
We can also observe the evolution from the Hubbard to
the tJ limits through the expectation value of the kinetic
energy per bond, ?K? = −?
?ikj?c†
iσ(Sk)σσ′cjσ′, but
this term is usually ignored, from the assumption that
σ?c†
iσcjσ+ c†
jσciσ?, which is
proportional to the density of free carriers in the system.

Page 15

15
Fig. 10 shows ?K? as a function of electron density for
the temperature 1/t = 0.2, calculated at several different
couplings U0/t. As U0/t is increased, the value of ?K?
at half-filling is reduced, and when U0/t = 1000 we are
close to the tJ limit, with the kinetic energy at half-filling
almost zero, indicating no available free carriers due to
the prohibitively high energy of double occupation. The
U0/t = 1000 curve almost exactly overlaps the result cal-
culated from the tJ model renormalization-group equa-
tions at the same temperature using the corresponding
coupling J/t = 4t/U0= 0.004.
Acknowledgments
This research was supported by the U.S. Department
of Energy under Grant No. DE-FG02-92ER-45473, by
the Scientific and Technical Research Council of Turkey
(T¨UBITAK) and by the Academy of Sciences of Turkey.
MH gratefully acknowledges the hospitality of the Feza
G¨ ursey Research Institute and of the Physics Department
of Istanbul Technical University.
APPENDIX A: DETERMINATION OF THE γp
IN TERMS OF THE MATRIX ELEMENTS
OF THE THREE-SITE HAMILTONIAN
Eq.(8) allows us to express the matrix elements γp≡
?φp|e−β′H′(i,k)|φp? of the renormalized, exponentiated
two-site Hamiltonian in terms of matrix elements of the
unrenormalized, exponentiated three-site Hamiltonian,
as given below. The γp, in turn, determine the renormal-
ized interaction constants, in Eq.(10). In the equations
below, ?ψq||ψ¯ q? denotes ?ψq|e−βH(i,j)−βH(j,k)|ψ¯ q?:
γ1=?ψ1||ψ1?+2?ψ2||ψ2?+?ψ9||ψ9?,
γ2=?ψ3||ψ3?+1
2?ψ8||ψ8?+3
2?ψ15||ψ15?+?ψ24||ψ24?,
γ4=?ψ6||ψ6?+1
2?ψ11||ψ11?+3
2?ψ20||ψ20?+?ψ32||ψ32?,
γ6=?ψ10||ψ10?+2?ψ26||ψ26?+?ψ44||ψ44?,
γ7=?ψ13||ψ13?+?ψ34||ψ34?+?ψ38||ψ38?+?ψ48||ψ48?,
γ8=?ψ12||ψ12?+2?ψ31||ψ31?+?ψ47||ψ47?,
γ9=?ψ14||ψ14?+2
3?ψ23||ψ23?+4
3?ψ39||ψ39?+?ψ49||ψ49?,
γ12=?ψ25||ψ25?+1
γ14=?ψ33||ψ33?+1
2?ψ45||ψ45?+3
2?ψ46||ψ46?+3
2?ψ50||ψ50?+?ψ59||ψ59?,
2?ψ55||ψ55?+?ψ62||ψ62?,
γ16=?ψ43||ψ43?+2?ψ58||ψ58?+?ψ64||ψ64?,
γ0≡?φ6|e−β′H′(i,k)|φ8?
=?ψ10||ψ12?+2?ψ26||ψ31?+?ψ44||ψ47?.
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