HTSC-CIF Framework: Pseudogap and Internal Fields

• The paper shows that the failure of AFM perturbation theory in the t–J model drives a pseudogap state with intertwined modulated AFM (SDW) and checkerboard CDW orders.
• HTSC-CIF is defined by dual internal fields that emerge from hole aggregation, leading to observable neutron scattering and STM signatures in cuprates.
• The framework employs aggregation operators and a mean-field Hamiltonian to reconstruct the Fermi surface, aligning theoretical predictions with ARPES and experimental data.

The HTSC-CIF framework encompasses multiple distinct research domains under a shared acronym, but most notably signifies (1) Condensed Internal Fields models for cuprate high-temperature superconductors and (2) Hierarchical Task Structure-Based Cross-Modal Causal Intervention architectures in machine learning. This article focuses on the original condensed-matter physics context ("Condensed Internal Fields") as specified in (Dayan, 2010), while integrating relevant recent generalizations and parallel nomenclature.

1. Theoretical Foundation: Breakdown of AFM Perturbation Theory

The HTSC-CIF framework originates from the failure of standard perturbation theory in the t–J model on quasi-2D Cu–O lattices, described by the Hamiltonian

$$H_{tJ} = -t\sum_{\langle ij\rangle, s} (\tilde c_{is}^\dagger \tilde c_{js} + \mathrm{h.c.}) + J\sum_{\langle ij\rangle} (\mathbf{S}_i\cdot \mathbf{S}_j - \tfrac{1}{4} n_in_j)$$

where projected fermion operators $\tilde c_{is}$ enforce single occupancy, and $\mathbf{S}_i$ is the spin-1/2 operator. In the undoped limit ($x=0$), the ground state is the uniform AFM (Néel) configuration. However, expanding in $H_t$ (kinetic term) about the AFM reference state yields catastrophically divergent series for large $t/J$, indicative of an instability and the formation of a new ground state, as shown formally by Gell-Mann–Low techniques: $H_t(t)\,|\Psi(t)\rangle = J|\Psi(t)\rangle$ This drives a quantum phase transition into a pseudogap state dominated by complex internal fields rather than simple AFM order.

2. Emergence of Dual Internal Fields: Modulated AFM and Checkerboard CDW

The pseudogap phase in cuprates features two intertwined internal fields:

• A modulated AFM field (spin-density wave, SDW), evidenced by neutron-scattering peaks,
• A checkerboard charge-density wave (CDW), observed as $4a \times 4a$ density modulations in STM experiments.

Define column operators for hole aggregation $C_{j,r}^\dagger$ and consider the expectation value for the column density,

$$\langle \hat{\rho}_c(j) \rangle = n/N + \frac{2}{N} \sum_k v_k w_k \cos (2 k_F j a)$$

with $k_F a = \pi (1-\delta)$, $\delta=x$ the doping. AFM sublattice symmetrization yields modulated staggered magnetization

$$P_m(j a) = \sum_k A_k \sin(2 k_F j a), \quad A_k = 2 v_k w_k$$

The modulated SDW components sum constructively in the spin channel, destructively in charge, while the CDW part (checkerboard order) reads

$P_c(j a) \propto \sum_k A_k \cos(2 k_F j a)$

These reproduce precisely the incommensurate magnetic ordering vectors found experimentally.

3. Ground State Construction via Aggregation Operators

Hole-induced ordering is formalized through row and column creation operators ($C_{j,c}^\dagger, C_{j,r}^\dagger$), with Fourier transforms

$$C_{k,c}^\dagger = \frac{1}{\sqrt{N-n}} \sum_{j=1}^{N-n} e^{-ikja} C_{j,c}^\dagger, \qquad C_{k,r}^\dagger = \frac{1}{\sqrt{N-n}} \sum_{j=1}^{N-n} e^{-ikja} C_{j,r}^\dagger$$

These obey Clifford algebra anticommutation relations. The ground state wavefunction in the CIF phase factorizes into independent row and column condensates: $$|\Psi_0\rangle = \prod_k \left(v_k + w_k C_{k,c}^\dagger C_{k,r}^\dagger\right) |0_c, 0_r\rangle$$ where $|0_c, 0_r\rangle$ is the AFM vacuum and $v_k^2 + w_k^2 = 1$.

4. Mean-Field Hamiltonian and Internal Field Coupling

The effective mean-field Hamiltonian for the condensed phase is written in terms of a four-component Nambu-like field: $$\Psi_k = (C_{k,c}, C_{k,c}^\dagger, C_{k,r}, C_{k,r}^\dagger)^T$$ yielding

$$H_{\rm eff} = \sum_k \Psi_k^\dagger \begin{pmatrix} \varepsilon_k & A_k & 0 & 0\ A_k & -\varepsilon_k & 0 & 0 \ 0 & 0 & \varepsilon_k & -A_k \ 0 & 0 & -A_k & -\varepsilon_k \end{pmatrix} \Psi_k - \sum_k(\varepsilon_k - A_k)$$

with $\varepsilon_k = 2J \cos (ka)$ and $A_k = 2v_kw_k$. The inclusion of $A_k$ in off-diagonal elements precisely mediates the modulated AFM field coupling in both directions. This model naturally accommodates both particle–hole mixing and momentum-shifted density modulations.

5. Green’s Functions and Excitation Spectrum

The time-ordered Green’s function,

$$G_{ij}(k, t-t') = -i\langle \Psi_{k,i}(t) \Psi_{k,j}^\dagger(t') \rangle$$

has both diagonal and anomalous (momentum-shifting) components. The diagonal frequency-domain propagator is

$$G_0(k, \omega) = \frac{\omega + \varepsilon_k \tau_3 + A_k \tau_1}{\omega^2 - [\varepsilon_k^2 + A_k^2] + i0^+}$$

with anomalous components at shifted momenta,

$$G_0(k, k\pm 2k_F, \omega) = G_0(k, \omega) \tau_1 \delta_{k, k\pm 2k_F}$$

These encode the checkerboard $2k_F$ superstructure, consistent with the presence of intertwined modulated AFM and CDW order.

6. Gapless Excitation Spectrum and Pseudogap Phenomenon

Diagonalization of the mean-field Hamiltonian yields two excitation branches,

$E_k^\pm = \pm \sqrt{\varepsilon_k^2 + A_k^2}$

At the “Fermi points” ($k=k_F$), $\varepsilon_{k_F}=0$ and $E_{k_F}^\pm = 0$, implying a gapless spectrum despite the existence of a finite order parameter ($A_k \neq 0$). This resolves the long-standing contradiction of the pseudogap: symmetry breaking and order coexist with zero-energy excitations at the Fermi surface.

7. Reconstructed Fermi Surface and Experimental Signatures

The CIF phase reconstructs the Fermi surface into four nodal hole arcs and two nearly nested antinodal segments of width $2\pi\delta/a$. This structured Fermi surface matches ARPES observations. The SDW peaks at wavevectors $(1\pm2\delta, 1)$ and $(1, 1\pm2\delta)$, exactly as predicted by the modulated AFM density formula above, are observed in neutron-scattering experiments. The four-unit-cell period checkerboard order ($2k_F = \pi/2a$), as seen in STM at low doping, is quantitatively explained by the periodicity of the CDW component.

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In summary, the HTSC-CIF framework attributes the pseudogap in underdoped cuprates to the breakdown of AFM perturbation theory in the t–J model, resulting in condensation into a phase with modulated internal fields. The ground state is characterized by the aggregation of doped holes into rows and columns, producing orthogonal modulated AFM and checkerboard CDW order. The mean-field Hamiltonian, excitation spectrum, and reconstructed Fermi surface are quantitatively consistent with experimental observations in neutron scattering, STM, and ARPES, resolving the pseudogap paradox within a nontrivial symmetry-breaking scenario.