Spin-charge split pairing in underdoped cuprate superconductors: support from low-$T$ specific heat

TL;DR

This paper models the specific heat of underdoped cuprates, supporting the split-pairing mechanism where spinons and holons pair separately before Bose condensation, explaining experimental specific heat data.

Contribution

It provides a theoretical calculation of specific heat consistent with experiments, supporting the split-pairing mechanism in underdoped cuprates.

Findings

Calculated specific heat matches experimental data.

Supports split-pairing of spinons and holons.

Explains the T-linear and T^3 terms in specific heat.

Abstract

We calculate the specific heat of a weakly interacting dilute system of bosons on a lattice and show that it is consistent with the measured electronic specific heat in the superconducting state of underdoped cuprates with boson concentration $\rho \sim x/2$, where $x$ is the hole (dopant) concentration. As usual, the $T^3$ term is due to Goldstone phonons. The zero-point energy, through its dependence on the condensate density $\rho_0(T)$, accounts for the anomalous $T$-linear term. These results support the split-pairing mechanism, in which spinons (pure spin) are paired at $T^*$ and holons (pure charge) form real-space pairs at $T_p < T^*$, creating a gauge-coupled physical pair of charge $+2e$ and concentration $x/2$ which Bose condenses below $T_c$, accounting for the observed phases.

Equations

$$G^{el-pair}(\mathbf{k},\omega )=-\frac{A^{2}Z(-\mathbf{k})}{i\omega +{ϵ}_b(-\mathbf{k})}.$$

$$c(T)=AT+B{T}^3.$$

$$H=\sum _{\mathbf{k}}ϵ(\mathbf{k})a_{\mathbf{k}}^{†}{a}_{\mathbf{k}}+\frac{1}{2N}\sum _{\mathbf{q},{\mathbf{k}}_1,{\mathbf{k}}_2}V(\mathbf{q})a_{{\mathbf{k}}_1+\mathbf{q}}^{†}{a}_{{\mathbf{k}}_2-\mathbf{q}}^{†}{a}_{{\mathbf{k}}_2}{a}_{{\mathbf{k}}_1},$$

$$H_B({\rho }_0)=\sum _{\mathbf{k}\ne 0}[(ϵ(\mathbf{k})+\Sigma -\mu )a_{\mathbf{k}}^{†}{a}_{\mathbf{k}}+\frac{1}{2}\Lambda (a_{\mathbf{k}}^{†}{a}_{-\mathbf{k}}^{†}+h.c.)],$$

$$H_B({\rho }_0)=U_0({\rho }_0)+\sum _{\mathbf{k}\ne 0}E(k){\alpha }_k^{†}{\alpha }_k.$$

$$U=U_0({\rho }_0)+\sum _{\mathbf{k}\ne 0}\frac{E(k)}{(e^{E/T}-1)}=U_0({\rho }_0)+U_1({\rho }_0,T),$$

$${\rho }_0=\rho -{\rho }_1,$$

$$\Lambda =V({\rho }_0+\sigma )=V(\rho -{\rho }_1+\sigma ),$$

$$\Lambda (T=0)={\Lambda }_0\approx \rho g(1+\eta /3),$$

$$u_2=\frac{V{\rho }^2}{2}+\frac{{\Lambda }^2}{2V}-V{\rho }_0^2.$$

$$c=\rho \eta \frac{{\pi }^2}{3g}\frac{T}{2{\Lambda }_0}[(V-g)+\frac{2{\pi }^2}{5}(5g-V)(\frac{T}{2{\Lambda }_0}{)}^2+O(T^4)],$$

$${\rho }_{11}(\Lambda ,T)=\rho (T/T_c{)}^{3/2}+(\frac{1}{\tau }{)}^{\frac{3}{2}}[-a_1{\Lambda }^{\frac{1}{2}}T+a_2\Lambda T^{\frac{1}{2}}],$$

$$\Lambda \approx g_2\rho (1-(T/T_c{)}^{3/2})\sim (g_2/T_c)(T_c-T),$$

$${\rho }_0=\rho -{\rho }_1\approx \rho (1-(T/T_c{)}^{3/2})+a_1\frac{{\Lambda }^{\frac{1}{2}}T}{{\tau }^{3/2}}\propto (T_c-T{)}^{\frac{1}{2}}.$$

$$u=\frac{1}{2}V{\rho }^2-V{a}_1^2\frac{T^2\Lambda }{{\tau }^3}+\frac{3}{2}\rho (T/T_c{)}^{\frac{3}{2}}[\frac{\zeta (5/2)}{\zeta (3/2)}T-\Lambda ],$$

$$c\approx c_B-(\frac{2V{a}_1^{2}T}{{\tau }^3}+\frac{9\rho }{4T_c}(\frac{T}{T_c}{)}^{\frac{1}{2}})\Lambda -(\frac{V{a}_1^2{T}^2}{{\tau }^3}+\frac{3\rho }{2}(\frac{T}{T_c}{)}^{\frac{3}{2}})\frac{d\Lambda }{dT},$$

$$\frac{c-c_B}{c_B}=\frac{2}{5}\frac{\zeta (3/2)}{\zeta (5/2)}\frac{g_2}{\tau }[\frac{3}{2}{\nu }^{2/3}{\rho }^{1/3}+\frac{V{a}_1^2}{\tau }(\rho /\nu {)}^{2/3}],$$

Full text

PACS numbers: 74.20.Mn, 74.72.-h, 71.27.+a, 74.20.-z.

Introduction.—In conventional superconductors electrons form spatially overlapping Cooper pairs below a temperature $T_{c}$. In contrast, the metallic phase of an underdoped cuprate superconductor with a small concentration $x$ of holes is not characterized by electron-like quasiparticles tim . Instead, spin and charge appear to be carried by separate excitations, consistent with Anderson’s proposal of spin-charge separation and . For example, with decreasing $T$ the paramagnetic susceptibility decreases rapidly below a pseudogap temperature $T^{*}>T_{c}$, consistent with gapping of spin excitations, even as charge propagation becomes more metallic, as evidenced by the appearance of a Drude peak syr . Nevertheless, the superconducting state is widely believed to be conventional, that is, a condensate of Cooper pairs with $d$-wave symmetry, primarily because of observations of electron-like peaks in the photoemission spectrum below $T_{c}$ shen ; shen2 .

The conventional model, however, has serious inconsistencies. It cannot explain why pairs continue to exist up to $T_{p}$ far above $T_{c}$, but below $T^{*}$ ong1 . Since a Cooper pair has zero momentum a macroscopic condensate of such pairs is superconducting and hence should not exist above $T_{c}$. The problem is partly circumvented if electrons form nonoverlapping (real-space) pairs at a high temperature, with superconductivity occuring by Bose-Einstein condensation (BEC) chen . However, cuprates are basically doped Mott insulators in which occupancy of a lattice site by two electrons is effectively forbidden due to large on-site repulsion $U$. Then, given the high electron concentration ($1-x$), there is little room for creating nonoverlapping pairs. More importantly, it will be almost impossible for two electrons to avoid double occupancy and propagate as a well-defined bound pair when even a single electron has a hard time propagating with a well-defined momentum ${\bf k}$. Indeed, electron pairing cannot explain why electrons are not seen when pairs break in the metal. Or, why $T_{c}$ decreases with decreasing $x$, and superfluid density $\rho_{s}\sim x$ orn1 , (and not $1-x$), consistent with the carrier density ($x$) of the metal. It seems, therefore, that despite the appearance of the electron peaks below $T_{c}$, neither electrons nor electron pairs are the true low-energy excitations in the metallic or the superconducting phase. Not suprisingly, a plausible microscopic theory connecting electron pairing with the actual excitations is yet to emerge.

Recently a different, logically consistent, pairing mechanism has been found in the spin-charge separated state of a renormalized $t$-$J$ model in which the spin and the charge of the electrons are paired separately sar9 ; sar10 . As an electron hops on the lattice with an amplitude $t$ avoiding double occupancy, its spin behaves like a localized moment, coupled with neighbors with an antiferromagnetic interaction $J$, as in the Mott insulator ($x=0$). In the quantum antiferromagnet, spins are known to be paired into singlets which form a BCS-like, spin RVB state of paired spinons aro . The charge ($+e$) is carried by a holon (concentration $x$), which is coupled to the spinon via a gauge field. The preformed pairs, Anderson argued and , could lead to superconductivity for $x>0$. But, (1) numerous attempts to construct an useful metallic state based on the bare $t$-$J$ model failed, and (2) the model does not have a mechanism to convert the chargeless RVB singlets into electron pairs. However, one-hole calculations kane have shown that for $x>0$, competition from AF correlation renormalizes the model with $t\gg J$ to one with $t_{eff}<J_{eff}$. This allows one to decouple the spinons perturbatively and obtain a reduced Hamiltonian sar9 , in which sublattice-mixing bare hopping $t$ is renormalized away and is replaced by sublattice-preserving one-holon and holon-pair hopping terms accompanied by spinon singlet backflows, which cures both problems. The in-plane amplitudes for both one-hole and the pair hopping are the same: $t_{s}\sim 4t_{eff}^{2}/J_{eff}\propto J$, and $\ll t$. The new Hamiltonian is short-range RVB type krs , with the pair hopping term supplying the missing superconducting mechanism.

By requiring that the spin phases are the same (by symmetry) as those in the insulator ($x=0$), one obtains a qualitatively correct phase diagram for underdoped cuprates, as detailed in sar9 ; sar10 . Briefly, the phase above $T^{*}$ - the strange metal - has no quasiparticles of any kind. The spinon singlets condense below $T^{*}$ in the spin RVB state, accounting for the observed suppression of paramagnetic susceptibility. The RVB order allows holons to propagate coherently and create a spinless Fermi liquid of concentration $x$ via the one holon hopping term, which is consistent with the observed transport properties syr ; ando ; pad and quantum oscillation in a magnetic field tai .

The smallness of $x$ causes holons to form real-space pairs below $T_{p}<T^{*}$ via the strong coupling term sar10 . As it propagates, the phase of the bound (holon) pair is locked to that of the spin RVB ordered state, creating a physical (gauge invariant) pair with a split character. It behaves like a mobile hole pair of charge $+2e$ with concentration $x/2$, which lives in a $d$-wave band; its spin part is an RVB singlet (concentration $1-x$), already ordered at $T^{*}$. This is seen in the calculated electron pair Green’s function sar10

[TABLE]

where $A$ is the spin RVB order parameter, $Z$ is the pole strength, and $\epsilon_{b}(-{\bf k})\geq 0$ is the energy of the bound holons. Thus the mobile part of the electron pair actually has negative energy and opposite momentum, i.e., it behaves like a hole pair. Superconducivity appears via BEC of these pairs below $T_{c}$, which accounts for the decline of $T_{c}$ with decreasing $x$ (for a noninteracting Bose gas $T_{c}\sim x^{2/3}$) and the fact that $\rho_{s}\sim x$.

Specific heat.—In this Letter we calculate the low-$T$ specific heat $c(T)$, a bulk property which directly reflects the dominant low-energy excitations in the superconductor, and thus provides a serious test of the theory. In cuprates, the background, including the lattice phonon contribution, is removed, e.g., by subtracting specific heat measured in a modest magnetic field huen ; marc . Since $T^{*}\gg T_{c}$ the field does not break the singlets, but removes contributions from the (chargeless) spin (zero) sector. Then, at low $T$, we can model the system with weakly interacting bosons of concentration $\rho=x/2\ll 1$ hopping on a lattice, and interacting via a weak short-ranged repulsion. Although cuprates are layered, low-$T$ physics for small $\rho$ can be described by scaling the wave vectors and using an isotropic 3D bosonic band of energy $\epsilon_{b}(k)=t_{b}k^{2}$, with $t_{b}=(t_{ip}^{2}t_{op})^{1/3}$, where $t_{ip}$ and $t_{op}<t_{ip}$ are the in-plane and out-of-plane boson hopping parameters which can be calculated from the renormalized model sar10 .

Above $T_{c}$ the measured $c(T)$ does not agree with the BCS theory. Instead, it declines slowly from a maximum as in a Bose gas, with an eventual crossover to metallic ($T$-linear) behavior above $T_{p}$ huen , as expected in our model. Moreover, the maximum (and the overall scale) increases with $x$, not $1-x$ loram . For $T\ll T_{c}$, the measured $c(T)$ in many cuprates behaves like marc

[TABLE]

If density of states of the relevant excitations behaves like $D(\epsilon)\propto\epsilon^{p}$, for small excitation energy $\epsilon(q)$, then $c(T)\propto T^{p+1}$. For the $d$-wave BCS state, the excitations are the nodal electrons with $p=1$, and the absence of a $T^{2}$ term in c(T) implies that the electron peaks in the photoemission spectrum do not behave like long-lived quasiparticles, effectively ruling out this state.

For phonons $\epsilon(q)\propto q$, giving $p=2$, and $c(T)\propto T^{3}$. However, a definite curvature in $\gamma=c/T$ remains even after lattice phonons are removed, indicating that $T^{3}$ term in (1) is of electronic origin. For a dilute Bose gas, Bogoliubov has shown that bog , coupling of bosons with the condensate leads to a depleted condensate density $\rho_{0}(T)$, and Goldstone phonons, which yields a $T^{3}$ specific heat yang . While this agreement with Eq. (1) is encouraging, the $T$-linear term poses a fundamental challenge, since it dominates at small $T$, and it has become increasingly clear that it cannot be attributed to disorder marc .

Weakly interacting bosons, low $T$.—This leads us to re-examine the Bogoliubov theory at low $T$. The point is, in addition to the direct ($\sim T^{4}$) contribution to the energy from Bogoliubov phonons, there is also a zero-point energy which depends on $T$ through its dependence on $\rho_{0}(T)$, and thus contributes a $T$-linear term to the specific heat. To see this, consider the Hamiltonian

[TABLE]

where $\epsilon_{b}({\bf k})=t_{b}k^{2}$, $V(q)$ is a weak repulsive interaction, and $a_{{\bf k}}$ destroys a boson of momentum ${\bf k}$. For uniform $\rho$, the Hartree ($V(0)$) term is a constant and is subtracted out. We also approximate $V(q)$ by a constant which does not change the results qualitatively. Symmetry is broken below $T_{c}$, so that $\langle a_{0}\rangle=(N\rho_{0})^{\frac{1}{2}}\neq 0$. Replacing $a_{0}$ by its average and retaining quadratic terms we obtain the Boguliobov Hamiltonian

[TABLE]

where $\Sigma=\rho_{0}V$ and $\Lambda(k)=\rho_{0}V$ are self energies, $\mu$ is the chemical potential. The partition function is calculated by using $H_{B}$, which depends on $\rho_{0}$ via $\Lambda$. Diagonalization leads to

[TABLE]

where $U_{0}$ is the zero point or “vacuum” energy, which arises due to the anomalous coupling, and $\alpha^{{\dagger}}_{k}$ creates an excitation (out of this vacuum) of energy $E(k)$, which depends on $\rho_{0}$. In the Bogoliubov theory bog ; FW the spectrum is taken to be gapless which gives $\mu=\Sigma-\Lambda=0$ and $E(k)=[2\Lambda\epsilon(k)+\epsilon^{2}(k)]^{1/2}$.

For a given $\rho_{0}$ and $T$, the average energy ($k_{B}=1$) is then

[TABLE]

where $U_{1}$ is excitation contribution which vanishes as $T^{4}$, yielding a $T^{3}$ term in $c$. $U_{0}(\rho_{0})$ is the ground-state energy, which is usually taken to be a constant. However, since $\rho_{0}$ represents the condensate it is a thermodynamic variable, and thus depends on $T$, as emphasized by Yukalov yuka . It is obtained from the thermodynamic relation

[TABLE]

where $\rho_{1}=\sum_{k\neq 0}n_{k}/N$ is the density of noncondensate bosons, with $n_{k}$ = $\langle a^{{\dagger}}_{k}a_{k}\rangle$, calculated as a function of $T$ and $\rho_{0}$ by using $H_{B}(\rho_{0})$. Solving Eq. (5) self-consistently one obtains $\rho_{0}(T)$ and hence thermodynamic properties as a function of $T$ and $\rho$. The true ground state energy is given by $U(\rho_{0}(0))$. However, the behavior for $T>0$ is not determined by the excitations about this $(T=0)$ vacuum, but by the true many-body states in the ensemble characterized by $\rho_{0}$ at this $T$ (i.e, approximately by those of $H_{B}(\rho_{0}(T)$); $U_{0}(\rho_{0}(T))$ is the ground state energy for this ensemble and hence will contribute to $c(T)$. This is a general result which would apply to other systems since it is a consequence of thermodynamics. In particular, using Bogoliubov low-$T$ result $\rho_{0}(T)\approx\rho_{0}(0)-aT^{2}$ (where $a$ is positive constant) in Eq. (4) one immediately obtains a $T$-linear term for $c(T)$. It arises from a nonperturbative change in phonon speed (not its $q$ dependence) in the presence of the condensate. The self-consistent procedure has been used in earlier $T>0$ studies of a dilute Bose gas yuka ; griff , but the effect on $U_{0}$ apparently has not been recognized.

Hartee-Fock corrections.—Although the Bogoliubov approximation agrees qualitatively with $c(T)$ at low $T$, it has well-known problems anders , particularly at higher-$T$, predicting a spurious first-order transition. This is corrected if the remaining interactions are treated by a self-consistent Hartree-Fock approximation. Our treatment is similar to earlier ones yuka ; griff , except that we calculate $c(T)$ and in our lattice problem the bandwidth $W$ acts as a natural cutoff. We also solve the HF problem analytically at low $T$ which allows as to obtain the crossover scale as a function of $\rho$ below which the linear term becomes important.

The HF decomposition leads to the same quadratic Hamiltonian (3) except now, $\Sigma=V(\rho_{0}+\rho_{1})=V\rho$, and

[TABLE]

where $\sigma=\sum_{k}\sigma_{k}/N$ is the anomalous density. It arises as a response to $\rho_{0}$, i.e., vanishes if $\rho_{0}$ does. Once $\mu$ is fixed by requiring gaplessness, the Hamiltonian depends on $\rho_{0}$ through $\Lambda$. These are determined as functions of $\rho$ and $T$ by solving Eqs. (5) and (6) self consistently.

As before, each of $\rho_{1}$, $\sigma$, and the energy per site $u=\langle H\rangle/N$ splits into vacuum and excitation terms; the latter depends on $T$. For example, $\rho_{1}=\rho_{10}(\Lambda)+\rho_{11}(\Lambda,T)$. For small $\rho$, the relevant energy scales $(\Lambda,T)\ll W$. The integrals are evaluated to leading orders in $(\Lambda/W)$ etc. We set $\int_{0}^{W}d\epsilon D(\epsilon)=1$, where $D(\epsilon)=\epsilon^{1/2}/\tau^{3/2}$ is the density of states, with $\tau=(4\pi^{2})^{2/3}t_{b}$, which gives $W=(3/2)^{2/3}\tau$. At low-$T$, HF theory provides a screened interaction $g=V/[1+VW^{1/2}/\tau^{3/2}]$. Then the solution at $T=0$ is

[TABLE]

where $\eta=(2\Lambda_{0}/\tau)^{3/2}/\rho\approx(2g/\tau)^{3/2}\rho^{\frac{1}{2}}$, emerges as the small parameter with the same $\rho^{\frac{1}{2}}$ dependence as in the Bogoliubov theory. For $V\sim t$ and $\rho<0.1$, $\eta<0.023$ is very small. The fractional depletion of the condensate is given by $\rho_{1}/\rho\approx\eta/6$.

The characteristic energy scale at low-$T$ is $\Lambda_{0}$. The energy per site is $u=u_{1}+u_{2}$, where $u_{1}=\sum_{k}\epsilon(k)n_{k}/N$, and the average interaction energy $u_{2}$ is given by

[TABLE]

At low-$T$, we have solved the consistency equations by expanding in powers of $(T/2\Lambda_{0})^{2}$, and obtaining $u$ up to $(T/2\Lambda_{0})^{4}$. This gives

[TABLE]

which is valid for $\Lambda_{0}\approx\rho g\gg T$, and has the observed form (Eq. (1)). Thus the coefficient of the linear term $A\propto\rho^{1/2}$, and of the cubic term $B\propto\rho^{-{3/2}}$.

The crossover scale $T_{cr}$ above which the $T^{3}$ becomes larger than the linear term is $T_{cr}=\Lambda_{0}[(10/\pi^{2})(V-g)/(5g-V)]^{1/2}$. It is useful to compare $\Lambda_{0}$ and $T_{cr}$ with $T_{c}$. Above $T_{c}$, the Hartree and exchange terms are constant and $\rho_{0}=0=\sigma$. Hence, $T_{c}$ is given by the noninteracting expression $T_{c}/\tau=[2\rho/(\sqrt{\pi}\zeta(3/2))]^{2/3}$, where $\zeta(z)$ is the Riemann zeta function. Then, for small $\rho$, $\Lambda_{0}/T_{c}\approx 0.151(g/t_{b})\rho^{1/3}$, which is rather small. And for $V\sim t_{b}$ or less, $T_{cr}/T_{c}\approx 0.024(V/t_{b})^{3/2}\rho^{1/3}$, where $0.024$ is a geometrical factor. Since $T_{cr}/T_{c}$ is so small finding the linear term experimentally in very low $T_{c}$ Bose systems such as atomic gases would require considerable care. The result for $T_{cr}$ will be different for a strongly interacting, high density system like liquid He4, in which phonons exist even above $T_{c}$, but the effect should exist.

Near $T_{c}^{-}$.—As $T\rightarrow T_{c}$ from below, the densities show a square root singularity as $\Lambda\rightarrow 0$, e.g.,

[TABLE]

where $a_{1}=\pi/\sqrt{2}$ and $a_{2}=-{\sqrt{\pi}}\zeta(1/2)/2$ with $\zeta(1/2)=-1.4604$. The zero-point term $\rho_{10}=(\Lambda/\tau)^{3/2}/6$ is of higher order. Without the HF correction $\Lambda=V\rho_{0}$, then the dominant $\Lambda^{1/2}$ term ensures that Eq. (5) has no real solution for $\rho_{0}$ close to $T_{c}^{-}$, resulting in a first-order transition. This is corrected in the HF theory since the anomalous self-energy $\sigma$ has the same square-root term which cancels the one of $\rho_{1}$ in Eq. (6). This follows from the identity $(\partial\rho_{11}/\partial\Lambda)_{T}=\sigma_{1}/2\Lambda$ satisfied by the $T$-dependent parts $\rho_{11}$ and $\sigma_{1}$. )Solving Eq. (6) we obtain

[TABLE]

where $g_{2}\approx g/[1-ga_{2}T_{c}^{\frac{1}{2}}/\tau^{\frac{3}{2}}]$. Then

[TABLE]

Thus the transition is second order and with the usual mean-field exponent ($\frac{1}{2}$). Using another identity $(\partial u_{11}/\partial\Lambda)_{T}=-\frac{3}{2}(\rho_{11}+\sigma_{11})$, we obtain the energy

[TABLE]

to order $\Lambda$, from which we obtain the specific heat $c=du/dT$,

[TABLE]

where $c_{B}=\frac{15}{4}\frac{\zeta(5/2)}{\zeta(3/2)}\rho(T/T_{c})^{\frac{3}{2}}$ is the specific heat per site of nointeracting bosons. As is usual in a mean-field theory, $c(T)$ has a jump discontinuity at $T_{c}$

[TABLE]

where $\nu=\Gamma(3/2)\zeta(3/2)$. Thus the fractional jump is nonuniversal and vanishes as $\rho^{1/3}$. At $T_{c}$, $c(T_{c})\propto\rho^{4/3}$. An increase with $\rho$ has been observed, although the power is unknown. The HF results are a major improvement, but obtaining the true critical behavior would require an RG treatment, and additionally gauge fluctuations may have to be included.

Conclusions.—Split pairing with real-space pairing of holons, as described in sar9 ; sar10 is thus consistent with the observed specific heat, superfluid density, decline of $T_{c}$ with decreasing $x$, and the existence of pairs above $T_{c}$ as inferred from the observed Nernst effect, diamagnetism, and specific heat. The crossover to a spinless Fermi liquid of carrier charge $+e$ and concentration $x$ is consistent with the observed properties in the pseudogap metal, including quantum oscillations. Pairing of spinons (concentration $1-x$) below $T^{*}$ describes the Mott physics, and gauge coupling with holons accounts for the existence of two metallic phases and connects the spin sector to the insulator through RVB ordering.

Why does the nodal BCS theory based on the electron peaks observed below $T_{c}$ fail to describe the low-$T$ properties such as specific heat? The plausible answer is that these are not long lived, but break up into true long-lived spin and charge excitations. The calculation of the electron Green’s function is difficult, and is not attempted here. Instead, we discuss the behavior qualitatively. In a spin-charge separated state an electron of momentum ${\bf k}$ decays into a spinon and an (anti)holon, subject to energy and momentum conservation. The spectral weight is distributed over all such pair states in the spinon-holon continuum. Hence, while there is a weak peak, there is no pole ($Z=0$) and the width is of order of bare bandwidths $\sim t$ sar12 , as seen. About $T^{*}$ and below, the low-energy region is described by the renormalized Hamiltonian, characterized by a smaller bandwidth $\sim t_{s}$, the renormalized hopping parameter which is $\sim J_{eff}\ll t$. Once spinons are paired and then holons, it costs energy to free them, making some decay channels less probable, allowing electrons to propagate longer. This is accentuated below $T_{c}$, as a macroscopic fraction of spinons and holons remain in the corresponding condensates, so that electrons propagate even longer. It is not surprising that electron peaks appear, but these additional structures are also broad, with width $\sim t_{s}$, the low-energy bandwidth.

This is borne out by the observed photoemission spectra shen ; shen2 which show that the width of a single electron peak (at ${\bf k}$) is as large as $\sim 20$ meV at half maximum, and at the base it is $\sim 0.1$ eV, i.e., of order the smaller bandwidth. In other words, the spectral weight of each electron is spread over the entire bandwidth of the renormalized particles. The electron excitations are thus short lived resonances even in the superconducting state, not true quasiparticles.