arXiv:2605.17227

Disorder effect on the superfluid density and the origin of the pseudogap end point in the cuprate superconductors

Rong Cheng, Tao Li, Jianhua Yang

Published May 17, 2026• 0 citations

Abstract

A major puzzle in the study of the cuprate superconductivity is the origin of the pseudogap end point. Intriguingly, such a critical doping is also where the superfluid density of the system reaches its maximum. A non-monotonic doping dependence of the superfluid density is rather unusual since the Drude weight of the cuprate system is found to increase monotonically with the doping concentration. It is generally believed that such a peculiar behavior should be attributed to both the strongly correlated nature of the cuprate system and the disorder effect. In this work, we develop a variational theory for the zero temperature superfluid density of the disordered $t-J$ model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state for the disordered $t-J$ model. Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density $ρ_{s}(0)$ is then extracted from the curvature of the variational ground state energy of the system as a function of the external electromagnetic field. We find that $ρ_{s}(0)$ computed in this way is remarkably robust against the disorder effect. More specifically, we find that $ρ_{s}(0)$ is a monotonically increasing function of doping concentration $x$ and scales linearly with the total optical weight. This is consistent with the observation in the underdoped cuprates but is strongly at odd with the behavior in the overdoped cuprates. The strong contrast between the disorder effect in the underdoped and the overdoped regime lends strong support to our previous proposal that there exist a Mott transition between a doped-Mott-insulating metal in the underdoped regime and a fermi-liquid-like metal in the overdoped regime around the pseudogap end point.

DOI: 10.48550/arxiv.2605.17227

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A tale of two pairs: on the origin of the pseudogap end point in the high-T$_{c}$ cuprate superconductors

Jianhua Yang, Tao Li

arXiv:2402.02785Feb 5, 20240 citations

Origin of the pseudogap end point in high- T c cuprate superconductors

Jianhua Yang, Tao Li

Physical review. B./Physical review. BJul 30, 20241 citation

The optical conductivity of the 2D $t-J$ model and the origin of electron incoherence in the high-T$_{c}$ cuprate superconductors: a variational study

Jianhua Yang, Tao Li

arXiv:2401.00118Dec 30, 20230 citations

Specific heat maximum as a signature of Mott physics in the two-dimensional Hubbard model

G. Sordi, C. Walsh, P. Sémon +1 more

arXiv:1905.02281Physical review. B./Physical review. BSep 12, 201916 citations

Low-temperature pseudogap phenomenon: precursor of high-Tcsuperconductivity

Yao Ma, Peng Ye, Zheng-Yu Weng

arXiv:1311.3395New Journal of PhysicsAug 26, 201422 citations

The Effects of Electronic Doping on Quantum Materials: Cuprates and Graphene

J. P. F. LeBlanc

May 1, 20120 citations

arXiv:2605.17227

Disorder effect on the superfluid density and the origin of the pseudogap end point in the cuprate superconductors

Rong Cheng, Tao Li, Jianhua Yang

Published May 17, 2026• 0 citations

arXiv HTML PDF

Abstract

t-J

model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state for the disordered

t-J

model. Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density

ρ_{s}(0)

is then extracted from the curvature of the variational ground state energy of the system as a function of the external electromagnetic field. We find that

ρ_{s}(0)

computed in this way is remarkably robust against the disorder effect. More specifically, we find that

ρ_{s}(0)

is a monotonically increasing function of doping concentration

x

and scales linearly with the total optical weight. This is consistent with the observation in the underdoped cuprates but is strongly at odd with the behavior in the overdoped cuprates. The strong contrast between the disorder effect in the underdoped and the overdoped regime lends strong support to our previous proposal that there exist a Mott transition between a doped-Mott-insulating metal in the underdoped regime and a fermi-liquid-like metal in the overdoped regime around the pseudogap end point.

DOI: 10.48550/arxiv.2605.17227

Related papers

A tale of two pairs: on the origin of the pseudogap end point in the high-T $_{c}$ cuprate superconductorsA tale of two pairs: on the origin of the pseudogap end point in the high-T$_{c}$ cuprate superconductors

Jianhua Yang, Tao Li

There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around $x_{c}$ , the end point of the superconducting dome on the overdoped side, where the cuparte is believed to be well described by the fermi liquid theory. This is the most evident in the observed $ρ_{s}(0)-T_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the d-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around $x^{*}$ , the so-called pseudogap end point. Through a systematic variational optimization of the disordered 2D $t-J$ model from the resonating valence bond(RVB) perspective, we find that the d-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional d-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the d-wave RVB pairing of the charge neutral spinons becomes essentially immune to the impurity potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at $x^{*}$ , where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for $x>x^{*}$ , whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around $x_{c}$ .There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around $x_{c}$, the end point of the superconducting dome on the overdoped side, where the cuparte is believed to be well described by the fermi liquid theory. This is the most evident in the observed $ρ_{s}(0)-T_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the d-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around $x^{*}$, the so-called pseudogap end point. Through a systematic variational optimization of the disordered 2D $t-J$ model from the resonating valence bond(RVB) perspective, we find that the d-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional d-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the d-wave RVB pairing of the charge neutral spinons becomes essentially immune to the impurity potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at $x^{*}$, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for $x>x^{*}$, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around $x_{c}$.

arXiv:2402.02785Feb 5, 20240 citations

Origin of the pseudogap end point in high- T c cuprate superconductors

Jianhua Yang, Tao Li

There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around ${x}_{c}$ , the end point of the superconducting dome on the overdoped side, where the cuprate is believed to be well described by the Fermi-liquid theory. This is the most evident in the observed ${\ensuremath{\rho}}_{s}(0)\ensuremath{-}{T}_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the $d$ -wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around ${x}^{*}$ , the so called pseudogap end point. Through a systematic variational Monte Carlo (VMC) study of the disordered 2D $t\ensuremath{-}J$ model from the resonating valence bond (RVB) perspective, we find that the $d$ -wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional $d$ -wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the $d$ -wave RVB pairing of the charge-neutral spinons becomes essentially immune to the disorder potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at ${x}^{*}$ , where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for $x>{x}^{*}$ , whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around ${x}_{c}$ .There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around ${x}_{c}$, the end point of the superconducting dome on the overdoped side, where the cuprate is believed to be well described by the Fermi-liquid theory. This is the most evident in the observed ${\ensuremath{\rho}}_{s}(0)\ensuremath{-}{T}_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the $d$-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around ${x}^{*}$, the so called pseudogap end point. Through a systematic variational Monte Carlo (VMC) study of the disordered 2D $t\ensuremath{-}J$ model from the resonating valence bond (RVB) perspective, we find that the $d$-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional $d$-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the $d$-wave RVB pairing of the charge-neutral spinons becomes essentially immune to the disorder potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at ${x}^{*}$, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for $x>{x}^{*}$, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around ${x}_{c}$.

Physical review. B./Physical review. BJul 30, 20241 citation

The optical conductivity of the 2D $t-J$ model and the origin of electron incoherence in the high-T $_{c}$ cuprate superconductors: a variational studyThe optical conductivity of the 2D $t-J$ model and the origin of electron incoherence in the high-T$_{c}$ cuprate superconductors: a variational study

Jianhua Yang, Tao Li

Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T $_{c}$ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D $t-J$ model through the variational calculation of its optical conductivity $σ(ω)$ . We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight $\bar{K}$ and the integrated incoherent optical weight $I$ increase monotonically with doping, with their ratio $R_{incoh}=I/\bar{K}$ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D $t-J$ model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from $D=\bar{K}-I$ scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T$_{c}$ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D $t-J$ model through the variational calculation of its optical conductivity $σ(ω)$. We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight $\bar{K}$ and the integrated incoherent optical weight $I$ increase monotonically with doping, with their ratio $R_{incoh}=I/\bar{K}$ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D $t-J$ model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from $D=\bar{K}-I$ scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.

arXiv:2401.00118Dec 30, 20230 citations

Specific heat maximum as a signature of Mott physics in the two-dimensional Hubbard model

G. Sordi, C. Walsh, P. Sémon +1 more

Recent experiments on cuprates show that as a function of doping, the normal-state specific heat sharply peaks at the doping ${\ensuremath{\delta}}^{*}$, where the pseudogap ends at low temperature. This finding is taken as the thermodynamic signature of a quantum critical point, whose nature has not yet been identified. Here we present calculations for the two-dimensional Hubbard model in the doped Mott insulator regime, which indicate that the specific heat anomaly can arise from the finite-temperature critical end point of a first-order transition between a pseudogap phase with dominant singlet correlations and a metal. As a function of doping at the temperature of the end point, the specific heat diverges. Upon increasing temperature, the peak becomes broader. The diverging correlation length is associated with uniform density fluctuations. No broken symmetries are needed. These anomalies also occur at half-filling as a function of interaction strength, and are relevant for organic superconductors and ultracold atoms.Recent experiments on cuprates show that as a function of doping, the normal-state specific heat sharply peaks at the doping ${\ensuremath{\delta}}^{*}$, where the pseudogap ends at low temperature. This finding is taken as the thermodynamic signature of a quantum critical point, whose nature has not yet been identified. Here we present calculations for the two-dimensional Hubbard model in the doped Mott insulator regime, which indicate that the specific heat anomaly can arise from the finite-temperature critical end point of a first-order transition between a pseudogap phase with dominant singlet correlations and a metal. As a function of doping at the temperature of the end point, the specific heat diverges. Upon increasing temperature, the peak becomes broader. The diverging correlation length is associated with uniform density fluctuations. No broken symmetries are needed. These anomalies also occur at half-filling as a function of interaction strength, and are relevant for organic superconductors and ultracold atoms.

arXiv:1905.02281Physical review. B./Physical review. BSep 12, 201916 citations

Low-temperature pseudogap phenomenon: precursor of high-Tcsuperconductivity

Yao Ma, Peng Ye, Zheng-Yu Weng

In this paper, we try to understand the pseudogap phenomenon observed in the cuprate superconductor through a model study. Specifically, we explore the so-called low-temperature pseudogap state by turning off the superconducting (SC) off diagonal long range order in an ansatz state for the t – J model (Weng 2011 New J. Phys. 13 103039 ). Besides strong non-Gaussian SC fluctuations, the resulting state also exhibits a systematic pseudogap behavior in both spin and charge degrees of freedom, manifested in the uniform spin susceptibility, specific heat, non-Drude resistivity, Nernst effect, as well as the quantum oscillation associated with small Fermi pockets emerging in strong magnetic fields, etc. These anomalous 'normal state' properties are found in qualitative consistency with experimental measurements in the cuprates. Such a model study establishes an intrinsic connection between the peculiar pseudogap properties and the non-BCS nature of the SC ground state. A critical comparison with other approaches to the doped Mott insulator is also made.In this paper, we try to understand the pseudogap phenomenon observed in the cuprate superconductor through a model study. Specifically, we explore the so-called low-temperature pseudogap state by turning off the superconducting (SC) off diagonal long range order in an ansatz state for the t – J model (Weng 2011 New J. Phys. 13 103039 ). Besides strong non-Gaussian SC fluctuations, the resulting state also exhibits a systematic pseudogap behavior in both spin and charge degrees of freedom, manifested in the uniform spin susceptibility, specific heat, non-Drude resistivity, Nernst effect, as well as the quantum oscillation associated with small Fermi pockets emerging in strong magnetic fields, etc. These anomalous 'normal state' properties are found in qualitative consistency with experimental measurements in the cuprates. Such a model study establishes an intrinsic connection between the peculiar pseudogap properties and the non-BCS nature of the SC ground state. A critical comparison with other approaches to the doped Mott insulator is also made.

arXiv:1311.3395New Journal of PhysicsAug 26, 201422 citations

The Effects of Electronic Doping on Quantum Materials: Cuprates and Graphene

J. P. F. LeBlanc

In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-T $_c$ ) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-T $_c$ cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-T$_c$) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-T$_c$ cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.

May 1, 20120 citations

t-J

model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state for the disordered

t-J

model. Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density

ρ_{s}(0)

is then extracted from the curvature of the variational ground state energy of the system as a function of the external electromagnetic field. We find that

ρ_{s}(0)

computed in this way is remarkably robust against the disorder effect. More specifically, we find that

ρ_{s}(0)

is a monotonically increasing function of doping concentration

x

A tale of two pairs: on the origin of the pseudogap end point in the high-T

_{c}

cuprate superconductors

There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around

x_{c}

, the end point of the superconducting dome on the overdoped side, where the cuparte is believed to be well described by the fermi liquid theory. This is the most evident in the observed

ρ_{s}(0)-T_{c}

scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the d-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around

x^{*}

, the so-called pseudogap end point. Through a systematic variational optimization of the disordered 2D

t-J

model from the resonating valence bond(RVB) perspective, we find that the d-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional d-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the d-wave RVB pairing of the charge neutral spinons becomes essentially immune to the impurity potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at

x^{*}

, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for

x>x^{*}

, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around

x_{c}

There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around

{x}_{c}

, the end point of the superconducting dome on the overdoped side, where the cuprate is believed to be well described by the Fermi-liquid theory. This is the most evident in the observed ${\ensuremath{\rho}}_{s}(0)\ensuremath{-}{T}_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the

d

-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around

{x}^{*}

, the so called pseudogap end point. Through a systematic variational Monte Carlo (VMC) study of the disordered 2D $t\ensuremath{-}J$ model from the resonating valence bond (RVB) perspective, we find that the

d

-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional

d

-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the

d

-wave RVB pairing of the charge-neutral spinons becomes essentially immune to the disorder potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at

{x}^{*}

, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for

x>{x}^{*}

, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around

{x}_{c}

The optical conductivity of the 2D

t-J

model and the origin of electron incoherence in the high-T

_{c}

cuprate superconductors: a variational study

Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T

_{c}

cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D

t-J

model through the variational calculation of its optical conductivity

σ(ω)

. We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight

\bar{K}

and the integrated incoherent optical weight

I

increase monotonically with doping, with their ratio

R_{incoh}=I/\bar{K}

decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D

t-J

model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from

D=\bar{K}-I

scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.

In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-T

_c

) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-T

_c

cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.