Ground-State Phase Diagram

Updated 13 January 2026

Ground-state phase diagrams are detailed maps of quantum phases at zero temperature, delineating phase boundaries, order parameters, and critical transitions as system parameters vary.
They employ a blend of analytical, variational, and numerical techniques—such as tensor networks, QMC, and DMRG—to precisely locate phase transitions and quantify key observables.
These diagrams guide experimental research in condensed matter and ultracold atom systems by predicting phenomena like Mott insulators, superfluids, and topological states.

A ground-state phase diagram provides a comprehensive map of quantum phases realized by many-body systems at zero temperature, as a function of system parameters such as interaction strength, filling, external fields, symmetry breaking, and dimensionality. It encodes the structure and boundaries of distinct quantum phases, their emergent order parameters, regimes of criticality, and the nature of transitions between them. The construction and interpretation of ground-state phase diagrams draws on a combination of analytical techniques, variational and tensor-network approaches, quantum Monte Carlo, exact diagonalization, renormalization-group flow, and symmetry/topology analyses. Characterization of quantum phases relies on the identification of robust signatures: spectral gaps, correlation functions, order parameters, topological invariants, and scaling dimensions. Research across quantum condensed matter, ultra-cold atomic gases, and quantum magnetism provides explicit archetypes, each delivering rich phase diagrams and novel physical phenomena.

1. Formulation and Methodologies for Ground-State Phase Diagrams

The construction of ground-state phase diagrams begins with a microscopic model Hamiltonian—typically in second-quantization form—for the system under study. For bosonic, fermionic, or spin systems, the Hamiltonian incorporates hopping (kinetic), interaction, chemical potential, anisotropy, symmetry-breaking, and external-field terms. Examples include the Bose-Hubbard model for lattice bosons (McBrian et al., 2019), the t–J model for correlated electrons (1012.40281001.3343Zhang et al., 2024), SU(N) Hubbard chains (Rapp et al., 2011), spin chains and ladders with anisotropy and bond alternation (Hida et al., 2016 Okamoto et al., 2019 Tonegawa et al., 2015 Hida, 2013), and models with long-range or cluster-forming interactions (Mendoza-Coto et al., 2020 Kroiss et al., 2016 Bisset et al., 2016).

To locate phase boundaries:

Variational approaches (Gutzwiller, tensor networks, neural-network ansatz) are employed for complex systems (1102.30531001.3343McBrian et al., 2019).
Numerical methods such as density-matrix renormalization group (DMRG), exact diagonalization (ED), quantum Monte Carlo (QMC), and pseudofermion functional RG (PFFRG) provide high-precision mapping, direct computation of order parameters, correlation functions, and excitation gaps (1012.40282207.03786Luo et al., 2017 Tonegawa et al., 2015 Kroiss et al., 2016).
Analytical techniques: Mean-field theory, bosonization, field-theory scaling dimensions, level spectroscopy, and perturbative mappings (e.g., S=1 chain to S=2 chain (Okamoto et al., 2019 Hida, 2013)) clarify underlying physics and limiting regimes.

Characteristic observables include compressibility, spectral/spin/charge gaps, superfluid density, structure factors, string order, topological invariants, and fidelity/entanglement in many-body wavefunctions.

2. Principal Quantum Phases and Order Parameters

Ground-state phase diagrams reveal both conventional and exotic quantum phases:

Mott Insulators: Identified by incompressibility ( $\kappa=0$ ), integer occupation plateaus, and gapped excitations—exemplified in the Bose–Hubbard model (McBrian et al., 2019), 2D Hubbard (Deng et al., 2014), SU(3) Hubbard (Rapp et al., 2011).
Superfluids and Superconductors: Marked by gapless excitations, algebraically decaying off-diagonal correlations, finite superfluid stiffness (Kroiss et al., 20161012.40281408.2088). Superconducting pairing symmetry (s, d, p-wave) differentiates phases (1001.33431408.2088).
Spin Liquids: Quantum-disordered states with fractionalized excitations, topological order, and absence of long-range magnetic order; prototypical in Kitaev–Heisenberg systems for low- $S$ (Fukui et al., 2022).
Charge/Spin Density Waves, Wigner Crystals: Broken translational symmetries and long-range order, detected by static structure factor peaks, e.g., CDW and SDW in t–J and extended Hubbard models (1001.33431102.3053Baguet et al., 2013).
Luttinger Liquids and Luther–Emery Liquids: Gapless 1D quantum liquids with power-law correlation decay; extended LE liquids with spin gaps and dominant pairing or molecular order (1012.40282409.09344).
Topological Phases: SPT states such as the Haldane phase in integer-spin chains, detected by nonlocal string order and robust edge states; transition boundaries marked by Gaussian criticality (1310.85861611.10050Okamoto et al., 2019 Tonegawa et al., 2015).
Ferrimagnetic and Partially Magnetized Phases: Spontaneous magnetization plateaux or continuous magnetization, often in mixed-spin or bond-alternating chains (Hida et al., 2016 Hida, 2023 Hida, 2013).

Order parameters are phase-specific: magnetization ( $m_\mathrm{AF}$ , $m_\mathrm{F}$ ), density-wave amplitudes ( $\Delta_{\rm CDW}$ , $\Delta_{\rm SDW}$ ), pairing amplitudes ( $\hat\Delta_s$ , $\hat\Delta_d$ , $\hat\Delta_{p_x}$ ), string order parameters, molecular (trionic) correlators.

3. Quantum Criticality and Nature of Phase Transitions

Phase boundaries in the ground-state phase diagram are governed by quantum critical points:

First-order (discontinuous): Level crossings in energy lead to abrupt changes in symmetry/order, e.g., phase separation between PM/AF/FM in SU(3) Hubbard (Rapp et al., 2011), QF/PF/TLL in diamond chains (Hida, 2023 Hida et al., 2016), domain boundaries in D(D₃) anyon chains (Finch et al., 2010).
Continuous (second-order, Gaussian, BKT, Ising): Universality classes are distinguished by critical exponents, scaling dimensions (e.g., 2D Ising for AFstN–Haldane boundary (Tonegawa et al., 2015), Gaussian lines for SPT transitions (1310.85861910.02456)), and emergent central charge ( $c=1$ TLL, $c=0$ gapped).
Crossover and Multicritical Point: Multiple phases coalesce at multicritical points, e.g., coexistence of p-, p′-, and d-wave superfluids in the 2D Hubbard model (Deng et al., 2014).

Topological transitions, protected by symmetry or nonlocal order, can be smooth (adiabatic) when connecting trivial SPT manifolds, or sharp when changing the projective representation of the symmetry group.

4. Dimensionality, Symmetry, and Topological Aspects

Dimensionality and symmetries fundamentally shape the diagram:

1D vs 2D: Luttinger liquid theory holds in 1D, with algebraic decay and scaling exponents, while higher-dimensional systems exhibit Fermi-surface instabilities and true long-range order (1408.20881012.4028).
SU(N) symmetry: Enhances richness: SU(3) t–J chains host trionic superfluids and molecular LE liquids (Zhang et al., 2024); SU(3) Hubbard supports orbital-selective Mott and color density waves (Rapp et al., 2011).
Topological invariants and edge states: Nonlocal string order and projective symmetry representations define SPT phases (Haldane, ID); mixed-diamond and bond-alternating chains provide explicit realization (Hida et al., 20161310.85862312.01630).
Frustration and chiral phases: Frustration, anisotropy, and alternation induce inversion phenomena, incommensurate order, or chiral phases with nonzero ground-state momentum (Tonegawa et al., 2015 Finch et al., 2010).

Level crossings, degeneracy, and topological charge sectors emerge naturally in systems with underlying non-Abelian symmetries (e.g., D(D₃) anyons (Finch et al., 2010)).

5. Case Studies from Quantum Models

RBM-optimized ground-state ansatz reveals characteristic Mott lobes and superfluid regions in the $(t/U,\mu/U)$ plane, with phase boundaries sharply determined via compressibility and equation of state. Limitations arise in approaching KT criticality and off-diagonal long-range order.

Two-Dimensional t–J and Hubbard Models (1001.33431408.2088)

Tensor-network calculations reveal regime-separated phases: CDW + SDW + triplet superconductivity, d+s-wave superconductivity, extended s-wave, and triplet–FM. Emergent BCS analysis maps precise p- and d-wave superfluid boundaries versus $n,U$ , with weak coupling eigenvalues $\lambda\ll1$ (Deng et al., 2014).

Mapping onto mixed-spin chains and leveraging conservation laws, phase diagrams display Haldane, LD, ID, SPT, trivial, and ferrimagnetic plateaux. Level spectroscopy and PRG techniques locate Gaussian, BKT, and Ising critical lines. Inversion phenomena are observed in anisotropic frustrated ladders (Tonegawa et al., 2015).

SU(3) Hubbard and t–J Models (1102.30532409.09344)

Gutzwiller variational approach finds PM, FM, AF2, AFMM, and CDW phases with extensive first-order transitions and phase separation. SU(3) t–J chains show LL, extended LE liquid (spin gap, trion quasi-long-range order), and phase separation; trionic correlators and scaling exponents are computed via DMRG (Zhang et al., 2024).

Supersolid, cluster-crystal, and reentrant superfluid phases mapped for ultrasoft and Gaussian-core potentials, employing mean-field and QMC. Coexistence regions (SC, SSC) signal solid-superfluid mixtures and supersolidity. Dipolar condensates with quantum fluctuations exhibit LDP–HDP transitions and droplets stabilized by LHY corrections.

6. Experimental Context and Limitations

Comparison with large-scale QMC (McBrian et al., 2019 Kroiss et al., 2016 Mendoza-Coto et al., 2020), cluster expansion, and perturbation theory affirms the reliability and scope of advanced numerical techniques. Experimentally accessible observables—compressibility, pairing, structure factor, photon-number/quadrature measurements (Ying et al., 2015)—permit direct mapping and identification of phases. Model-specific limitations stem from finite size, truncation, neglect of correlation energy, or simplified variational ansätze.

7. Outlook and Research Directions

The ground-state phase diagram remains a central framework for exploring and classifying quantum matter—its precision maps quantum criticality, SPT and topological orders, and the interplay between symmetry, dimensionality, and frustration. Ongoing research includes: optimization of neural-network quantum states, investigation of multi-mode and higher-spin generalizations, exploration of non-Abelian anyonic chains, and experimental realization in ultracold atoms, Rydberg arrays, material systems, and quantum simulators. The extension to dynamical phase diagrams, disorder, or systems with non-equilibrium driving and quantum heating remains an active domain.

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