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arXiv:2606.07609

Finite-temperature formation of magnetic plateaus and simplex liquid states on the frustrated ruby lattice

Antonio Francesco Mello, E. Miles Stoudenmire, Joseph Tindall

cond-mat.str-elquant-phSubmitted May 29, 2026• Announced Jun 1, 2026• 0 citations
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Abstract

Geometric frustration in quantum systems can stabilize unconventional phases of matter that avoid traditional magnetic ordering at low temperatures. Here, we observe this phenomenon while mapping out the finite temperature phase diagram of the spin-1/2 Heisenberg antiferromagnet on the ruby lattice with next-nearest-neighbor interactions. Using an infinite tensor network state (iTNS) optimized and measured with belief propagation (BP) and corrections to BP, we observe the low temperature formation of stable magnetic plateaus at various magnetic field strengths. We find these plateaus host a novel `simplex liquid state' -- a disordered phase involving strongly paired spin simplices that retains non-zero residual entropy due to an exponentially large subspace of crystalline configurations. We accurately quantify the energy gap associated with these states and show that, as the temperature of the system is lowered, it does not go through a phase transition to reach them: the heat capacity remains finite and continuous at all observed temperatures. Our work demonstrates how BP-based tensor network techniques provide a powerful route to understanding frustrated quantum magnets at finite temperature.

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Main text: 6 pages, 4 figures. Comments are welcome

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Geometric frustration in quantum systems can stabilize unconventional phases of matter that avoid traditional magnetic ordering at low temperatures. Here, we observe this phenomenon while mapping out the finite temperature phase diagram of the spin-1/2 Heisenberg antiferromagnet on the ruby lattice with next-nearest-neighbor interactions. Using an infinite tensor network state (iTNS) optimized and measured with belief propagation (BP) and corrections to BP, we observe the low temperature formation of stable magnetic plateaus at various magnetic field strengths. We find these plateaus host a novel `simplex liquid state' -- a disordered phase involving strongly paired spin simplices that retains non-zero residual entropy due to an exponentially large subspace of crystalline configurations. We accurately quantify the energy gap associated with these states and show that, as the temperature of the system is lowered, it does not go through a phase transition to reach them: the heat capacity remains finite and continuous at all observed temperatures. Our work demonstrates how BP-based tensor network techniques provide a powerful route to understanding frustrated quantum magnets at finite temperature.
We investigate the spin-1/21/21/2 Heisenberg antiferromagnet on the ruby lattice with uniform first- and second-neighbor interactions, which forms a two-dimensional network of corner-sharing tetrahedra. Using infinite projected entangled pair states (iPEPS), we study the ground state of the system to find that it assumes a gapped threefold-degenerate simplex crystal ground state, with strong singlets formed on pairs of neighboring triangles. We argue that the formation of the simplex singlet ground state at the isotropic point relates to the weak inter-triangle coupling limit where an effective spin-chirality Hamiltonian on the honeycomb lattice exhibits an extensively degenerate ground state manifold of singlet coverings at the mean-field level. Under an applied Zeeman field, the iPEPS simulations uncover magnetization plateaus at m/ms=0,1/3,1/2,m/m_s = 0, 1/3, 1/2,m/ms​=0,1/3,1/2, and 2/32/32/3, separated by intermediate supersolid phases, all breaking the sixfold rotational symmetry of the lattice. Unlike the checkerboard lattice, these plateaus cannot be described by strongly localized magnons.
By utilizing tensor-network-based methods, we investigate the zero- and finite-temperature properties of the spin-1/2 Heisenberg antiferromagnetic (HAF) model on an infinite Husimi lattice that contains 3/2 sites per triangle. The ground state of this model is found to possess vanishing local magnetization and is featureless; the spin-spin and dimer-dimer correlation functions are verified to decay exponentially, and its ground-state energy per site is determined to be ${e}_{0}=\ensuremath{-}0.4343(1)$, which is very close to that [${e}_{0}=\ensuremath{-}0.4386(5)$] of the intriguing kagome HAF model. The magnetization curve shows the absence of a zero-magnetization plateau, implying a gapless excitation. A 1/31/31/3-magnetization plateau with spin-up-up-down state is observed, which is selected and stabilized by quantum fluctuations. A ground-state phase diagram under magnetic fields is presented. Moreover, both magnetic susceptibility and the specific heat are studied, whose low-temperature behaviors reinforce the conclusion that the HAF model on the infinite Husimi lattice owns a gapless and featureless spin-liquid ground state.
The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor S=1/2 antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date. We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement. By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid. We discuss the comparison with other numerical studies and the physical interpretation of this result.
The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor S ¼ 1=2 antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date.We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement.By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid.We discuss the comparison with other numerical studies and the physical interpretation of this result.
Aimed at a more realistic classical description of natural quantum systems, we present a two-dimensional tensor network algorithm to study finite temperature properties of frustrated model quantum systems and real quantum materials. For this purpose, we introduce the infinite projected entangled simplex operator ansatz to study thermodynamic properties. To obtain state-of-the-art benchmarking results, we explore the highly challenging spin-1/2 Heisenberg antiferromagnet on the Kagome lattice, a system for which we investigate the melting of the magnetization plateaus at finite magnetic field and temperature. Making a close connection to actual experimental data of real quantum materials, we go on to studying the finite temperature properties of Ca10Cr7O28{\mathrm{Ca}}_{10}{\mathrm{Cr}}_{7}{\mathrm{O}}_{28}Ca10​Cr7​O28​. We compare the magnetization curve of this material in the presence of an external magnetic field at finite temperature with classically simulated data. As the first theoretical tool that incorporates both thermal fluctuations as well as quantum correlations in the study of this material, our work contributes to settling the existing controversy between the experimental data and previous theoretical works on the magnetization process.
Quantum fluctuations from frustration can trigger quantum spin liquids (QSLs) at zero temperature. However, it is unclear how thermal fluctuations affect a QSL. We employ state-of-the-art tensor network-based methods to explore the spin-1/21/21/2 kagom\'e Heisenberg antiferromagnet (KHA). Its ground state is shown to be a gapless U(1) QSL. By studying the behaviors of specific heat and magnetic susceptibility, we propose a finite-temperature phase diagram for the spin-1/2 KHA in a magnetic field. It is unveiled that there exists an algebraic paramagnetic liquid (APL) that possesses both paramagnetic and algebraic behaviors inherited from the QSL. The APL is induced under the interplay between quantum fluctuations from geometrical frustration and thermal fluctuations. Our work suggests that even when the QSL at zero temperature is frozen into a solid state by a magnetic field, proper thermal fluctuations could melt the system into a liquid-like state. We speculate that an APL phase could generally emerge after proper thermal fluctuations intervene in a gapless QSL.