Frustration from Localized Zhang-Rice States: A Unified Theory of Doping-Driven Magnetic Transitions in Cuprates

The gist: Doped holes in cuprates form spatially localized Zhang-Rice singlets that act as active intermediate states, mediating emergent longer-range superexchange interactions (J2 and J3) which rapidly disrupt antiferromagnetic order and drive the emergence of a robust spin-glass phase upon hole doping.

Microscopic Mechanism

The fundamental mechanism proposed is that doped holes do not act as simple non-magnetic vacancies but form spatially localized Zhang-Rice singlets which actively mediate emergent spin exchange (Page 1). These localized states introduce emergent next-nearest J2 and third-nearest J3 neighbor superexchanges (Page 1). This dopant-induced exchange pathway generates significant magnetic frustration, which naturally explains the rapid collapse of the Neel AFM order and the emergence of a spin-glass phase on the hole-doped side (Page 1).

Effective Spin Hamiltonian

The magnetism in lightly doped cuprates is described by an effective spin Hamiltonian: H = J1 X′⟨i j⟩1 Si·Sj + J2 X ZR⟨i j⟩2 Si·Sj + J3 X ZR⟨i j⟩3 Si·Sj (Page 1). The first term describes the intact NN bonds on the square lattice, excluding those disrupted by dopants (Page 1), while the latter terms describe emergent longer-range superexchange couplings mediated by localized ZR singlet states, which are activated only under hole doping (Page 1).

Asymmetric Magnetic Response

The results demonstrate a striking asymmetry in the magnetic response between electron- and hole-doped regimes (Page 2). On the electron-doped side (δ < 0), the magnetization Mz decreases slowly, consistent with the 'bare vacancy' picture, where doped carriers effectively act as non-magnetic site dilutions (Page 2). In sharp contrast, hole doping (δ > 0) severely and rapidly suppresses the AFM order because of the dressed ZR vacancies whose emergent frustrating exchange pathways strongly disrupt the Neel background (Page 2).

Emergence of Phases

The suppression of long-range AFM order leads to two distinct magnetic outcomes depending on temperature and doping. The system exhibits a finite Neel temperature TN stabilized by weak interlayer coupling in 3D systems, where TN is determined by the crossing point of Binder ratio curves B(T) (Page 4). Upon further hole doping, the system enters a spin-glass (SG) phase characterized by the squared Edwards-Anderson order parameter q(2)SG, which exhibits a sharp onset at low temperatures and yields a non-zero intercept in the thermodynamic limit (Page 4).

Role of Frustration

The frustration is directly linked to the geometry of the localized states. The third-nearest-neighbor exchange J3 is significantly larger than J1 because it is mediated by virtual hopping through intermediate states involving up to four holes in the central plaquette, and Jξp is much larger than JU, making J3 comparable to, or even larger than the undoped J1 (Page 12). Furthermore, the geometric orthogonality between the e-orbital channels for J3 and J2 explains why J2 relies solely on the a orbital and the heavily suppressed b orbital (Page 12).

Numerical Validation

The VMC approach is validated against DMRG calculations on a 12 × 12 lattice with open boundary conditions, showing quantitative agreement in ground-state energetics and the evolution of macroscopic magnetic order (Page 15). The VMC method successfully reproduces the highly inhomogeneous, fragmented domain-wall structures predicted by DMRG, with the local magnetic moments matching quantitatively across the lattice (Page 15).

Conclusion

The paper concludes that localized ZR singlets are not inert vacancies, but active mediators of emergent longer-range superexchange interactions (J2 and J3) (Page 1). The dopant potentials are identified as the microscopic origin of the pronounced magnetic electron–hole asymmetry in cuprates (Page 1).

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Xiaodong Wang, Ping Xu, Jiong Mei, Shao-Hang Shi, Zi-Xiang Li, Mingpu Qin, Kun Jiang and Hui-Ke Jin. Frustration from Localized Zhang-Rice States: A Unified Theory of Doping-Driven Magnetic Transitions in Cuprates. 1 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 3 Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 4 Hefei National Laboratory, Hefei 230088, China 5 School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210. (Dated: June 9, 2026) The microscopic mechanism by which doped holes disrupt the antiferromagnetic order is one of the fundamental questions in cuprates. In this work, we propose a unified microscopic theory in which doped holes form spatially localized Zhang-Rice singlets which actively mediate emergent spin exchange. Rather than acting as simple non-magnetic vacancies, these localized states introduce emergent next-nearest J2 and third-nearest J3 neighbor superexchanges. This dopant-induced exchange pathway generates significant magnetic frustration, naturally explaining the rapid collapse of the Neel AFM order and the emergence of a spin-glass phase on the hole-doped side. Our findings provide a comprehensive framework for understanding the complex dopingdriven magnetic phase transitions and magnetic electron-hole asymmetry in lightly doped cuprates.

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Introduction.— Hole doping in a charge-transfer insulator such as cuprates introduces carriers into a strongly correlated background, rapidly suppressing antiferromagnetic (AFM) order and ultimately giving rise to superconductivity [1–5]. Within the standard Hubbard or t–J framework, these doped holes are typically treated as itinerant carriers—such as Zhang–Rice singlets—whose motion renormalizes the underlying magnetic correlations [6–15]. In real materials, however, dopant atoms associated with chemical substitution or oxygen nonstoichiometry inevitably generate spatially inhomogeneous potentials [16–23]. Accumulating evidence from scanning tunneling microscopy (STM) indicates that, particularly in the lightly doped regime, doped carriers can become locally trapped, forming spatially localized Zhang–Rice (ZR) singlet states rather than itinerant carriers [24–30]. With the kinetic motion of doped holes quenched, how such localized ZR singlet states disrupt the magnetic properties of cuprates remains an open question [24, 25]. Early theories proposed that bare oxygen holes might introduce magnetic frustration effects via local ferromagnetic bonds [16, 17] or impurities [31–34]. Here we reveal a fundamentally different mechanism. We demonstrate that the fully formed, localized ZR singlets act as active intermediate states for virtual hopping. This dopant-mediated channel generates emergent longer-range superexchange interactions (J2 and J3) between copper spins. Employing large-scale Monte Carlo simulations and density matrix renormalization group (DMRG) calculations [35, 36], we demonstrate that this dopant-induced frustration strongly disrupts the Neel background, naturally explaining the rapid collapse of the AFM order and the subsequent emergence of a robust spin-glass phase upon hole doping [37–40]. When contrasted with the simple spin-dilution mechanism in electron-doped cuprates [18, 26, 41], our results provide a unified perspective on the role of dopants. In particular, we identify the dopant potential as the microscopic origin of the pronounced magnetic electron-hole asymmetry—a long-standing puzzle highlighted in early theories [42–44] and reviews [5]—as recently suggested in our previous work [26].

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Microscopic Model.— Microscopically, the essential physics of cuprates originates from their charge-transfer nature, which can be captured by the three-band Emery model in the hole representation [6, 7]. In the undoped limit, there is one hole per unit cell, predominantly residing on the Cu sites. The low-energy physics then reduces to an effective Heisenberg Hamiltonian with S = 1/2 spins on the Cu lattice, which interact via nearest-neighbor (NN) superexchange and stabilize a Neel-ordered ground state [´ 6, 45]. The dominant superexchange interaction, mediated by oxygen orbitals, is given by J1 ≈

4t

4

pd

ξ

2

p

1

Ud + 1

ξp

where tpd denotes the Cu–O hybridization, ξp is the charge-transfer energy, and Ud is the on-site Coulomb repulsion on Cu d Ud [46]. Upon hole doping, it is well established that the additional holes primarily reside on oxygen orbitals and bind with Cu spins to form ZR singlet states [6]. However, the STM experiments have shown that these doped holes are locally trapped because of the local attractive dopant potential Eloc < 0, and form spatially localized ZR singlets. These localized singlets act as bound states whose excitation energies are reduced by a factor of 2 ∼ 4 compared with the charge-transfer gap [24, 25, 29]. Given that the lightly doped system remains insulating, it is natural to ask whether these localized states can act as intermediate states for virtual hopping processes and thereby mediate additional superexchange interactions [45]. Owing to their reduced excitation energy, this dopant-assisted exchange channel is substantially enhanced. In particular, virarXiv:2605.

1. naturally explain rapid collapse of Neel AFM order and emergence of a spin-glass phase on the hole-doped side by identifying localized ZR singlet states as active mediators of emergent next-nearest J2 and third-nearest J3 neighbor superexchanges. This allows the AI to model magnetic transitions not just as simple site dilution, but as a complex frustration mechanism driven by localized electronic states.

2. Predict magnetic electron-hole asymmetry in lightly doped cuprates by identifying the dopant potential as the microscopic origin of the pronounced magnetic electron-hole asymmetry, distinguishing this from simple spin-dilution mechanism in electron-doped cuprates. This enables the AI to generate physically accurate phase diagrams differentiating between hole-doped and electron-doped doping effects.

3. Model complex magnetic ground states by utilizing a unified effective spin Hamiltonian: H = J1 X′ ⟨i j⟩1 Si · Sj + J2 X ZR ⟨i j⟩2 Si · Sj + J3 X ZR ⟨i j⟩3 Si · Sj. This allows the AI to simulate the resulting magnetic phase diagram and critical temperatures (like the critical doping δc ≈ 0.05) based on emergent interactions rather than standard models.

4. Identify and quantify emergent longer-range superexchange couplings (J2 and J3) mediated by localized ZR singlets, noting that the dominant contribution across a ZR singlet yields J3 ≈ 4t4 pd/ξ3 e. This enables the AI to calculate specific exchange coupling strengths based on material parameters, providing quantitative predictions for magnetic interactions.

5. Develop a robust predictive framework for finite-temperature states by incorporating the weak interlayer coupling present in bulk cuprates and analyzing the Binder ratio B(T) to determine critical temperatures (like TN) via its crossing point, rather than relying solely on the Mermin-Wagner theorem limitations in 2D. This allows for accurate prediction of phase boundaries in simulated 3D systems.

6. Simulate spin-glass phases by analyzing the squared Edwards-Anderson order parameter [57], q(2)SG = ⟨q2⟩T, and confirming its robustness via finite-size scaling of q(2)SG versus 1/L at T = 0.1 [inset of Fig. 3(a)] yields a non-zero intercept as L → ∞. This enables the AI to accurately predict the emergence and temperature dependence of the spin-glass phase upon hole doping.

7. Analyze local magnetic inhomogeneity by using real-space distribution of the local staggered magnetization mzi obtained by VMC, which shows that the magnitude of the local staggered magnetization remains robust (m zi ∼ 0.2), but it undergoes a π phase shift across spatial boundaries, resulting in a static domain-wall structure. This allows the AI to predict and visualize spatially inhomogeneous magnetic textures induced by defects.

8. Quantify exchange coupling hierarchies by comparing ED Perturbation Theory results with exact diagonalization (ED) benchmarks, showing that the perturbative results capture the hierarchy "J3 > J1 ≫ J2 in excellent quantitative agreement with the ED benchmarks." This ensures the AI's derived interaction strengths are physically accurate and correctly ordered.

9. Optimize variational wavefunctions using stochastic reconfiguration (SR) to minimize energy by following a gradient defined as f⃗ ≡ −∇θE(θ) = −2 Re ⟨(∇θ ln Ψ)∗Eloc⟩ − ⟨Eloc⟩⟨(∇θ ln Ψ)∗⟩. This allows the AI to efficiently search for the true ground state of complex, correlated spin systems.

10. Construct a mean-field ansatz that decomposes the Hamiltonian into Uniform Background Terms (Hbg), Inhomogeneous Magnetic Terms (Hmag), and Defect-Induced Local Terms (Hdef), allowing for explicit modeling of both global order and localized frustration simultaneously. This enables the AI to capture the full physics of lightly doped cuprates, from long-range AFM order to spin-glass freezing.