# Fractional statistics of charge carriers in the one- and two-dimensional t-J
model. A hint for the cuprates?

P. A. Marchetti∗ Dipartimento di Fisica e Astronomia, Universitá di Padova,
INFN,
Padova, I-35131,Italy
∗E-mail<EMAIL_ADDRESS>
###### Abstract

We show that we can interpret the exact solution of the one-dimensional t-J
model in the limit of small J in terms of charge carriers with both exchange
(braid) and exclusion (Haldane) statistics with parameter 1/2. We discuss an
implementation of the same statistics in the two-dimensional t-J model,
emphasizing similarities and differences with respect to one dimension. In
both cases the exclusion statistics is a consequence of the no-double
occupation constraint. We argue that the application of this formalism to
hole-doped high Tc cuprates and the derived composite nature of the hole give
a hint to grasp many unusual properties of these materials.

##  1 Introduction

This paper is a brief review of the attempt to assign 1/2 Haldane statistics
to the charge carriers of the one- and two-dimensional $t$-$J$ model,
comparing the two cases and arguing that the second case is relevant for the
low-energy physics of hole-doped high $T_{c}$ cuprates. The emphasis is on
ideas and only few mathematical details are given.

In quantum systems of identical particles one can define two kinds of
statistics. One is the exchange (or braid) statistics: the ($\bf C$-valued)
many-body wave-function acquires a phase factor $e^{i(1-\alpha)\pi}$, with
$\alpha\in[0,2[$, when one performs a positively oriented exchange between two
particles and the inverse phase factor for a negatively oriented exchange. The
parameter $\alpha$ is 0 for fermions, 1 for bosons, and for the other values
the corresponding particles are generically called anyons (see e.g.
[Wilczek(1990)]), in particular for $1-\alpha=\pm 1/2$ they are called semions
and in this paper we will be mainly interested in this case. The same phase
factors arise in the quantum field theory setting if we perform an equal-time
oriented exchange of the fields creating the corresponding particles
[Fröhlich(1989)]. Anyons require an hard-core exclusion condition and can
exist only in space-dimension $D<3$, where the exchange of two identical
hardcore particles can be oriented, namely, one can distinguish an exchange
and its inverse. This is possible in one dimension (1D) because the real line
is oriented and in two dimensions (2D) because one cannot continuously connect
exchanges with opposite orientation. Since oriented exchanges generate the
braid groups [Artin(1947)], this statistics is also called braid statistics.
This connection can be made more intuitive using the path-integral formalism,
where, due to the boundary conditions in time, each configuration of the
worldlines of the particles form a braid if there are no intersections.
(Actually, using the fact that the probability for two brownian paths in $\bf
R^{2}$ to intersect each other at a fixed time is zero, the hard-core
condition is not necessary in 2D in the continuum [Fröhlich(1992)])

Another kind of statistics can be defined for quasi-particles in finite-
density quantum systems of identical particles: the Haldane’s fractional
exclusion statistics, generalizing Pauli exclusion principle for fermions
[Haldane(1991)]. It is identified by a parameter $g$ measuring the effective
interaction among quasi-particles occupying the same state in the one-particle
Hilbert space. A way of characterizing this statistics at $T=0$ is the
following [Wu(1994)]: A quasi-particle obeys exclusion statistics with
parameter $g$ if the quasi-particle density, denoted by $n_{g}$, having the
same volume enclosed by the Fermi surface of the Fermi gas with density
$n_{0}$, satisfies

$n_{0}/n_{g}=(1-g).$ (1)

Clearly, $g=0$ for standard fermions and other intermediate values of $g$ in
$(0,1)$ define a fractional exclusion statistics. In particular, and this will
be the case of interest here, this implies that at a fixed momentum
(neglecting other internal degrees of freedom) a quasi-particle with exclusion
statistics 1/2 can have an occupation number twice that of a free fermion, so
that the volume of its Fermi surface is half of that of a Fermi gas with the
same density.

In this paper, following [Marchetti(1996), Weng(1997), Marchetti(2019)], we
show that we can attribute consistently both exchange and exclusion statistics
1/2 to the charge carriers of the one- and two- dimensional $t$-$J$ model. A
quite interesting physical application of these ideas is a derived explanation
of many features of the low-energy physics of high $T_{c}$ cuprates. In fact,
most of the researchers believe that for the hole-doped cuprates the key
actors in the superconducting transition are the so-called Zhang-Rice singlets
[Zhang(1988)], that can be modeled as empty sites of a 2D $t$-$J$ model on a
square lattice representing the Cu sites in the CuO planes of the cuprates.
The model hamiltonian for a plane is then given by:

$H_{t-J}=\sum_{\langle
i,j\rangle}P_{G}\left[-tc^{*}_{i\alpha}c_{j\alpha}+J{\vec{S}}_{i}\cdot{\vec{S}}_{j}\right]P_{G},$
(2)

where $i,j$ denote nearest neighbour (nn) sites of the lattice, $c$ the hole
field operator, $P_{G}$ the Gutzwiller projection eliminating double
occupation, $\alpha$ the spin indices and summation over repeated spin (and
vector) indices is understood, here and in the following. For the cuprates a
typical value for the nn hopping is $t\approx 0.3\hskip 2.0pt\mathrm{eV}$ and
for the anti-ferromagnetic Heisenberg coupling is $J\approx 0.1\hskip
2.0pt\mathrm{eV}$. (Actually to get a reasonable shape of the Fermi surface in
the tight binding approximation one adds to the nn hopping term at least a nnn
term with coefficient $t^{\prime}$, $|t^{\prime}|\sim 0.03-0.1\hskip
2.0pt\mathrm{eV}$, strongly material dependent, but it will not be considered
here, except in the final section). As already mentioned we will be interested
also in the 1D version of the model which can be exactly solved by Bethe
ansatz [Ogata(1990)] and conformal field theory techniques [Ren(1993)], but,
following [Marchetti(1996), Weng(1997)], here will be solved via spin-charge
decomposition. This solution suggests the main strategy to treat the
physically more relevant 2D version.

Probably the most interesting features of this approach applied to the
cuprates are the following: First, the holes are composites made of only
weakly bound spinless holons, charge carriers with Fermi surface, and spinons,
spin carriers without Fermi surface, so that some physical responses dominated
by the spinons have a totally non-Fermi liquid character. This occurs in spite
of the fact that the holes have, in the overdoped region, a Fermi surface
satisfying Luttinger theorem, precisely due to the 1/2 Haldane statistics of
the holons. Second, the semionic statistics of the holons, generated by
attached charge vortices, implies the appearance, in the spin counterpart, of
antiferromagnetic spin vortices with opposite chirality when centered on
holons in two different Néel sublattices. Lowering the temperature such gas of
vortices undergoes a Kosterlitz-Thouless-like transition, with the formation
of a finite density of vortex-antivortex pairs and since the vortices are
centered on the charge carriers, this provides a novel topological mechanism
of charge-pairing finally leading to superconductivity [Marchetti(2011)].

##  2 The spin-charge decomposition

Let us outline the formalism of spin-charge decomposition that we use.
Inspired by an idea pioneered by Anderson [Anderson(1987)] and Kivelson
[Kivelson(1987)], suggested for the cuprates by the rather different response
of charge and spin degrees of freedom in many experiments, one can tackle the
no-double occupation constraint enforced by the Gutzwiller projector $P_{G}$
by rewriting the hole field $c_{\alpha}$ as a product of a charged spinless
fermion field $h$, the holon, and a neutral spin 1/2 boson field
$\tilde{s}_{\alpha}$, the spinon, as $c_{\alpha}=h^{*}\tilde{s}_{\alpha}$
[Marchetti(1998)]. (We use the tilde here because we will denote by
$s_{\alpha}$ a slightly different field). Being spinless, by the Pauli
principle the holon $h$ implements exactly the Gutzwiller constraint.
Furthermore if one imposes the constraint
$\tilde{s}^{*}_{\alpha}\tilde{s}_{\alpha}=1$, then, since
$c^{*}_{\alpha}c_{\alpha}=1-h^{*}h$, one sees that $h^{*}h$ is just the
density of empty sites in the model. However, if we treat the holon in mean-
field, because it is spinless the Fermi surface that we get is not the desired
one. In 1D , from the exact solution, and in 2D, from experiments for
overdoped cuprates (see e.g. [Hussey(2003)]) , we know that the Fermi surface
satisfies the Luttinger theorem, hence the D-volume enclosed by the Fermi
surface of fermionic holons is twice the desired one. This suggests that an
exclusion statistics with parameter 1/2 would be the appropriate solution to
this problem. We now prove that precisely this exclusion statistics appears if
the holons are not fermions but semions and show that this exchange statistics
can be supported by an argument of energy optimization. To implement the
semionic exchange statistics in the path-integral formalism we start from 2D
and make use of the following

Theorem [Fröhlich(1992)][Marchetti(1996)] We embed the lattice of the 2D
$t$-$J$ model in a 2-dimensional space and denote by $x=(x^{0},x^{1},x^{2})$
the coordinates of the corresponding 2+1 euclidean space-time, $x^{0}$ being
the euclidean time. We couple the fermions of the $t$-$J$ model to a $U(1)$
gauge field, $B_{\mu}$, gauging the global charge symmetry, and to an $SU(2)$
gauge field, $V_{\mu}$, gauging the global spin symmetry of the model, and we
assume that the dynamics of the gauge fields is described by the Chern-Simons
action $-2S_{c.s.}^{U(1)}(B)+S_{c.s.}^{SU(2)}(V)$, with

$S_{c.s.}^{U(1)}(B)=\frac{1}{4\pi}\int
d^{3}x\epsilon_{\mu\nu\rho}B_{\mu}\partial_{\nu}B_{\rho}(x),$
$S_{c.s.}^{SU(2)}(V)=\frac{1}{4\pi}\int d^{3}x{\rm
Tr}\epsilon_{\mu\nu\rho}[V_{\mu}\partial_{\nu}V_{\rho}+\frac{2}{3}V_{\mu}V_{\nu}V_{\rho}](x),$
(3)

where $\epsilon_{\mu\nu\rho}$ is the Levi-Civita anti-symmetric tensor in 3D
and $\partial_{\nu}=\partial/\partial x^{\nu}$. Then the spin-charge (or
$SU(2)\times U(1)$) gauged model so obtained is exactly equivalent to the
original $t$-$J$ model. In particular the spin and charge invariant euclidean
correlation functions of the fermions fields $c_{j\alpha}$ of the $t$-$J$
model are exactly equal to the correlation functions of the fields
$\exp[i\int_{\gamma_{j}}B]P(\exp[i\int_{\gamma_{j}}V])_{\alpha\beta}c_{j\beta}$
, where $c$ denotes now the fermion field of the gauged model, ${\gamma_{j}}$
a string at constant euclidean time connecting the point $j$ to infinity and
$P(\cdot)$ the path-ordering, which amounts to the usual time ordering
$T(\cdot)$, when “time” is used to parametrize the curve along which one
integrates.

The two crucial points in the proof of the above theorem are: First, due to a
result of Witten [Witten(1989)] the integration of the two above gauge fields
on a set of closed string forming a braid averaged with the above Chern-Simons
actions exactly cancel against each other. Second, the worldlines of the
fermions of the $t$-$J$ model in the partition function (joined to the strings
in the correlation functions) form braids , because they have no
intersections, due to the Gutzwiller constraint.

To the fermion field of the gauged model we now apply the spin-charge
decomposition discussed above, rewriting it as a product of a holon and a
spinon field. A good feature of introducing the above gauge fields is that
they allow a more flexible treatment of charge and spin responses within a
spin-charge decomposition scheme, attaching the ”charge” $B$-string to the
holon and the ”spin” $V$-string to the spinon. Spin and charge responses are
in fact quite independent in 1D and this turns out to be partially true even
in 2D. One can verify [Fröhlich(1992), Marchetti(1996)] that the non-local
quantum field operator corresponding to the product of the fermionic holon
field $h$ and the ”charge gauge string” $\exp[i\int_{\gamma_{j}}B]$ obeys
semionic statistics and the same is true for the product of the bosonic spinon
field $\tilde{s}$ and the ”spin gauge string” $P\exp[i\int_{\gamma_{j}}V]$.
After the holon-spinon decomposition introduced above has been implemented the
action of the gauged $t$-$J$ model at doping $\delta$ is given by
[Marchetti(1998)] :

$\displaystyle
S(h,h^{*},\tilde{s}^{*},\tilde{s},B,V)=\int_{0}^{\beta}dx^{0}\sum_{j}h^{*}_{j}[\partial_{0}-iB_{0}(j)-2t\delta]h_{j}+iB_{0}(j)+i(1-$
(4) $\displaystyle
h^{*}_{j}h_{j})(\tilde{s}_{j\alpha}^{*}V_{0}(j)_{\alpha\delta}\tilde{s}_{j\delta})-\sum_{\langle
i,j\rangle}th^{*}_{j}\exp[i\int_{\langle
i,j\rangle}B]h_{i}(\tilde{s}_{i\alpha}^{*}P(\exp[i\int_{\langle
i,j\rangle}V])_{\alpha\delta}\tilde{s}_{j\delta})]$
$\displaystyle+\frac{J}{2}(1-h^{*}_{j}h_{j})(1-h^{*}_{i}h_{i})[|(\tilde{s}_{i\alpha}^{*}P(\exp[i\int_{\langle
i,j\rangle}V])_{\alpha\delta}\tilde{s}_{j\delta})|^{2}-\frac{1}{2}]-2S_{c.s.}^{U(1)}(B)+S_{c.s.}^{SU(2)}(V),$

where $\beta=1/k_{B}T$. The two-point euclidean correlation function of the
hole can be written as:

$\displaystyle G(x,y)=\langle
c^{*}_{x\alpha}(x^{0})c_{y\alpha}(y^{0})\rangle=\int\mathcal{D}h\mathcal{D}h^{*}\mathcal{D}\tilde{s}\mathcal{D}\tilde{s}^{*}\mathcal{D}B\mathcal{D}V\exp[-S(h,h^{*},\tilde{s},\tilde{s}^{*},B,V)]$
$\displaystyle
h_{x}\exp[-i\int_{\gamma_{x}}B](x^{0})\exp[i\int_{\gamma_{y}}B]h^{*}_{y}(y^{0})(\tilde{s}_{x\beta}^{*}P(\exp[-i\int_{\gamma_{x}}V])_{\beta\alpha})(x^{0})(P(\exp[i\int_{\gamma_{y}}V])_{\alpha\delta}\tilde{s}_{y\delta})(y^{0})$
$\displaystyle[\int\mathcal{D}h\mathcal{D}h^{*}\mathcal{D}\tilde{s}\mathcal{D}\tilde{s}^{*}\mathcal{D}B\mathcal{D}V\exp[-S(h,h^{*},\tilde{s},\tilde{s}^{*},B,V)]]^{-1},$
(5)

where $x,y$ are lattice sites and $x^{0},y^{0}$ are euclidean times, which in
the following will be often understood, if no confusion can arise.

##  3 The one-dimensional $t$-$J$ model

To discuss the one-dimensional model we perform a dimensional reduction of the
gauged model defined in Theorem 1, by restricting the spatial support of the
hole fields to a line, let’s say along $x^{1}$, keeping the strings $\gamma$
not along this line to avoid intersections with the hole worldlines making the
previous theorem inapplicable.

In 1D one can give an intuitive picture of holons and spinons as follow (see
e.g.[Giamarchi(2003)]): Consider the Heisenberg spin 1/2 chain, describing the
1D $t$-$J$ model in the limit of zero doping and take as reference state the
one with the spin antiferromagnetically ordered, mimicking the Néel order
appearing in 2D. If we insert a dopant by removing the spin from a site then
the two neighbouring spins will be ferromagnetically aligned. Let then the
empty site to hop by a simultaneous opposite hopping of the spin, then we get
two separate excitations. There is an empty site, but with neighbouring spins
antiferromagnetically aligned, thus carrying charge but not spin; the
corresponding excitation is the holon. There is another site where one finds a
domain wall between two different Néel sublattices, hence carrying spin 1/2
but neutral; the corresponding excitation is the spinon. Notice, however, that
attached to the site with a spin mismatch of 1/2 there is a string of spins
flipped w.r.t. the reference state from that site to the holon position, which
is integral part of the spinon excitation and corresponds to the ”spin string”
introduced in Theorem 1. In the continuum semiclassical limit one can view the
spinon as a spin kink, continuum analog of the above domain wall.

Let’s make the above picture mathematically precise in the spin-charge
decomposition formalism previously outlined. Both in 1 and 2 D the gauged
model equivalent to the $t$-$J$ model has three local gauge invariances: the
”charge” $U(1)$, the ”spin” $SU(2)$ and the slave-particle corresponding to
the multiplication of the holon field $h$ and the spinon field
$\tilde{s}_{\alpha}$ by the same local phase, thus leaving the hole field
unchanged.

In 1D [Marchetti(1996)] we gauge-fix the first symmetry by setting $B_{2}=0$,
the second one, as suggested by the previous ”picture”, by setting
$\tilde{s}_{j\alpha}=\sigma_{x}^{j}(10)^{t}$, where $\sigma_{x}$ denotes the
Pauli matrix and the superscript $t$ the transpose. Finally the slave-particle
gauge symmetry is fixed by imposing a kind of 1D Coulomb gauge:
$\arg(\tilde{s}_{\alpha i}^{*}P(\exp[i\int_{\langle
i,j\rangle}V])_{\alpha\beta}\tilde{s}_{\beta j})=0$. We now turn to the
discussion of charge and spin carriers. We start with the ”charge carrier”.

Since the 0-component of $B$ appears linearly in (4) , we can safely integrate
it out getting the constraint (with $\mu,\nu=1,2,z\in\mathbf{R}^{3}$):

$\displaystyle\epsilon_{\mu\nu}\partial_{\mu}B_{\nu}(z)=-\pi\delta(z^{2})[\sum_{j}\delta(z^{1}-j)(1-h^{*}_{j}h_{j})(z^{0})].$
(6)

By imposing the gauge-fixing $B_{2}=0$ one finally gets

$\displaystyle B_{1}(z)=-\frac{\pi}{2}{\rm
sgn}(z^{2})[\sum_{j}\delta(z^{1}-j)(1-h^{*}_{j}h_{j})(z^{0}))],$ (7)

with the convention ${\rm sgn}(0)=0$, so that the holon field turns out to be
given by

$\displaystyle\exp[i\int_{\gamma_{j}}B]h_{j}=\exp[\pm
i\frac{\pi}{2}\sum_{\ell>j}(1-h^{*}_{\ell}h_{\ell})]h_{j},$ (8)

with the sign in the exponent being positive (negative) if the angle between
the $x^{1}$ axis and $\gamma$ is positive (negative) in $]-\pi,\pi[$.
Therefore the sign depends on the choice of the orientation of the $x^{1}$
line in the spatial plane; since the original model is independent of this
choice we take an average of the two possibilities. If the doping is $\delta$,
the fermion $c_{\alpha}$ of the 1D $t$-$J$ model in the tight-binding
approximation has a Fermi momentum $\pi(1-\delta)/2$ since two fermions with
opposite spin can have the same momentum. For the spinless fermion $h^{*}$ the
Fermi momentum would be $\pi(1-\delta)$ since only one spinless fermion con
have a fixed momentum. However, if we consider the phase string attached to
the spinless fermion in (8) we see that, since the expectation value of
$h^{*}_{\ell}h_{\ell}$ is $\delta$, in the leading term it contributes to the
Fermi momentum a term $-\pi(1-\delta)/2$. Hence the distance between the two
Fermi point of the semionic holon of (8) is $\pi(1-\delta)$ and it obeys an
Haldane statistics of parameter 1/2.

To derive the holon contribution to the euclidean $c$-correlator $G(x,y)$ in
the scaling limit we decompose each $h$ into right and left movers as
$h(x)\sim h_{R}(x)e^{i\pi(1-\delta)x^{1}}+h_{L}(x)e^{-i\pi(1-\delta)x^{1}}$ ,
the spinor doublet $H=(h_{R},h_{L})^{t}$ having the dynamics of a massless
Dirac fermion . Using as in [Ye(2017)] the Schwinger formula [Schwinger(1962)]
for the two-point correlation functions of massless 2D Dirac fields coupled to
gauge fields, one finds as leading holon contributions corresponding to the
two string directions: $e^{\pm i\pi(1-\delta)x^{1}/2}[(x^{1}-y^{1})\pm
iv_{c}(x^{0}-y^{0})]^{-1/2}[(x^{1}-y^{1})^{2}+v^{2}_{c}(x^{0}-y^{0})^{2}]^{-1/16},$
where $v_{c}\sim t_{R}$ denotes the charge velocity.

Let us turn to the ”spin carrier”. Since we have already gauge-fixed the
$SU(2)$ gauge-invariance using the spinons, the gauge field $V$ has to be
integrated without gauge-fixing. This can be implemented by splitting the
integration over $V$ into an integration over a field $V^{m}$, satisfying the
gauge-fixing condition $V^{m}_{2}=0$ and its gauge transformations expressed
in terms of an $SU(2)$-valued scalar field $g$, i.e.,
$V_{\mu}=g^{\dagger}V^{m}_{\mu}g+g^{\dagger}\partial_{\mu}g,\mu=0,1,2$. We now
find the configuration of $g$ that optimize the partition function of holons
in a fixed, but holon-dependent, $g$-background. It is rigorously proved in
Ref.[Marchetti(1996)] that such configuration is given by
$g^{m}_{j}=\exp[-i\frac{\pi}{2}\sigma_{x}\sum_{\ell>j}h^{*}_{\ell}h_{\ell}]$.
This is the “string” of spin flips that we have encountered in the previous
”pictorial” discussion. Finally we set $g=Ug^{m}$, with $U\in SU(2)$
describing the fluctuations around the optimal configuration. We write

$\displaystyle U=\begin{pmatrix}s_{1}&-s_{2}^{*}\\\
s_{2}&s_{1}^{*}\end{pmatrix},$ (9)

and we will call in the following the $s_{\alpha}$ again “spinons”. As a first
“mean field” approximation we neglect in the calculation of $V^{m}$ the
fluctuations of the spinons in the directions orthogonal to the spins of the
optimal configuration. Proceeding now as done for the $B$ field and
integrating over $V^{m}_{0}$ one finds

$\displaystyle V^{m}_{1}(z)=\frac{\pi}{2}{\rm
sgn}(z^{2})[\sum_{j}\delta(z^{1}-j)(1-h^{*}_{j}h_{j})(z^{0}))(s_{1j}^{*}s_{1j}-s_{2j}^{*}s_{j2})]\sigma_{3}.$
(10)

Let us remark an interesting feature [Marchetti(2007)] of the above approach:
thanks to the holon-depending spin flips, since the holon in a oriented
hopping link is at its end, $(\tilde{s}_{i\alpha}^{*}P(\exp[i\int_{\langle
i,j\rangle}V]))_{\alpha\delta}\tilde{s}_{j\delta})$ in the $t$-term of (4)
equals
$(\sigma_{x}^{|i|}U_{i}^{\dagger}U_{j}\sigma_{x}^{|i|})_{11}=s^{*}_{\alpha
i}s_{\alpha j}$, whereas in the $J$-term, where there are no holons, it equals
$(\sigma_{x}^{|i|}U_{i}^{\dagger}U_{j}\sigma_{x}^{|j|})_{11}=\epsilon_{\alpha\beta}s_{\alpha
i}s_{\beta j}$. But under the constraint $s^{*}_{\alpha j}s_{\alpha j}=1$ the
following identity holds: $|s^{*}_{\alpha i}s_{\alpha
j}|^{2}+|\epsilon_{\alpha\beta}s_{\alpha i}s_{\beta j}|^{2}=1$ so that if one
optimizes the $t$-term choosing $|s^{*}_{\alpha i}s_{\alpha j}|=1$,
simultaneously one optimizes also the $J$-term. This optimizing property is
somewhat strange, since the same expression in terms of $V$ gives rise to
different expressions in terms of the spinons $s$ in the $t$\- and the
$J$-terms. It is intrinsically due to the $SU(2)$ gauge degrees of freedom,
absent in the more standard slave-particle approaches involving only $U(1)$
gauge degrees of freedom. An exception is the ”string formalism” of Weng
[Weng(1997)], in 1D essentially equivalent to the charge-spin decomposition
discussed here. The optimization is achieved there using the Gutzwiller
projection in the squeezed chain (see below) of occupied sites, the only ones
in which spinons are defined in this Hamiltonian formalism, a feature that
however has no natural extension to 2D.

As a consequence of the spin flips the gauge-fixing for the slave-particle
gauge symmetry in the present formalism becomes $\arg(s_{\alpha
i}^{*}s_{\alpha j})=0$ in the $t$-term, but in the links of the $J$-term it is
given by $\arg(s_{\alpha i}^{*}\epsilon_{\alpha\beta}s_{\beta j})=0$. This
fact is used in the following further approximations. We assume that the spin
fluctuations can be treated in mean field in the hopping term because the
corresponding term is real due to the discussed slave-particle gauge fixing
and we denote by $t_{R}$ the renormalized hopping. Furthermore since the paths
of spinons cannot overlap due to the Gutzwiller constraint we replace for the
spinon motion the original chain by a squeezed chain of lattice spacing
$(1-\delta)$ with single occupancy constraint. We redefine the spinons in the
squeezed chain, denoted by $\mathrm{s}$, through $\mathrm{s}_{\alpha
j}=(U_{j}\sigma_{x}^{|j|})_{\alpha 1}$. Written in terms of $\mathrm{s}$ the
$J$-term is just the Heisenberg model in the Schwinger-boson representation.
The slave-particle gauge fixing becomes $\arg(\mathrm{s}_{\alpha
i}^{*}\mathrm{s}_{\alpha j})=0$. Then approximating the term quartic in
$\mathrm{s}$ by mean field and making using of the reality condition imposed
by the gauge-fixing, the $J$-term can be rewritten as

$\displaystyle\sum_{i}\mathrm{s}_{\alpha
i}^{*}\partial_{\tau}\mathrm{s}_{\alpha i}+J_{R}\sum_{\langle
i,j\rangle}(\mathrm{s}_{\alpha i}^{*}\mathrm{s}_{\alpha
j}+h.c.)+i\sum_{i}\lambda_{i}(\mathrm{s}_{\alpha i}^{*}\mathrm{s}_{\alpha
i}-1)$ (11)

with $\lambda$ a Lagrange multiplier field enforcing the single-occupancy
constraint and $J_{R}$ a renormalized spin coupling constant. (The absence of
a possible gap term derived from umklapp is justified by its absence in the
Heisenberg model, see e.g. [Fradkin(2013)]).

One can now compute the long-wavelength continuum limit of the spinon
contribution to the euclidean correlator $G(x,y)$ of the fermion $c$ of the
$t$-$J$ model. Motivated by [Marchetti(1996)], but following closer the
formalism of [Weng(1997)], one first fermionize separately the two components
of $\mathrm{s}$ by a Jordan-Wigner transformation and we denote by
$f_{\alpha}$ the corresponding fields. Reinserting the ”spin string”, the
semionic spinon field in the squeezed chain is written in terms of $f$ as
$f_{\alpha
j}\exp[(-1)^{\alpha}(\pm)i\frac{\pi}{2}\sum_{\ell>j}f^{*}_{\beta\ell}f_{\beta\ell})]$,
where the sign $\pm$ depend on the direction of the ”spin string” w.r.t. the
$x^{1}$ axis, as discussed for the holon. Similarly, in the scaling limit we
decompose each $f_{\alpha}$ into right and left movers as

$\displaystyle f_{\alpha}(x)\sim f_{\alpha R}(x)e^{ik_{f}x^{1}}+f_{\alpha
L}(x)e^{-ik_{f}x^{1}},$ (12)

$k_{f}$ being the spinon Fermi momentum in the squeezed chain, with massless
Dirac spinor doublets $F_{\alpha}=(f_{\alpha R},f_{\alpha L})^{t}$, plus the
single-occupancy constraint. In the leading term the mean-field contribution
of the exponential in the ”spin string” exactly cancels one of the two $k_{f}$
exponentials in (12) so that in this approach the spinons do not contribute to
the Fermi momenta of the fermion $c$ of the $t$-$J$ model. Hence, taking into
account the Haldane statistics of the holon, when the electron is
reconstructed combining the holon with the spinon, it has the same Fermi
surface of the tight-binding approximation for the fermion, as we wanted to
prove, thus satisfying Luttinger theorem.

For completeness we now outline how to finalize the computation of the $c$
correlator. To implement exactly the single-occupancy constraint we use the
technique of duality (equivalent to bosonization in 1D [Burgess(1994),
Fröhlich(1995), Marchetti(1995)]): we couple minimally each $F_{\alpha}$ with
a $U(1)$ gauge field $A_{\alpha\mu},\mu=0,1$ and impose the constraint that
the field strengths of $A_{\alpha}$ are zero, by inserting in the Lagrangian a
term with multiplier fields, denoted by $\phi_{\alpha}$, given by
$i\epsilon_{\mu\nu}\partial_{\nu}\phi_{\alpha}A_{\alpha\mu}$. Defining
$\phi_{\pm}=(\phi_{1}\pm\phi_{2})/2$ a change of variable, re-adsorbing
$\lambda$ into the gauge fields $A_{\alpha}$ produces the constraint
$\phi_{+}=0$. Then using the procedure of [Ye(2017)] integrating out
$A_{\alpha}$ and $\phi_{-}$ one gets for the spinon contributions with the two
directions of the strings: $[(x^{1}-y^{1})\pm iv_{s}(x^{0}-y^{0})]^{-1/2}$,
where $v_{s}\sim J_{R}$ denotes the spin velocity. Combining the holon and
spinon contribution one thus reproduces [Marchetti(1996), Weng(1997)] the
correct long-wavelength limit of the correlation function $G(x,y)$ of the 1D
$t$-$J$ previously obtained by Bethe ansatz and conformal field theory
techniques .

##  4 The two-dimensional $t$-$J$ model

If we try naively to export to 2D the string mechanism discussed in 1D we
immediately find a big difference already in the intuitive picture: in
correspondence to the string of spin flips between the holon and spinon
positions in a chain, the two parallel adjacent chains have the spin with the
same orientation of those of the string. and since the spin interaction is
antiferromagnetic this configuration costs an energy proportional to the
length of the string, i.e. holon and spinon are confined: this is a
manifestation of the slave-particle gauge force. However, this is not an
optimal solution from the point of view of energy. To get a suggestion for
improvement one notices that the kink corresponding to the continuum limit of
the spin string is the typical semiclassical excitation in 1D, but not in 2D,
where its role is played by the vortex. If we accept the suggestion coming
from the 1D model for the exchange statistics of holon and spinon, one should
search for a semionic representation of the hole field also in 2D and
precisely a charge-vortex attached to a fermionic ”bare” holon converts it
into a semion. As explained later, the spin-vortex generated by consistency
will also play a key role in the formalism. These vortices are somewhat
analogous to those introduced by Laughlin in the Fractional Quantum Hall
Effect and in fact a semionic representation of the hole was advocated by him
[Laughlin(1988)] quite soon after the discovery of high $T_{c}$. As we sketch
below, in 2D, as in 1D, this semionic holon obeys a 1/2 exclusion statistics.

We now make these ideas precise in the spin-charge gauge formalism. For
convenience we start with the spin excitation, since some of its features will
be needed in the description of the charge carrier, but we try to follow as
close as possible the procedure discussed in 1D. Analogously we introduce the
gauge-fixing $\tilde{s}_{j\alpha}=\sigma_{x}^{j}(10)^{t}$ for the ”spin” gauge
symmetry, the field $V^{m}$ satisfying the Coulomb gauge-fixing condition
$\partial_{\mu}V^{m}_{\mu}=0,\mu=1,2$ and its gauge transformations, again
written in terms of an $SU(2)$-valued $g$ field. We didn’t succeeded to find
rigorously, as in 1D, a configuration of $g$, depending on the holon
configuration, optimizing the holon-partition function in that $g$ background,
but we still found a configuration $g^{m}$ optimal “on average”, in a Born-
Oppenheimer approximation [Marchetti(1998)]. As in 1D in this configuration
the spinons $\tilde{s}$ are antiferromagnetically ordered but there is in
addition a spin flip on the sites where holons are present, as for the final
site of a hopping link of holons at the time of hopping. Hence, neglecting
$V^{m}$, the argument given in the previous section on the simultaneous
optimization in this formalism of the $t$ and $J$ terms holds also in 2D,
giving a motivation “a priori” for this spin-charge gauge approach.

Spin fluctuations around the ”optimal” configuration are described by the
$SU(2)$\- matrix field $U$ whose components are written, as in 1D, in terms of
spinon fields $s_{\alpha}$. At least in the long wavelength continuum limit it
is convenient to make the slave-particle gauge symmetry explicit by
introducing the related gauge field that we denote by $A_{\mu}$. This can be
done in absence of holons simply by assuming a continuum limit of the spinon
field $s$ of the form $s_{x\alpha}(x^{0})\longrightarrow
s_{\alpha}(x)+(-1)^{|x|}p_{\alpha}(x)$, where $s$ and $p$ in the r.h.s. are
continuum fields and $|x|=x^{1}+x^{2}$. Integrating out the ferromagnetic
component $p$, as shown in [Read(1989)], both in 1D and 2D we obtain a low-
energy spinon Lagrangian in the form of a CP1 model:

$\displaystyle\mathcal{L}_{s}=\frac{1}{g}\left[\left|\left(\partial_{0}-iA_{0}\right)s_{\alpha}\right|^{2}+v_{s}^{2}\left|\left(\vec{\nabla}-i\vec{A}\right)s_{\alpha}\right|^{2}\right],$
(13)

where $g\sim J$ and $v_{s}\sim J$ is the spinon velocity, with the lattice
spacing set to 1. Furthermore
$A_{\mu}(x)=(-1)^{|x|}s^{*}_{\alpha}(x)\partial_{\mu}s_{\alpha}(x)$, with
$\mu$ a space-time index, and the implicit constraint
$s^{*}_{\alpha}s_{\alpha}=1$ is understood. For later purposes we remark that
in 1D there is an additional $\Theta$-term

$\displaystyle\mathcal{L}_{\Theta}=i\frac{\Theta}{2\pi}\epsilon_{\mu\nu}\partial_{\mu}A_{\nu},$
(14)

with $\Theta=\pi$.

In 2D neglecting, with a stronger approximation w.r.t. 1D , the spinon
fluctuations $U$ in the computation of $V^{m}$ one gets ( with $\mu=1,2)$:

$\displaystyle
V^{m}_{\mu}(z)=-\frac{1}{2}\sum_{j}(-1)^{|j|}\partial_{\mu}\arg(\vec{z}-j)h^{*}_{j}h_{j}(z^{0})\sigma_{z},$
(15)

with $|j|=j^{1}+j^{2}$. We recognize in the term
$(-1)^{|j|}\partial_{\mu}\arg(\vec{z}-j)$ the vector potential of a vortex
centered on the holon position $j$, with opposite vorticity (or chirality) for
the center in opposite Néel sublattices. We call these vortices
antiferromagnetic spin vortices; they are the topological excitations of the
$U(1)$ subgroup of the $SU(2)$ spin group unbroken in the antiferromagnetic
phase. Hence they are still a peculiar manifestation of the antiferromagnetic
interaction, like the more standard antiferromagnetic spin waves, but they
appear only in 2D. These vortices are of Aharonov-Bohm type, hence a purely
quantum effect, inducing a topological effect far away from the position of
the holon itself, where their classically observable field strength is
supported.

In the low-energy continuum limit one can see from (4), expanding to the
leading power the exponentials in the links, that $V^{m}$ appears linearly in
the $t$-term of the action; since its spatial average vanishes in a “mean
field” treatment we ignore it. With the same approximation in the $J$-term
$V^{m}$ appears instead quadratically and assuming self-consistently that for
sufficiently large $\delta$ the full $SU(2)$ spin symmetry is restored at
large scales, one finds an interaction between vortices and spinons
proportional to:

$\displaystyle\int
d^{3}x(V^{m}_{\mu}V^{m}_{\mu})(x)s^{*}_{\alpha}s_{\alpha}(x).$ (16)

A quenched average, $\langle\cdot\rangle$, over the positions of the center of
the antiferromagnetic spin-vortices yields the following estimate
[Marchetti(1998)]: $\langle
V^{m}_{\mu}V^{m}_{\mu}\rangle\approx\delta|\log\delta|$, thus providing a
mass-gap to the spinons. Hence the gapless spinons $s$ forming the spin waves
of the CP1 (or equivalently $O(3)$) model describing the undoped system,
traveling in a gas of antiferromagnetic spin vortices centered on holons
acquire a gap $m_{s}$ , converting the long-range AF of the undoped model in
the short-range AF when doping exceeds a critical value. This is
selfconsistent with the previous assumption of the $SU(2)$ symmetry
restoration at large scales. Leaving aside for the moment the monomial quartic
in the holons of the $J$-term in (4), we treat in mean field the monomial
quadratic in the holons, so that the antiferromagnetic coupling is
renormalized to $J(1-2\delta)$. We see here the effect of strong reduction of
antiferromagnetism due to the increase of the density of empty sites,
corresponding to Zhang-Rice singlets in the cuprates. As a result one finds
that the continuum limit of the term

$\displaystyle\int dx^{0}$
$\displaystyle\Big{[}\sum_{j}i(\sigma_{x}^{|j|}U^{\dagger}_{j}\partial_{0}U_{j}\sigma_{x}^{|j|})_{11}+\sum_{\langle
i,j\rangle}\frac{J}{2}(1-h^{*}_{j}h_{j}-h^{*}_{i}h_{i})$ (17)
$\displaystyle[|(\sigma_{x}^{|i|}U^{\dagger}_{i}P(\exp[i\int_{\langle
i,j\rangle}V^{m}])U_{j}\sigma_{x}^{|j|})_{11}|^{2}-\frac{1}{2}]\Big{]}(x^{0})$

within the above approximations is given by

$\displaystyle\int
d^{3}x\frac{1}{g}\left[\left|\left(\partial_{0}-ia_{0}\right)s_{\alpha}\right|^{2}+v_{s}^{2}\left|\left(\vec{\nabla}-i\vec{a}\right)s_{\alpha}\right|^{2}+m_{s}^{2}s^{*}_{\alpha}s_{\alpha}\right](x),$
(18)

where $m_{s}\approx J(1-2\delta)\sqrt{\delta|\log\delta|}$. In view of the
action (18) within this formalism it is natural to consider the bosonic
spinons $s_{\alpha}$ as the spin carriers at large scales, at odds with the
semionic statistics of the spinons found in 1D also in the continuum limit.
However, still a ”spin gauge string” will be necessary to reconstruct the
hole, as discussed below. The spinon system behaves as a spin liquid since the
spinon confinement which would appear with the massive action (18) is avoided
by the interaction with the gapless holons, as briefly discussed in the next
section.

We now turn to the charge carrier. We use the Coulomb gauge-fixing
$\partial_{\mu}B_{\mu}=0,\mu=1,2$ for the ”charge” gauge symmetry and
integrating the $B_{0}$ field we obtain $B_{\mu}(z)=B^{m}_{\mu}+b_{\mu}(z)$
where $B^{m}_{\mu}$ introduces a $\pi$-flux phase, i.e. $\exp[\int_{\partial
p}B^{m}]=-1$ for every plaquette $p$ and

$\displaystyle
b_{\mu}(z)=\frac{1}{2}[\sum_{j}\partial_{\mu}\arg(\vec{z}-j)(h^{*}_{j}h_{j})(z^{0}))].$
(19)

We can recognize in $\partial_{\mu}\arg(\vec{z}-j)$ the vector potential of a
vortex centered on the holon position $j$ , i.e. centered on an empty site of
the $t$-$J$ model. At first we neglect the $b$-field. Then, as discussed in
[Affleck(1988)], through Hofstadter mechanism [Hofstadter(1976)] the
$\pi$-flux (chosen in staggered form) converts the holon with tight-binding
dispersion $\omega_{h}\sim 2t[(\cos k_{x}+\cos k_{y})-\delta]$ into a pair of
lattice “Dirac fields”, $\psi_{a}$, with pseudospin index $a$ related to the
two Néel sublattices and with dispersion: $\omega_{h}\sim
2t[\sqrt{\cos^{2}k_{x}+\cos^{2}k_{y}}-\delta]$ restricted to the magnetic
Brillouin zone. One thus obtain two “small FS” centred at
$(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ with Fermi momenta $k_{F}\sim\delta$.

Above a crossover temperature $T^{*}$ we find that the optimal configuration
$g^{m}$ involves also a phase factor cancelling the contribution of $B^{m}$ in
the loops of hopping links of holons, so that the hopping holons feel an
approximately zero flux [Marchetti(2005)] and one recovers the tight-binding
Fermi surface. By analogy with the phase diagram of hole-doped cuprates, we
call pseudogap (PG) the parameter region in the $\delta$-$T$ plane below
$T^{*}$ and strange metal (SM) the region above.

Reintroducing the $b$-field, the non-local field
$h_{j}\exp[\int_{\gamma_{j}}b]$ obeys semionic statistics. We now use a result
proven in [Ye(2015)]:

Theorem In an incompressible 2D Fermi system with Hall conductivity
$\sigma_{H}=1/(2\pi)$, if its fermions are coupled to a Chern-Simons gauge
field $b$ with action $-\frac{1}{\alpha}S_{c.s.}^{U(1)}(b)$ with the
statistical parameter $\alpha=1/2$ turning them into semions, then these
semions obey the Haldane statistics with parameter 1/2.

Let’s sketch the proof. From the Chern-Simons coupling with parameter $\alpha$
we get $\epsilon_{\mu\nu}\partial_{\mu}b_{\nu}=-2\pi\alpha j_{0}$ with $j_{0}$
the charge density. Inside a region $S$ we turn on the Chern-Simons coupling
$\alpha$ adiabatically from 0 to 1/2. For an incompressible Hall system the
generated statistical flux in turn induces a charge current perpendicular to
the boundary of $S$. Denoting by $N(\alpha)$ the particle number in $S$ with
Chern-Simons coupling $\alpha$ and using the continuity equation
$\partial_{0}j_{0}=-\partial_{\mu}j_{\mu},\mu=1,2$ for the charge current
$j_{\mu}$ and the Hall relation
$j_{\mu}=\sigma_{H}\epsilon_{\mu\nu}\partial_{0}b_{\nu}$, we derive

$\displaystyle\partial_{0}N(\alpha)=\int_{S}d^{2}x\partial_{0}j_{0}(x)=-\int_{S}d^{2}x\partial_{\mu}\sigma_{H}\epsilon_{\mu\nu}\partial_{0}b_{\nu}(x)=$
$\displaystyle\int_{S}d^{2}x2\pi\alpha\sigma_{H}j_{0}(x)=2\pi\sigma_{H}\partial_{0}\alpha
N(\alpha).$ (20)

Integrating over the time of the adiabatic process from (4) we get
$N(1/2)(1-\pi\sigma_{H})=N(0)$ and using $\sigma_{H}=1/(2\pi)$ comparing with
(1) we get $g=1/2$.

Hence, in order to show that the semionic charge carriers of the 2D $t$-$J$
model in the present formalism have an exclusion statistics parameter 1/2 we
need to prove that the ”bare” fermionic holon system has Hall conductivity
$1/(2\pi)$ and is incompressible. To prove the first property we start
noticing that if the constraint
$\tilde{s}^{*}_{\alpha}\tilde{s}_{\alpha}=s^{*}_{\alpha}s_{\alpha}=1$ is
imposed, since $c^{*}_{\alpha}c_{\alpha}=1-h^{*}h$, one finds that
$(\exp[i\int_{\gamma_{j}}B]h_{j})^{*}(\exp[i\int_{\gamma_{j}}B]h_{j})=h_{j}^{*}h_{j}$
is just the density of empty sites in the model and the Gutzwiller constraint
is exactly implemented. However, in the large-scale continuum limit we have
seen that the interaction with antiferromagnetic spin vortices makes the
spinon $s(x)$ gapped. This implies that in this limit the single-occupancy
constraint on $s$ is not fully satisfied, as the spinon mass gap is
incompatible with it, so the Gutzwiller projection is not anymore exactly
implemented.

To understand the implication let us at first consider the free fermionic
holons at doping concentration $\delta=0$ in the presence of the staggered
$\pi$-flux implemented by $B^{m}_{\langle i,j\rangle}$. As mentioned above,
they are described by massless Dirac fields with two flavors, $\psi_{a}$, with
dispersion given by two double-cones with vertices at $(\pm\pi/2,\pi/2)$ in a
suitably defined magnetic Brillouin zone. Since the holons are spinless if
they obey fermionic exclusion statistics they fill both the sheets of the
double-cones, hence no dopants can be introduced for $\delta\neq 0$. However,
if the semionic holons obtained by coupling to the $b$-field satisfy the
exclusion statistics with parameter 1/2, at half filling they fill only the
lower bands. Still the holon density doesn’t vanishes even if there are no
empty sites in the original $t$-$J$ model at $\delta=0$. The lower bands are
thus an artifact produced by the violation of the Gutzwiller constraint in the
large-scale continuum limit. Since the semionic holons in the lower bands are
a result of relaxing the Gutzwiller projection, they are “spurious” and
describe the singly occupied sites in the original unprojected lattice model.
When the doping holes are introduced in the $t$-$J$ model, the corresponding
“physical” holons partially fill the upper bands of the double-cones and are
responsible for the low energy physics.

Although the spurious lower bands of holons are not directly relevant to the
low energy physics in the scaling limit, they are responsible for the 1/2
exclusion statistics. In fact, due to parity anomaly [Redlich(1984)], to
compute the Hall conductivity of massless Dirac fields we need to introduce an
infrared regulator (like a mass) with a parameter $m$, respecting the symmetry
of the system; at the end of the computation one takes the limit $m\rightarrow
0$. The mass regulator breaks parity and even after it is sent to zero, for
half-filling it induces a non-vanishing Hall conductivity sign$(m)/4\pi$ for
each double-cone, generated by its lower filled band. In the standard case of
pure $\pi$ flux the overall system preserves time reversal and the signs in
the two cones are opposite, in agreement with Nielsen-Ninomyia theorem
[Nielsen(1981)]. For the holon system in consideration, however, time reversal
is broken by the staggered structure of the optimal spinon configuration and
by the chirality of the antiferromagnetic spin vortices. (The full spinon-
holon system is nevertheless invariant under time-reversal and parity due to
the opposite signs of the two Chern-Simons actions). The symmetry structure of
the holon system is however preserved by time-reversal combined with parity
w.r.t. a line between the lattice sites realized by exchange of the two Néel
sublattices. As a consequence when a regulator respecting this symmetry is
introduced, still denoted $m$, the contributions of the two cones to the Hall
conductivity add up [Marchetti(2019)] to sign$(m)/2\pi$, as in the case of
topological insulators [Haldane(1988)]. Taking $m>0$, in terms of effective
action this corresponds to a contribution $S_{c.s.}^{U(1)}(b)$.

In the standard free case if the upper bands are partially filled, their
contribution to the Hall conductivity exactly cancel that of the lower bands.
In our case, however, the situation is different, because the interaction with
the slave-particle gauge field makes the holon liquid of the upper bands
incompressible and with vanishing contribution to the Hall conductivity. In
fact, one should not introduce a coupling of the “spurious” holons in the
lower bands with spinons so that the density of “physical” holons coupled to
spinons still correctly vanishes at $\delta=0$; a coupling between spinons and
lower bands would introduce unphysical interactions in the model. Hence the
contribution to the Hall conductivity of the lower bands is correctly computed
as above. The upper bands, however, have additional couplings to the slave-
particle gauge field, generated by spinons, and to $V^{m}$. In doing the
computation it will be convenient to take into account the approximation of
neglecting $U$ in $V^{m}$, by replacing $V^{m}$ in the action by a $U(1)$
gauge field $v$ corresponding to the $U(1)$ subgroup of the spin group $SU(2)$
previously selected by choosing the directions of the spinons. Accordingly, we
need also to replace the $SU(2)$ Chern-Simons action of $V$ by
$2S_{c.s.}^{U(1)}(v)$, where the factor 2 is due to a normalization needed
passing from $SU(2)$ to its $U(1)$ subgroup. Then the interaction with $A$ and
$v$, upon integration of “physical” fermionic holons, produce as leading term
of the effective action the Chern-Simons action
$-S_{c.s.}^{U(1)}(b+A)-S_{c.s.}^{U(1)}(v)$ . We now need to integrate $A$ to
find both the Hall conductivity and the compressibility of the holon system.

Since the upper bands are partially filled, the leading contribution comes
from a region near the Fermi surface. Then at $\omega=0$ in the limit
$k\rightarrow 0$ in the Coulomb gauge its polarization bubble matrix is given
by:

$\displaystyle\pi^{h}(\vec{k})=\begin{pmatrix}\chi_{0}^{h}&-k_{2}\sigma_{H}^{h}&k_{1}\sigma_{H}^{h}\\\
k_{2}\sigma_{H}^{h}&\vec{k}^{2}\chi_{\perp}^{h}&0\\\
-k_{1}\sigma_{H}^{h}&0&\vec{k}^{2}\chi_{\perp}^{h}\end{pmatrix},$ (21)

where $\chi_{0}^{h},\chi_{\perp}^{h},\sigma_{H}^{h}=-1/(2\pi)$ are the density
of states at the Fermi energy, the diamagnetic susceptibility and the Hall
conductivity, respectively.

Since the spinons are gapped, integrating them out one obtains a Maxwell
effective action for the slave-particle gauge field $A_{\mu}$; the spinon
polarization bubble matrix at $\omega=0$ is then given by the diagonal matrix

$\displaystyle\pi^{s}(\vec{k})={\rm
diag}(\chi_{0}^{s}k^{2},\vec{k}^{2}\chi_{\perp}^{s},\vec{k}^{2}\chi_{\perp}^{s}),$
(22)

with $\chi_{0}^{s},\chi_{\perp}^{s}$ the electric and the diamagnetic
susceptibility of the spinon system.

We now compute the polarization bubble of the holons dressed by the spinon
interaction in RPA approximation, which should give the leading contribution
since the slave-particle interaction is of long-range. In the small $k$ limit
it is then given by

$\displaystyle\Pi^{h}(\vec{k})=\pi^{h}[1+(\pi^{s})^{-1}\pi^{h}]^{-1},$ (23)

where the scalar component (corresponding to compressibility) is given by

$\displaystyle\Pi^{h}_{0}(\vec{k})=\chi_{0}^{s}k^{2},$ (24)

and the Hall polarization bubble by

$\displaystyle\sigma_{H}(\vec{k})=\frac{\sigma_{H}^{h}k^{2}\chi_{0}^{s}\chi_{\perp}^{s}}{(\sigma_{H}^{h})^{2}+\chi_{0}^{h}(\chi_{\perp}^{s}+\chi_{\perp}^{h})};$
(25)

therefore both vanish at $k=0$, implying that the Fermi liquid in the upper
bands of “physical” holons is incompressible and does not contribute to the
Hall conductivity. Since the lower holon bands are completely filled we get as
result that the total holon system before it is coupled to the Chern-Simons
$b$-field is incompressible and with Hall conductivity $1/(2\pi)$. Hence,
according to the result stated in Theorem 2, after coupling with $b$ the
resulting semionic holons have exclusion statistics parameter 1/2, as
anticipated above.

Since the crossover from the pseudogap PG to the strange metal SM is due to
spinons, the contribution of the lower bands remain unchanged in the two
”phases”. In the partially filled band in SM qualitatively the situation is as
discussed above, except for the value of the Fermi momenta, which is now that
of the tight-binding Fermi surface. For later purposes we finally remark that
the total effective Chern-Simons action given by the original $U(1)\times
SU(2)$ term of the gauged $t$-$J$ model plus the induced action obtained
integrating out the holons according to the above computation is given by
$-S_{c.s.}^{U(1)}(b)+S_{c.s.}^{U(1)}(v)$.

##  5 The retarded 2D hole correlator

In this section we sketch the computation [Bertolini(2019)], within some
approximation, of the scaling limit of the retarded two-point correlation
function of the hole reconstructed from spinons and holons in the PG ”phase”,
near the Fermi surface. At first we neglect the contribution of the $v$ and
$A$ fields, which will be re-inserted later on.

We start noticing that in the scaling limit the contribution of the lower band
of ”spurious” holons is just a matrix element suppressing the spectral weight
of the hole outside of the magnetic Brillouin zone. It will not be discussed
anymore, for details see [Marchetti(2004)].

To derive the contribution of the ”physical” holons to the euclidean
$c$-correlator in the scaling limit we use a ”tomographic decomposition”
[Fröhlich(1997)] near the Fermi surface, a sort of dimensional reduction
angle-dependent allowing to rewrite the holon correlator as a sum of
contributions from 1+1 massless Dirac fields. This procedure is an analogue of
the decomposition in 1D in terms of left and right movers. More precisely let
$\vec{n}_{\theta}$ a unit vector from the center of the considered Fermi
surface to a point on the Fermi surface at an angle $\theta$ say respect to
the $x^{1}$ positive direction. We set
$\vec{n}_{\theta}\cdot\vec{x}=x_{\parallel}^{\theta}$ and
$\vec{n}_{\theta}\wedge\vec{x}=x_{\perp}^{\theta}$. Notice that
$\vec{n}_{\theta+\pi}=-\vec{n}_{\theta}$ and,
accordingly,$x_{\parallel}^{\theta+\pi}=-x_{\parallel}^{\theta}$. Let’s for
simplicity omit in the following the flavor index of the Dirac fields
$\psi_{a}$. Following ideas developed in [Fröhlich(1997), Marchetti(2004)] in
the scaling limit we write $\psi(x)\exp[i\int_{\gamma_{x}}b]\sim\int
d\theta\exp[ik_{F\theta}x_{\parallel}^{\theta}]\exp[i\int_{\gamma_{x}}b]\psi_{\theta}(x)$,
where the $\psi_{\theta}$ can be identified as the quasi-particle fields in
the sense of Landau. Here $k_{F\theta}$ is the Fermi momenta along the
$\theta$ direction of the Fermi surface of a spin 1/2 Fermion, in spite of the
fact that the holon is spinless, since, as discussed above, the semionic
holons turn out to obey the Haldane statistics of parameter 1/2. The fields
$\psi_{\theta},\psi_{\theta+\pi}$ are the analogue of the 1D chiral components
$h_{R},h_{L}$ and we call them sector fields. The field $\psi_{\theta}$ can
then be decomposed as a product of two fields: a fermionic
$\psi_{\parallel\theta}(x^{0},x_{\parallel}^{\theta})$ and a bosonic
$\psi_{\perp\theta}(x^{0},x_{\perp}^{\theta})$. The second field has a
propagator given simply by an approximate Dirac-delta function in
$x_{\perp}^{\theta}$. As in 1D one can then group together
$\psi_{\parallel\theta}$ and $\psi_{\parallel\theta+\pi}$ into a two-component
massless Dirac field which we denote by $\Psi_{\theta}$ and we call radial
sector field. In the scaling limit $b$ is minimally coupled only to these
fields. Taking into account the renormalization of the coefficient of the
Chern-Simons action discussed at the end of the previous section, the
integration of the $b_{0}$ component produces the constraints

$\displaystyle\frac{1}{2\pi}\epsilon_{\mu\nu}\partial_{\mu}b_{\nu}(x^{0},x_{\parallel}^{\theta},x_{\perp}^{\theta})=\delta(x_{\perp}^{\theta})\Psi_{\theta}^{\dagger}\Psi_{\theta}(x^{0},x_{\parallel}^{\theta})$
(26)

with $\partial_{\mu}=\partial/\partial x^{0},\partial/\partial
x_{\parallel}^{\theta}$. After choosing the gauge $b_{\perp}^{\theta}=0$
equation (26) is solved by

$\displaystyle
b_{\parallel}^{\theta}(x^{0},x_{\parallel}^{\theta},x_{\perp}^{\theta})=\pi{\rm
sgn}(x_{\perp}^{\theta})\Psi_{\theta}^{\dagger}\Psi_{\theta}(x^{0},x_{\parallel}^{\theta})$
(27)

with the convention ${\rm sgn}(0)=0$.

Since in $\Psi_{\theta}$ we have assembled together the fields corresponding
to angles $\theta$ and $\theta+\pi$, naively one expects that the $\theta$
parameter for the radial sector fields should range over an interval of length
$\pi$. However, when we include the strings the result would depend, through a
sign, on which specific interval we choose in the natural range of $2\pi$.
This situation is the analogue of the choice in 1D of the relative angle
between the direction of the string and the $x^{1}$ axis. As we have done
there we should avoid this unphysical feature and integrate the contributions
of the $\Psi_{\theta}$ fields with $\theta$ running in the full $2\pi$ range.

Using the Schwinger formula as in [Ye(2017)] and restoring physical units
$x^{0}\to v_{f}x^{0}$ with $v_{f}$ the Fermi velocity, we find for the leading
contribution of the euclidean correlator of the radial sector fields:

$\displaystyle\langle\exp[i\int_{\gamma_{x}}b]\psi^{*}_{\theta}(x^{0},x_{\parallel}^{\theta})\exp[-i\int_{\gamma_{0}}b]\psi_{\theta}(0)\rangle=\frac{1}{\sqrt[4]{(v_{f}x^{0})^{2}+(x_{\parallel}^{\theta})^{2}}}.$
(28)

We can see that because of the Chern-Simons coefficient $-1$ we get the well
known expression of the propagators of hard-core bosons in one dimensions: the
$b$ field has ”bosonized” the radial sector fields. This change of statistics
from the semionic one for the holon fields to the bosonic one for the quasi-
particle excitations over their ground state is reminiscent of the change of
statistics appearing in the fractional quantum Hall effect, where the ground
state is described in terms of fermionic electrons, but the quasi-particle
excitations above it are the anyonic Laughlin vortices (see e.g.
[Jain(2007)]).

Since the hole must still be a fermion and, as discussed above, the large-
scale field for the spinon is bosonic, it is up to the spin gauge field $v$ to
restore the right statistics of holon quasiparticles. In fact if we add the
$v$-strings and use the straight-line Gorkov approximation (see e.g.
[Lee(1992)]) for the coupling of holons to $v$, then the action
$+S_{c.s.}^{U(1)}(v)$ has precisely the right coefficient to transmute back to
fermionic the exchange statistics of the quasi-particle holon fields dressed
by the $v$-strings, without changing, however, the power law of the sector
fields.

We finally need to re-insert the coupling to the slave-particle gauge field
$A$ and to the spinon. Preliminarily we remark that in the 1D systems of the
radial sector fields, because of the appearance of the factor ${\rm
sgn}(x_{\perp}^{\theta})$ with ${\rm sgn}(0)=0$, all the local gauge-invariant
correlation functions not involving gauge strings are unmodified by the
coupling to the Chern-Simons fields. Since the leading term of the effective
action of the slave-particle gauge field obtained integrating the sector
fields has as coefficient polarization bubbles that are gauge-invariant
without gauge strings, its scaling behaviour is the same of a fermionic system
without Chern-Simons couplings.

In the Coulomb gauge, the Coulomb interaction of the scalar component is
screened by the finite density of holons. The dominating term is then that of
the transverse field $A^{T}$, with correlator given for
$\omega,|\vec{k}|,\omega/|\vec{k}|\sim 0,$ by

$\displaystyle\langle
A^{T}A^{T}\rangle(\omega,\vec{k})\sim(-\chi|\vec{k}|^{2}+i\kappa\frac{\omega}{|\vec{q}|})^{-1},$
(29)

where $\chi\equiv\chi_{\perp}$ is the diamagnetic susceptibility and $\kappa$
the Landau damping. This behavior dominates at large scales over the
Maxwellian term due to the massive spinons, destroying confinement, as
anticipated [Ioffe(1989)]. Nevertheless the attraction generated by $A^{T}$ in
the spinon-holon pairs is sufficient to produce near the Fermi surface a
resonance with the quantum numbers of the hole. Thus in 2D a true spin-charge
separation is not realized in this approach.

Unfortunately a perturbative treatment of gauge fluctuations would be
insufficient to get a resonance out of a spinon and a holon, but a possible
way out is to implement the binding using an eikonal approach
[Marchetti(2004)]. The eikonal resummation is obtained by treating first
$A^{T}$ as an external field, using the Gorkov approximation for the holon and
expanding the correlation function of the spinons in terms of first-
quantization Feynman paths. One then integrates out $A^{T}$ to obtain an
interaction between these paths and the straight path of the Gorkov
approximation, interaction which is then treated in the eikonal approximation.
Finally a Fourier transform is performed to get the retarded correlation
function.

Plugging as typical energy scale $\mathrm{Max}(T,\omega)$, the transverse
gauge propagator has a typical momentum scale: the so-called Reizer
[Reizer(1989)] momentum,
$Q(T,\omega)\approx(\kappa\mathrm{Max}(T,\omega)/\chi)^{1/3}$. It turns out
numerically that in the PG ”phase” the spatial Fourier transform is dominated
by a nontrivial complex $|\vec{x}|$-saddle point precisely near the inverse
Reizer momentum, due to the effect of gauge fluctuations [Marchetti(2004)].
Below this momentum scale, or alternatively beyond the inverse momentum scale
in space, there is an effective attraction mediated by $A^{T}$ between
opposite charges relative to the slave-particle $U(1)$ group, yielding in
particular a hole resonance. The main effect of the complex saddle point in
the retarded hole correlator is to induce a shift in the mass of spinons,
$m_{s}\rightarrow
M_{s}(T,\omega)\approx(m_{s}^{2}-ic\mathrm{Max}(T,\omega)/\chi)^{1/2}$, where
$c$ is a real constant $O(1)$, which in turn adds to the chemical potential of
the holon. Since in the range considered $T,\omega<<m_{s}$, the real part of
$M_{s}\approx m_{s}$ essentially renormalizes the chemical potential and the
imaginary part introduces a dissipation $\Gamma(T,\omega)$, proportional to
$\mathrm{Max}(T,\omega)$.

A last effect that we need to consider in the PG ”phase” is the attraction
between the holons, and, as a consequence of the binding to spinons, between
the holes due to the attraction between the attached antiferromagnetic spin
vortices centered on opposite Néel sublattices. Such attraction mathematically
is generated from the term (16) averaging the spinons: $\int
d^{2}xV^{m}_{\mu}V^{m}_{\mu}\langle
s^{*}_{\alpha}s_{\alpha}\rangle\sim\sum_{i,j}(-1)^{|i|+|j|}h^{*}_{i}h_{i}h^{*}_{j}h_{j}\Delta_{(2)}^{-1}(i-j)$,
where $\Delta_{(2)}$ denotes the 2 D lattice laplacian. This attraction
induces the formation of charge pairs, the spin degrees of freedom being still
unpaired above a crossover spin-pairing temperature. The non-vanishing density
of charge pairs is described by a $d$-wave order parameter, $\Delta_{h}$,
whose modulus (up to its $d$-structure) we can keep constant near the Fermi
surface, but whose phase is strongly fluctuating, since the charge pairs are
not condensed. In fact the field describing the phase fluctuations has a gap,
which we denote by $m_{\phi}(T)$ decreasing with $T$, that modify the standard
BCS form of self-energy near the hole Fermi surface to

$\displaystyle\Sigma(\omega,\vec{k})\approx\frac{|\Delta_{h}(\vec{k},T)|^{2}}{\omega-\omega_{h}(\vec{k})-m_{s}+i\Gamma(\omega,T)}[1-\frac{m_{\phi}(T)}{\sqrt{-(\omega+i\Gamma(\omega,T))^{2}+(\omega_{h}(\vec{k})+m_{s})^{2}+m_{\phi}(T)^{2}}}],$
(30)

where $\Delta_{h}(\vec{k})$ is the $d$-wave holon-pair order parameter and
$\omega_{h}(\vec{k})$ the holon dispersion [Marchetti(2012)]. Hence finally
the retarded hole correlator near the Fermi surface in the PG ”phase” is given
by

$\displaystyle G^{R}(\omega,\vec{k})\approx
Z(\vec{k})\frac{\sqrt{m_{s}\kappa}}{\omega-\omega_{h}(\vec{k})-m_{s}+i\Gamma(\omega,T)-\Sigma(\omega,\vec{k})},$
(31)

with
$Z(\vec{k})=1-(\cos{k_{x}}-\cos{k_{y}})/(\sqrt{2}(\cos{k_{x}}^{2}+\cos{k_{y}}^{2}))$
the direction-dependent wave-function renormalization constant arising from
the mentioned matrix element due to the $\pi$-flux.

The independence of the numerator of (31) from $T$ and $\omega$ is due to the
semionic nature of the holons. If the effect of the $b$ field is neglected, as
in the more approximate treatment of [Marchetti(2012)], there is a dependence
in the form Max$(T,\omega)^{1/6}$, which actually disagree with experimental
data in the cuprates. Let us just mention that, since the $\pi$-flux is no
more felt by physical holons, in the SM ”phase” $Z(\vec{k})$ is absent and the
holon dispersion is the tight-binding one.

##  6 Comparison between 1D and 2D

The spin-charge gauge approach to the $t$-$J$ model outlined in the previous
sections allows a unified treatment of the one- and two-dimensional cases. Its
justification in terms of optimization of energy is the same in both.
Furthermore the charge carriers in 1D and 2D obey an exclusion statistics with
parameter 1/2, as a consequence of the no-double occupation constraint, though
the origin of this property is somewhat different. In 1D it originates from
the cancellation of the spinon Fermi momenta in the two chiralities due to the
spin string where the constraint is used. In 2D it arises from the
incompressibility of the holon fluid due to the Coulomb interaction generated
by the spinons, remnant of the Gutzwiller constraint at large scales.

However, as we have seen, the final physical result of this approach are quite
different in the two cases. In 1D spinon and holon are decoupled, both
exhibiting semionic braid statistics, furthermore the spinon is gapless. In 2D
spinon and holon are bound together at low energy, the exchange statistics of
the quasi-particle bounded together are bosonic and fermionic for the spin and
charge degrees of freedom, respectively. Furthermore the spin carriers are
gapped. Let us trace the origin of these differences.

First of all in 1D a gauge field has no transverse component, while in 2D it
does have one. The disappearance of this degree of freedom in 1D w.r.t. 2D
induces the following effects in one-dimension: $B^{T}$ =0, hence there is no
Hofstatder mechanism generated by $B^{m}$ and the holon doesn’t have a Dirac
structure; $V^{T}$=0, hence there is no spinon mass generation, which in 2D is
generated by the ”magnetic” field strength of $V$; $A^{T}$=0, hence spinons
and holons are decoupled, whereas in 2D the transverse slave-particle gauge
field is able to introduce a weak binding. In fact, the other natural possible
source of binding is the Coulomb interaction mediated by the scalar component
of the slave-particle gauge field, but it is screened. In 2D this occurs
trivially because of a Fermi surface of finite length of the holons; in 1D if
one considers the CP1 non-linear sigma model describing the large-scale
behaviour of spinons we have seen in equation (14) that it has a
$\Theta$-angle of $\pi$. Precisely at this value of $\Theta$ in 1D there are
two degenerate ground states connected by a parity transformation (see e.g.
Ref. [Affleck(1988)]) as in the Heisenberg chain. Kinks interpolating between
the two destroy the Coulomb attraction generated by $A_{\mu}$, which would be
massless since the global $O(3)$ spin symmetry is unbroken, in agreement with
the Mermin-Wagner theorem. (A hedgehog gas discussed in Ref.[Read(1989)] would
appear in 2D instead of the $\Theta$-term, actually inducing confinement, but
it disappears when we add the holons [Ioffe(1989)] and therefore here will not
be considered.)

Finally in 2D w.r.t. 1D in the considered approximation, including the
interaction with the spin gauge field $v$, the braid statistics of the quasi-
particle charge carriers is modified from semionic to fermionic because the
finite density of semionic holons introduce a renormalization of the effective
Chern-Simons coefficient for the charge and spin gauge fields $b$ and $v$ at
large scales, a phenomenon discussed also in [Ye(2015)], which has no analogue
in 1D.

We close this section noticing that whereas in 1D the string approach of Weng
[Weng(1997), Weng(2007)] is quite similar to the one reviewed here, in 2D the
two approaches are instead quite different, in spite of some surviving
similarities as the $\pi$-flux, the spin-vortices, not generating however a
charge-pairing mechanism, the appearence of an half-filling RVB structure
playing a role in some aspect analogous to the ”spurious” holon bands and the
recently advocated resonance nature of the hole [Weng(2019)]. In fact, the two
approaches give rise to different phase diagrams of the crossovers in the
$\delta$-$T$ plane, in particular in Weng’s string approach there is no
analogue of the pseudogap crossover line $T^{*}$ crossing the superconducting
dome. Furthermore no fractional Haldane statistics appears in the string
approach in 2D, at odds with its appearance in 1D.

##  7 Relevance for the cuprates

In this section we briefly outline properties of hole-doped cuprates that
found a natural explanation within the spin-charge gauge approach to the 2D
$t$-$J$ model presented above, in particular due to the composite nature of
the hole. Of course to describe at least semi-realistically the cuprates we
need to add to the $t$-$J$ Hamiltonian at least a next-nearest-neighbour
hopping term, with coefficient denoted by $t^{\prime}$, but it does not modify
qualitatively the general features previously discussed [Marchetti(2012)]. We
also remark that, according to the observations made in Sect. 5, for the
physical quantities not involving directly the hole resonance, one can
consider still valid the computations made in a previous approximate treatment
where the 1/2 Haldane statistics was assumed, but the effect of the $b$ field
was somewhat inconsistently neglected.

We have seen that the 1/2 Haldane statistics of charge carriers is strictly
linked to the presence of the charge vortices, converting spinless fermions
into semions. In turn, due to the fermionic statistics of the hole, the charge
vortices imply the presence of the antiferromagnetic spin vortices that
provide a somewhat novel mechanism of superconductivity. In fact, in this
approach superconductivity arises through a three-step mechanism.

1) At a temperature, denoted by $T_{ph}$, charge pairing occurs as a
consequence of the Kosterlitz-Thouless like attraction between
antiferromagnetic spin vortices centered on holons, corresponding to empty
sites in the $t$-$J$ model, on opposite Néel sublattices. From the explicit
expression (30)(31) of the hole Green function, since the value of $m_{\phi}$
decreases with $T$, one finds that lowering the temperature there is a gradual
reduction of the spectral weight on the Fermi surface at small frequency as we
move away from the diagonals of the Brillouin zone, due to the $d$-wave
structure of $\Delta_{h}(\vec{k})$. Simultaneously at larger frequencies we
have the formation and increase of two peaks of intensity precursors of the
excitations in the superconducting phase [Marchetti(2012)]. Thus the main
effect of charge-pairing is the generation of a phenomenology of Fermi arcs
near the diagonals, qualitatively consistent with ARPES experiments in
underdoped cuprates (see e.g. [Norman(2007)])

2) At a temperature denoted by $T_{ps}$, lower than $T_{ph}$, using the holon-
pairs as sources of attraction, the slave-particle gauge attraction between
holon and spinon induces the formation of short-range spin-singlet (RVB)
spinon pairs. Hence, below $T_{ps}$ there is a finite density of hole pairs.

3) Finally, at an even lower temperature, $T_{c}$, the hole pairs become
coherent and a $d$-wave hole condensate (in BCS approximation) appears,
leading to superconductivity.

Hence, in the $\delta$-$T$ phase diagram of the present approach to the 2D
$t$-$J$ model, besides the antiferromagnetic transition not considered here,
there is a phase transition, the superconducting one, and three crossovers
corresponding to charge-pairing, spin pairing and transition from a ”Dirac”
Fermi surface to a tight-binding one due to an effective ”melting” of the
$\pi$-flux. A comparison of the emerging structure of the phase diagram with
experimental data in hole-doped cuprates is given in Fig.1. The theoretical
computations whose result are presented in Fig.1 are performed with
experimental inputs for one doping to fix parameters that are then used
consistently in all the calculations in each ”phase”. $T_{ph}$ and $T_{ps}$
are evaluated solving gap equations in Mean-Field approximation, see
[Marchetti(2011)].

a

b

Figure 1: (a) Theoretically derived phase diagram from Ref. [Marchetti(2016)]:
holon pairing temperature $T_{ph}$ (yellow line) , spinon pairing temperature
$T_{ps}$ (red line) , $T_{c}$ (green line). The Nèel temperature (dot-dashed
line) is qualitative from experiments not derived theoretically. (b)
Experimental data for $T_{c}$ (yellow squares) and onset of Nernst signal
(green diamonds) in LSCO from Ref.[Li(2010)], “low pseudogap” (red triangles)
in LSCO from Ref.[Honma(2004)] and “high pseudogap” (blue circles) in YBCO
from Ref.[Barisic(2013)].

The comparison between the experimental data and the theoretical curves
suggests the identification of $T_{ph}$ with the ”upper pseudogap temperature”
corresponding for example to the deviation from linearity of the in-plane
resistivity (see e.g. [Ando(2004)]). This crossover line does not cross the
superconducting dome and the interpretation in terms of charge-pairing shares
some analogy with Uemura’s approach [Uemura(1997)], at odds instead with the
RVB interpretation (see e.g. [Lee(2006)]) in terms of spin-pairing. Similarly
$T_{ps}$ appears to be identifiable with the onset of magnetic-field induced
diamagnetic and Nerst signals due to vortices [Li(2010)], appearing in presece
of ”preformed hole pairs”, in agreement with many other interpretations. The
line of ”melting” of the $\pi$-flux is seen to correspond to the ”lower
pseudogap temperature” marked for example by the inflection point in the in-
plane resistivity [Ando(2004)]. It crosses the superconducting dome as
advocated in many approaches based on the existence of a critical point (see
e.g.[Tallon(2001)]). The only experimentally observed crossover that is
completely missed in this approach is the onset of charge-density waves peaked
around $\delta=1/8$ (see e.g. [Taillefer(2016)]); we conjecture that this is
due to a contribution of the oxygen orbitals in the cuprates not taken into
account by the $t$-$t^{\prime}$-$J$ model.

Besides the structure of the phase diagram there is another general feature of
data of the cuprates that found a natural explanation within our approach
[Marchetti(2016)]: a non Fermi-liquid universality [Xiang(2008)] of many
suitably normalized experimental curves, indicating an independence of details
of the Fermi surface. In fact, if the quantity we compute in the spin-charge
gauge approach depends essentially on spinons and holons by themselves and not
on the hole as a resonance and if the spinon contribution is the dominating
one, universality is a natural consequence of the fact that the spinon
propagator does not depend on details of the Fermi surface. The first of the
above requirements holds if the Ioffe-Larkin composition rule [Ioffe(1989)] is
valid. Both the requirements are satisfied for example in the case of in-plane
resistivity and superfluid density in the PG ”phase”. In fact the
theoretically derived universality curves are in a quite satisfactory also
quantitative agreement with the corresponding experimental data, as shown in
Fig.2, with the experimentally observed ”3/2” critical exponent for the
superfluid density [Hardy(1998)] well reproduced, due to the 3DXY nature of
the spinon-triggered superconductivity transition [Marchetti(2015)].

a

b

Figure 2: (a) The normalized in-plane resistivity $\rho_{n}$ theoretically
calculated, including a holon contribution with relative weight $r$ w.r.t. the
spinon contribution, from Ref. [Marchetti(2016)], compared with experimental
data from Ref. [Ando(2004)]. The discrepancy at low $T$ might be due to the
missing account of holon and spinon pair formation in the above treatment. (b)
The normalized superfluid density $\rho_{s}$ from Ref. [Marchetti(2016)],
compared with experimental data for underdoped (LSCO, YBCO) and optimally
doped (BSCCO, YBCO) samples, from Refs. [Hardy(1998)][Jacobs(1995),
Panagopoulos(1999)].

Among other experimental features of cuprates that find a natural explanation
are crossovers appearing in transport quantities in the PG ”phase” and the
emergence of Fermi arc phenomenology, briefly discussed in the previous
section. Concerning the first feature, examples are the metal-insulator
crossover of the in-plane resistivity [Marchetti(2001), Marchetti(2016)] and
the peak in the ${}^{63}(1/T_{1}T)$ spin-lattice relaxation rate
[Marchetti(2000)]. Both are interpreted as due to the competition between the
spinon gap dominating at lower $T$ and the dissipation growing with $T$
introduced by the couping of the spinons to the slave-particle gauge field.

We are presently analyzing [Bertolini(2019)] within this approach another
quite remarkable feature of the cuprates: the coexistence of the non-Fermi
liquid character of many quantities with a Fermi-liquid-like behaviour of
other quantities, in particular in the SM phase, as for example for the
uniform magnetic susceptibility. We conjecture that this occurs because in
these last cases the dominant contribution comes from the hole resonance and
not from the spinons.

The above overlook to the comparison between the theoretical scheme sketched
in this review and experimental data in the cuprates shows that, although
exhibiting an admittedly complicated structure, the approach based on the 1/2
exclusion statistics for the charge carriers of the 2D $t$-$J$ model is able
to explain naturally many unusual features of the hole-doped cuprates.

Acknowlegments. I gratefully acknowledge Jürg Fröhlich whose original
intuition was crucial for the origin of this journey attempting to understand
the physics of the cuprates, Su Zhao-Bin and Yu Lu, for the pleasure of a
collaboration since the beginning of this journey, Ye Fei who later
brilliantly joined us and the many other collaborators involved in the
project, in particular Dai Jian-Hui, Lorenzo De Leo, Giuliano Orso, Michele
Gambaccini, Alberto Ambrosetti, Giacomo Bighin and Tommaso Bertolini.

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