Mixed-Field Ising Model

• Mixed-field Ising model is a framework describing Ising systems influenced by both transverse and longitudinal fields, capturing diverse quantum phenomena.
• It serves as a benchmark for studying quantum phase transitions and thermalization, with distinct regimes ranging from solvable to mixed-order criticality.
• Its varied formulations enable precise exploration of phase structure and dynamics, guiding experimental validations and theoretical advancements.

The mixed-field Ising model denotes an Ising system in which more than one field contribution acts on the spins. In the standard quantum usage, it is the Ising chain or lattice with both transverse and longitudinal fields, for example

$$\hat{H} = \sum_{i=1}^L J \sigma_i^z \sigma_{i+1}^z -\sum_{i=1}^L \Gamma \sigma_i^x -h\sum_{i=1}^L \sigma_i^z,$$

or, in spin-1 form,

$$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$

In adjacent literatures, the label is also used for Ising systems with an external field of mixed signs, and for mixed-spin ferrimagnets studied in simultaneous longitudinal magnetic and crystal fields. The resulting terminology is not uniform, and the distinction between mixed-field, mixed-spin, crystal-field, and mixed-state usages is therefore part of the subject itself (Lajkó et al., 2020, Aizenman, 2 Sep 2025, Karimou et al., 2016).

1. Terminology and scope

In the literature surveyed here, “mixed-field Ising model” has more than one established meaning. The dominant quantum meaning is an Ising model with simultaneous transverse and longitudinal fields. A second rigorous usage concerns an external field of mixed signs, treated geometrically through the ghost-spin trick and frustration-adjusted random current methods. A broader statistical-mechanics usage applies the phrase to mixed-spin ferrimagnets in the presence of both a longitudinal magnetic field and a crystal-field term (Lajkó et al., 2020, Aizenman, 2 Sep 2025, Eddahri et al., 2018).

Usage	Representative form	Representative papers
Quantum mixed-field Ising model	$J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$	(Lajkó et al., 2020, Wurtz et al., 2020, Vallejo-Fabila et al., 5 Jan 2026)
External field of mixed signs	$- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$	(Aizenman, 2 Sep 2025)
Mixed-spin system with magnetic and crystal fields	$-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$	(Eddahri et al., 2018, Karimou et al., 2016)

This terminological spread is important because several nearby expressions are not equivalent. A mixed-spin Ising model refers to different spin magnitudes on different sublattices; a crystal-field model introduces a single-ion term such as $D\sum_i S_i^2$; and “mixed states” in the Ising field-theory literature refer to density matrices rather than multiple external fields (Ertaş et al., 2014, Chen et al., 2013). A related but distinct construction is the feedback Ising model, where a state-dependent coupling induces a nonlinear internal field; it is explicitly described as not being a conventional mixed-field Ising model (Ma et al., 10 Jun 2025).

2. Canonical quantum formulations

A central formulation is the one-dimensional mixed-field Ising chain

$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$

used as a benchmark nonintegrable model with both transverse and longitudinal fields (Wurtz et al., 2020). Three simple limits are emphasized. At $h_z=0$, the model reduces to the transverse-field Ising chain and is solvable by a Jordan-Wigner transformation to free fermions. At $h_x=0$, it is classical in the $\sigma^z$ basis. In the limit $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$0, it becomes a set of decoupled spins. The point $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$1 is identified as a first-order multicritical point, and for small $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$2 near that regime the low-energy effective Hamiltonian is the PXP model (Wurtz et al., 2020).

The antiferromagnetic convention is written as

$$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$3

with a critical line separating an AFM phase from a field-polarized FM phase for finite $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$4 (Lajkó et al., 2020). A staggered gauge transformation,

$$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$5

followed by an $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$6 rotation maps the AFM chain to a ferromagnetic-looking Ising coupling with a staggered field term,

$$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$7

which makes explicit that the AFM mixed-field chain is not trivially equivalent to the standard uniform ferromagnetic mixed-field chain (Lajkó et al., 2020).

Beyond one dimension, a square-lattice mixed-field Ising model with a sign-inverted next-nearest-neighbor interaction is defined by

$$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$8

Here $$H= -J \sum_{j=1}^{L-1} S_j^z S_{j+1}^z -g\sum_{j=1}^L S_j^z - h\sum_{j=1}^L S_j^x.$$9 is antiferromagnetic and $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$0 enters with opposite sign, so the NN coupling is AFM and the NNN coupling is FM (Nakamura et al., 2024). The same paper gives a Rydberg-atom route to implementing the sign-inverted NNN term by weakly coupling one Rydberg state to another (Nakamura et al., 2024).

A further generalization is the isolated quantum and classical one-dimensional spin-1 Ising model with transverse and longitudinal fields,

$J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$1

studied with open boundary conditions and with $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$2 throughout the analysis (Vallejo-Fabila et al., 5 Jan 2026).

3. Phase structure and critical behavior

The one-dimensional antiferromagnetic mixed-field chain exhibits a critical line that connects the TIM critical point at $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$3 to the classical first-order endpoint at $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$4 (Lajkó et al., 2020). The paper’s central claim is that for any finite ratio

$J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$5

the AFM–FM transition is mixed-order. The gap, entanglement entropy, and correlation length scaling remain TIM-like; the entanglement data are consistent with $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$6, and the critical exponents extracted from the ordered and disordered sides are $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$7 and $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$8. At the same time, the bulk correlation function has a finite jump at the transition and the end-to-end correlation function has a discontinuous derivative (Lajkó et al., 2020). In this sense the transition is neither purely second-order nor purely first-order.

In the square-lattice model with sign-inverted NNN coupling, the AFM–PM boundary changes order. For $J\sigma^z\sigma^z-\Gamma\sigma^x-h\sigma^z$9, the AFM–PM transition is second order when $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$0 and first order when $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$1; the two regimes meet at a QTCP located at

$- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$2

The associated coarse-grained theory is a relativistic-type GL equation,

$- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$3

with coefficients $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$4 obtained from the microscopic model (Nakamura et al., 2024).

The most distinctive result in that two-dimensional setting is surface criticality at a first-order quantum transition. With a boundary condition $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$5, the order-parameter profile yields a healing length defined from the inflection point, and near coexistence

$- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$6

The logarithmic divergence of the healing length signals surface criticality even though the bulk transition is first order (Nakamura et al., 2024). The paper interprets this as a boundary-induced disordered layer that thickens as the coexistence line is approached (Nakamura et al., 2024).

4. Dynamics, thermalization, and nonthermal sectors

A substantial part of the mixed-field literature concerns nonequilibrium dynamics and weakly broken integrability. In the one-dimensional spin-$- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$7 mixed-field chain, approximate adiabatic gauge potentials are used to construct dressed quasiparticles, approximate eigenstates, and quasi-local almost-conserved operators (Wurtz et al., 2020). At the representative point $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$8, the rotated truncated spectrum approximation reproduces exact-diagonalization spectra on 18 sites with fidelities normally $- \sum J_{x,y}\sigma_x\sigma_y-\sum h_x\sigma_x$9 (Wurtz et al., 2020). In the FM convention $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$0, the longitudinal field acts as a constant attractive force between domain walls and produces confined bound states (“mesons”). In the AFM convention $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$1, the dressed quasiparticle analysis yields two species with masses

$-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$2

The same framework also constructs high-energy atypical states. The standout example is the dressed all-up state, which has half-chain entanglement entropy $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$3 bits, fidelity $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$4 with an exact eigenstate, and lifetime estimate $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$5, despite lying at finite energy density rather than near a spectral edge (Wurtz et al., 2020).

The same paper builds an approximate conserved operator by dressing the domain-wall-number observable

$-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$6

At $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$7, the dressed operator is quasi-local and its infinite-temperature decay time is improved by about a factor of 40 relative to the bare operator; the paper quotes $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$8 and $-J s_i\sigma_j-D\sigma_j^2-h(s_i+\sigma_j)$9 in the corresponding commutator diagnostic (Wurtz et al., 2020). The physical conclusion is that the mixed-field chain can retain robust approximate conservation laws even where the model is generally quantum chaotic (Wurtz et al., 2020).

Thermalization in the spin-1 mixed-field Ising chain is studied through the occupation number of the local magnetization sublevels $D\sum_i S_i^2$0,

$D\sum_i S_i^2$1

Thermalization is defined by convergence of the long-time average of $D\sum_i S_i^2$2 to the microcanonical prediction as $D\sum_i S_i^2$3 increases (Vallejo-Fabila et al., 5 Jan 2026). In the strongly chaotic regime

$D\sum_i S_i^2$4

the quantum model shows GOE-like level statistics, delocalized eigenstates, ETH-like behavior, and relaxation of occupation numbers toward the microcanonical expectation (Vallejo-Fabila et al., 5 Jan 2026). Because exact diagonalization is limited by the $D\sum_i S_i^2$5 Hilbert-space growth, the analysis turns to the classical counterpart and formulates ergodicity on the single-spin Bloch sphere through a $D\sum_i S_i^2$6 test against the uniform distribution $D\sum_i S_i^2$7. The deviation from the classical ergodic threshold decays algebraically as

$D\sum_i S_i^2$8

with fitted exponents $D\sum_i S_i^2$9 for $$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$0, $$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$1 for $$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$2, and $$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$3 for $$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$4 (Vallejo-Fabila et al., 5 Jan 2026). The paper interprets this power law as a lower bound on the rate at which the quantum model approaches thermal equilibrium (Vallejo-Fabila et al., 5 Jan 2026).

A broader nonequilibrium usage appears in the mixed spin-$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$5 Ising model under an oscillating field,

$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$6

where EFT with Glauber dynamics yields dynamic phase diagrams with paramagnetic, ferrimagnetic, and coexistence phases, dynamic tricritical behavior, multicritical and zero-temperature critical points, and strong frequency dependence (Ertaş et al., 2014). That work is best understood as a mixed-spin Ising model under a time-dependent external field rather than as a random-field or sign-mixed-field model (Ertaş et al., 2014).

5. Geometric and continuum frameworks

A rigorous geometric treatment of mixed-sign external fields is provided by the extension of the random current representation to systems with “a mild amount of frustration, such as generated by disorder operators and external field of mixed signs” (Aizenman, 2 Sep 2025). For the finite-volume Ising model on a graph,

$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$7

mixed-sign fields are handled by the Griffiths ghost-spin trick, which converts field terms into pair interactions on an enlarged graph (Aizenman, 2 Sep 2025). The resulting frustration-adjusted random-current formalism expresses the partition-function ratio as

$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$8

and the two-point function as a conditional expectation over frustration-free configurations (Aizenman, 2 Sep 2025). One concrete output is the Ding–Song–Sun inequality, written for a field decomposition

$$H = \sum_i^N J\sigma_z^i\sigma_z^{i+1} + h_x\sigma_x^i + h_z\sigma_z^i,$$9

as

$h_z=0$0

This places mixed-sign-field Ising systems within a positive geometric framework based on duplication, switching, and conditioning on the frustration-free sector (Aizenman, 2 Sep 2025).

At the continuum level, the GL theory derived for the square-lattice model with sign-inverted NNN interaction gives a microscopic-to-mesoscopic description of both statics and dynamics,

$h_z=0$1

and provides explicit formulas for the boundary profile, the bulk order parameter

$h_z=0$2

and the logarithmically diverging healing length near the first-order line (Nakamura et al., 2024). In this literature, the mixed-field Ising model is therefore studied not only as a lattice spin system but also as a field theory with boundary critical behavior (Nakamura et al., 2024).

6. Related model classes and recurrent confusions

A recurring source of ambiguity is the overlap between mixed-field and mixed-spin usages. On the Bethe lattice, the mixed spin-$h_z=0$3 and spin-$h_z=0$4 model in a longitudinal magnetic field is defined by

$h_z=0$5

and is explicitly described as a mixed-spin ferrimagnet with a uniform crystal field and a longitudinal magnetic field (Eddahri et al., 2018). A closely related Bethe-lattice study of the mixed spin-$h_z=0$6 and spin-$h_z=0$7 system calls the model “mixed-field” because it contains both a longitudinal magnetic field $h_z=0$8 and a crystal-field term $h_z=0$9, but its temperature phase diagrams are drawn mainly for $h_x=0$0 and it reports only second-order ferrimagnetic–paramagnetic transitions in the zero-field phase diagrams (Karimou et al., 2016). These papers use “mixed-field” in a broader statistical-mechanics sense than the transverse-plus-longitudinal quantum-chain literature (Eddahri et al., 2018, Karimou et al., 2016).

The expression must also be separated from “mixed states” of the Ising model. In the massive Ising quantum field theory, “mixed states” refers to density matrices,

$h_x=0$1

and the subject is mixed-state form factors of order and disorder fields in thermal, generalized Gibbs, and nonequilibrium steady states; it is explicitly not the lattice mixed-field Ising model with both $h_x=0$2 and $h_x=0$3 (Chen et al., 2013).

Another formal issue is that field parameters may be basis-dependent under binary recoding. For mixed spin-class Ising systems, exact nodewise affine transformations preserve the full-state probabilities while changing both pair couplings and field terms. In the paper’s notation,

$h_x=0$4

and the transformed field $h_x=0$5 contains both a direct rescaling and induced neighbor contributions (Kruis, 2020). This suggests that some apparent heterogeneity in “field” terms is representation-dependent rather than intrinsic.

Finally, the feedback Ising model replaces the constant Curie–Weiss coupling by a magnetization-dependent function $h_x=0$6,

$h_x=0$7

and generates intermediate mixed phases through a nonlinear self-field $h_x=0$8. The paper states explicitly that this is not a standard mixed-field Ising model with multiple externally imposed fields (Ma et al., 10 Jun 2025). The distinction is conceptually useful because it separates externally mixed fields from self-generated internal fields.

The mixed-field Ising model is therefore best regarded as a family of closely related constructions rather than a single universally standardized Hamiltonian. In the narrow quantum sense, it is the Ising model with simultaneous transverse and longitudinal fields and is a principal laboratory for mixed-order quantum criticality, weakly broken integrability, quasi-local approximate conservation laws, and thermalization. In broader usage, it includes Ising systems with external field of mixed signs and mixed-spin ferrimagnets in simultaneous magnetic and crystal fields. Much of the technical literature is devoted precisely to keeping these meanings distinct while exploiting the mathematical structures they share (Lajkó et al., 2020, Wurtz et al., 2020, Aizenman, 2 Sep 2025).