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Transverse-field Ising model

The transverse-field Ising model (TFIM) is a paradigmatic quantum spin model consisting of spin-1/2 particles arranged on a lattice, where nearest-neighbor spins interact via Ising-type couplings and each spin is subjected to a uniform magnetic field applied transverse to the interaction axis.^[1] Its standard Hamiltonian takes the form H = -J \sum_{\langle i,j \rangle} \sigma_i^z \sigma_j^z - h \sum_i \sigma_i^x, with J > 0 denoting the ferromagnetic coupling strength between neighboring sites \langle i,j \rangle, h the transverse field strength, and \sigma^{x,z}_i the Pauli matrices acting on site i.^[1] This model captures the competition between quantum fluctuations induced by the transverse field and classical ordering favored by the interactions, making it a fundamental framework for studying quantum many-body phenomena.^[2] The TFIM traces its origins to the mid-20th century, building on the classical Ising model introduced by Ernst Ising in 1925 to describe ferromagnetism, but extended to the quantum regime to model systems like ferroelectric transitions in materials such as KDP crystals.^[1] A key milestone was the exact solution of the one-dimensional (1D) case in 1961 by Lieb, Schultz, and Mattis using the Jordan-Wigner fermionization transformation, which maps the spin operators to non-interacting fermions, allowing full diagonalization via Bogoliubov transformations. Pierre Pfeuty further elaborated this solution in 1970, deriving explicit expressions for the ground-state energy, magnetization, and correlation functions, revealing a zero-temperature quantum phase transition at h_c = J.^[3] In higher dimensions, the model lacks an exact solution but can be analyzed using renormalization group methods, mean-field approximations, or quantum Monte Carlo simulations, with connections to the (d+1)-dimensional classical Ising model via the Suzuki-Trotter decomposition.^[1]^[2] At its core, the TFIM exhibits a quantum phase transition (QPT) driven by quantum fluctuations at absolute zero temperature, separating a low-field ferromagnetic phase with long-range order (spontaneous magnetization along the z-axis for h < h_c) from a high-field paramagnetic phase where the transverse field aligns spins along the x-axis, suppressing order.^[1] In 1D, the transition is continuous with critical exponents \nu = 1 (correlation length) and z = 1 (dynamic), belonging to the 2D classical Ising universality class, while in higher dimensions it features mean-field behavior above the upper critical dimension d = 3.^[2] The model also displays rich non-equilibrium dynamics, such as dynamical QPTs during quenches across the critical point, Kibble-Zurek scaling of defect density n \sim \tau^{-1/2} (where \tau is the quench time), and entanglement growth near criticality.^[1] These properties arise from the non-conserved order parameter and the role of the transverse field in tuning quantum disorder.^[2] Beyond theory, the TFIM has profound applications in quantum information science, where entanglement measures like concurrence and negativity diverge at the QPT, enabling studies of quantum correlations and fidelity susceptibility as probes of criticality.^[1] It serves as a benchmark for quantum annealing algorithms in optimization problems, mimicking the transverse field as a quantum tunneling driver analogous to temperature in classical simulated annealing.^[1] Experimentally, the model is realized in diverse platforms, including dipolar spin systems like LiHoF₄ and CoNb₂O₆ crystals, ultracold atoms in optical lattices, trapped ions, and superconducting qubits, allowing direct observation of QPTs and quantum simulation of otherwise intractable Hamiltonians.^[1]

Definition and Basics

Hamiltonian

The one-dimensional transverse-field Ising model is mathematically formulated through its Hamiltonian

H = -J \sum_{i=1}^N \sigma_i^z \sigma_{i+1}^z - h \sum_{i=1}^N \sigma_i^x,

where \sigma_i^z and \sigma_i^x denote the Pauli matrices representing the z and x components of the spin operator at lattice site i, J > 0 is the ferromagnetic coupling constant for nearest-neighbor interactions, and h is the strength of the uniform transverse magnetic field applied along the x-direction. The first sum runs over the interactions along a linear chain of N sites, while the second sum accounts for the field at each site. This Hamiltonian describes a quantum spin-1/2 system and serves as the foundational equation for studying quantum phase transitions in the model. Periodic boundary conditions are typically imposed, such that \sigma_{N+1}^z = \sigma_1^z, to model a ring-like lattice geometry, although open boundary conditions (where the sum over interactions goes to N-1) are also considered in finite-size analyses. The formulation generalizes naturally to arbitrary N, with thermodynamic properties emerging in the limit N \to \infty. In the special case h = 0, the Hamiltonian coincides with that of the classical Ising model. The equation is expressed in natural units where \hbar = 1, rendering J and h dimensionful with units of energy (or frequency). To simplify studies, J is frequently normalized to 1, leaving the ratio h/J as the sole tunable parameter that governs the competition between ordering and disordering tendencies.

Physical Interpretation

The transverse-field Ising model describes a system of quantum spins-1/2 arranged on a lattice, featuring nearest-neighbor interactions that couple the spins along the z-axis and a uniform magnetic field applied transversely along the x-axis. This setup captures the quantum mechanical nature of spin degrees of freedom, where each spin can exist in a superposition of up and down states, unlike purely classical models. The model is particularly relevant for one-dimensional chains but extends to higher dimensions, providing insights into quantum many-body phenomena.^[4] In the Hamiltonian, the term proportional to the coupling constant J represents the ferromagnetic exchange interaction between neighboring spins, which energetically favors their alignment parallel to the z-direction to minimize the overall energy. Conversely, the transverse field term, scaled by the field strength h, introduces quantum dynamics by coupling the eigenstates of the z-interactions; specifically, it acts like an external magnetic field that flips spins via quantum tunneling, allowing transitions between aligned (up-up or down-down) and antialigned configurations without thermal activation. This tunneling effect arises because the transverse field mixes the localized spin states, generating coherent superpositions and quantum fluctuations that compete with the ordering tendency of the interactions.^[4]^[5] The model's significance lies in its role as a paradigmatic system for investigating quantum phase transitions, where varying the ratio h/J tunes the system across a critical point at zero temperature. At weak fields (h \ll J), quantum fluctuations are suppressed, yielding an ordered phase with long-range correlations akin to ferromagnetism. As the field strengthens, these fluctuations dominate, disordering the system into a paramagnetic phase. This transition is driven purely by quantum effects, offering a controlled framework to study critical phenomena such as scaling laws and universality classes in quantum many-body physics. Historically, the transverse-field Ising model was introduced by P. G. de Gennes in 1963 to model order-disorder transitions in hydrogen-bonded ferroelectric materials, interpreting the pseudospins as proton displacements in double-well potentials, with the transverse field accounting for tunneling between wells and the interactions reflecting electrostatic couplings. This application highlighted its utility in describing dielectric properties and collective excitations in insulators. The model gained renewed prominence in the 1970s following exact solutions in one dimension, establishing it as a cornerstone for quantum criticality studies in condensed matter systems, including heavy-fermion materials and quantum magnets.^[6]^[7]

Relation to Classical Ising Model

Classical Ising Model

The classical Ising model is a foundational statistical mechanics model introduced by Wilhelm Lenz in 1920 to describe ferromagnetic phenomena in solids through a simplified lattice of interacting magnetic moments.^[8] In this model, each site on a d-dimensional lattice hosts a classical spin variable s_i = \pm 1, representing the two possible orientations of a microscopic magnetic dipole. The interactions favor alignment of neighboring spins, captured by the Hamiltonian

H = -J \sum_{\langle i,j \rangle} s_i s_j ,

where J > 0 denotes the ferromagnetic exchange coupling strength, and the sum runs over all pairs of nearest-neighbor sites \langle i,j \rangle. For the one-dimensional chain with N sites and periodic boundaries, this reduces to

H = -J \sum_{i=1}^N s_i s_{i+1},

with s_{N+1} = s_1.^[9] The model's equilibrium properties are obtained via the canonical partition function

Z = \sum_{\{s_i = \pm 1\}} \exp(-\beta H),

where \beta = 1/(k_B T), k_B is Boltzmann's constant, and T is the temperature; observables such as the free energy F = -k_B T \ln Z and magnetization follow from derivatives of \ln Z.^[9] Ernst Ising provided the exact solution for the one-dimensional case in his 1925 doctoral thesis, using a recursive method to compute Z and related quantities. This solution reveals the absence of spontaneous magnetization m = \frac{1}{N} \left\langle \sum_i s_i \right\rangle = 0 at any finite temperature in zero external field, indicating no phase transition to a ferromagnetically ordered state for T > 0. The two-point spin correlation function \left\langle s_i s_{i+r} \right\rangle decays exponentially as (\tanh \beta J)^r for large separation r, yielding a correlation length \xi = -1 / \ln (\tanh \beta J) that remains finite at all T > 0 but grows as \xi \sim \exp(2J/T) in the low-temperature limit.^[9] In one dimension, thermal fluctuations destroy long-range order at any finite temperature, with ferromagnetic alignment persisting only at T = [0](/page/0). Higher-dimensional variants exhibit richer behavior, including finite-temperature phase transitions; for instance, Lars Onsager's exact solution of the two-dimensional square-lattice model in 1944 establishes a critical point separating disordered and ordered phases at k_B T_c = 2J / \ln(1 + \sqrt{2}) \approx 2.269 J, below which spontaneous magnetization emerges.^[10]

Transverse Field Extension

The transverse-field Ising model represents a quantum extension of the classical Ising model, where a uniform transverse magnetic field is applied perpendicular to the direction of spin-spin interactions. This addition transforms the purely classical spin system into a quantum one, enabling the study of quantum fluctuations and coherence effects that are absent in the classical counterpart. Applied by de Gennes in 1963 to model order-disorder transitions in hydrogen-bonded ferroelectrics such as KDP crystals, the model captures essential quantum phenomena through the interplay between interaction strength J and field strength h.^[7]^[11] Unlike the classical model, which relies on scalar spin variables, here the spins are represented by Pauli matrices, with the transverse field coupling to the \sigma^x components. In the classical limit, setting h = 0 recovers the familiar Ising Hamiltonian of nearest-neighbor interactions, but with quantum spins whose \sigma^z eigenvalues remain \pm 1, preserving the ground-state degeneracy characteristic of the ferromagnet below the critical temperature. However, the transverse field h introduces non-commuting terms, such as [\sigma^z_i \sigma^z_{i+1}, \sigma^x_j] \neq 0, which generate quantum tunneling between spin configurations and induce coherent dynamics. This non-commutativity fundamentally enhances the model with quantum effects, leading to zero-temperature phase transitions driven by quantum fluctuations rather than thermal disorder. As detailed in early exact treatments, these features make the model a paradigmatic example for understanding quantum criticality.^[7] The behavior of the transverse-field Ising model varies significantly with the ratio h/J. For h \ll J, the interaction term dominates, and the system approximates a classical ferromagnet with long-range order in the [z](/page/Z)-direction, where quantum effects are perturbative. Conversely, for h \gg J, the transverse field overwhelms the interactions, aligning spins along the x-direction and yielding a quantum paramagnetic state with gapped excitations and no spontaneous symmetry breaking. This tunable competition between ordering and disordering highlights the quantum nature of the model. The TFIM in d dimensions maps to the (d+1)-dimensional classical Ising model via the Suzuki-Trotter decomposition, linking quantum and classical critical behaviors.^[7]^[2] A key distinction from the classical Ising model lies in the type of phase transitions: the classical version undergoes thermal transitions at finite temperature due to entropy, whereas the transverse-field extension features quantum critical points at absolute zero, controlled by varying h/J. These quantum transitions arise from the competition between the commuting interaction terms and the non-commuting field, resulting in universal scaling behaviors that map onto classical criticality in higher dimensions. This quantum-to-classical correspondence underscores the model's importance in theoretical physics.^[7]

Phases in One Dimension

Ordered Phase

In the one-dimensional transverse-field Ising model, the ordered phase emerges in the regime where the transverse field strength h is less than the ferromagnetic coupling J, specifically for h < h_c = J. Here, the ground state displays long-range ferromagnetic order, with a non-zero average spin alignment \langle \sigma^z_i \rangle \neq 0 along the interaction direction, breaking the \mathbb{Z}_2 symmetry of the Hamiltonian H = -J \sum_i \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma^x_i. This phase is characterized by dominant Ising interactions that favor coherent spin alignment over quantum delocalization induced by the transverse field.^[12]90115-4) The ground state in this regime is doubly degenerate in the thermodynamic limit, consisting of two configurations where all spins are predominantly aligned either up or down along the z-direction. The transverse field perturbs these classical-like states by introducing small admixtures of flipped spins, effectively tilting the local moments slightly away from perfect z-alignment. Quantum tunneling between the two degenerate states is present but strongly suppressed by the energetic cost of domain walls in the ordered background, ensuring the stability of the long-range order. This behavior mirrors the low-temperature ferromagnetic phase of the classical Ising model, where thermal fluctuations are replaced by quantum ones.^[12]90115-4) Excitations above the ground state take the form of gapped magnons, representing collective spin-wave modes or propagating domain walls. The excitation spectrum features a finite energy gap that protects the ordered state from low-energy perturbations, with the gap closing only at the critical point h = J. These magnons arise from the exact diagonalization via Jordan-Wigner fermions, where the ordered phase corresponds to a filled Fermi sea with a gapped single-particle spectrum.^[12]90270-8) The primary order parameter quantifying the ferromagnetic order is the spontaneous magnetization m = \lim_{h_z \to 0^+} \langle \sigma^z_i \rangle, obtained by applying an infinitesimal longitudinal field h_z along the z-direction to select one of the degenerate states. In the ordered phase, m = \left[1 - \left(h/J\right)^2\right]^{1/8}, which approaches zero continuously as h \to J from below, signaling the quantum phase transition to the disordered phase. This exact expression, derived from the model's solution, highlights the essential role of quantum fluctuations in reducing the magnetization from its classical value of 1.^[12]90270-8)

Disordered Phase

In the disordered phase of the one-dimensional transverse-field Ising model, which occurs for transverse field strengths h > J, the ground state is unique and paramagnetic, consisting of all spins aligned along the positive x-direction in the limit of large h/J.90270-8) The expectation value of the transverse magnetization approaches \langle \sigma^x \rangle \approx 1, while the longitudinal magnetization vanishes as \langle \sigma^z \rangle = 0.90270-8) Excitations above the ground state are gapped, primarily involving single-spin flips with an energy cost of approximately $2(h - J), which renders the system akin to non-interacting paramagnets.90270-8) The two-point correlation functions in the [z](/page/Z)-direction, \langle \sigma^z_i \sigma^z_j \rangle, exhibit short-ranged exponential decay with distance |i - j|, governed by a finite correlation length \xi \sim 1 / \ln(h/J).90270-8) This absence of long-range order arises from the dominant quantum fluctuations driven by the transverse field, which fully disrupt any potential alignment in the [z](/page/Z)-direction.90270-8)

Critical Point

The quantum critical point in the one-dimensional transverse-field Ising model occurs at the transverse field strength h_c = J, where J is the ferromagnetic coupling constant, marking the boundary between the ordered and disordered phases at zero temperature.^[13] At this point, the energy gap \Delta closes, with the excitation spectrum exhibiting \Delta \sim |h - h_c|^{\nu z} and critical exponents \nu = 1, z = 1. This behavior places the transition in the universality class of the two-dimensional classical Ising model. In the ground state at criticality, the system is gapless, characterized by power-law decaying correlations of the order parameter \langle \sigma^z_i \sigma^z_j \rangle \sim 1/|i - j|^{1/4}. The low-energy excitations are described by non-interacting gapless fermions obtained via the Jordan-Wigner transformation, featuring linear dispersion near the momentum where the gap closes.^[13] The quantum phase transition is driven by the ratio h/J [at

T](/page/AT&T) = 0

, separating the spontaneously broken symmetry phase for h < h_c from the paramagnetic phase for h > h_c.^[13] Near the critical point, the specific heat exhibits logarithmic divergences, reflecting the singular nature of the transition.

Exact Solutions

Jordan-Wigner Transformation

The Jordan-Wigner transformation provides an exact mapping of the one-dimensional transverse-field Ising model to a system of non-interacting fermions, enabling its complete solvability, as solved by Lieb, Schultz, and Mattis in 1961.^[14] Introduced originally for representing spin operators in terms of fermionic creation and annihilation operators, the transformation is defined for the Pauli spin operators at site i as

\sigma^z_i = 1 - 2 c_i^\dagger c_i, \quad \sigma^x_i = (c_i^\dagger + c_i) \prod_{j < i} (1 - 2 c_j^\dagger c_j),

where c_i^\dagger and c_i are fermionic operators satisfying the anticommutation relations \{c_i, c_j^\dagger\} = \delta_{ij}, \{c_i, c_j\} = \{c_i^\dagger, c_j^\dagger\} = 0. The product in \sigma^x_i introduces a nonlocal "string" operator, akin to a Majorana fermion representation, which ensures the correct fermionic statistics for the spins. Substituting these into the Hamiltonian H = -J \sum_i \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma^x_i (with J > 0 the ferromagnetic coupling and h > 0 the transverse field strength) yields a quadratic form in the fermion operators:

H = -\frac{J N}{2} + \sum_i \left[ (h - J) (2 c_i^\dagger c_i - 1) + J (c_i^\dagger - c_i)(c_{i+1}^\dagger + c_{i+1}) \right],

up to boundary terms that vanish in the thermodynamic limit N \to \infty. The bilinear terms, including pairing contributions like c_i^\dagger c_{i+1}^\dagger + \text{h.c.}, arise from the cancellation of strings between adjacent sites in the \sigma^z_i \sigma^z_{i+1} interaction, rendering the model free-fermionic.^[14] To diagonalize this quadratic Hamiltonian, one first performs a Fourier transform to momentum space:

c_k = \frac{1}{\sqrt{N}} \sum_j e^{-i k j} c_j, \quad k = \frac{2\pi m}{N}, \quad m = 0, 1, \dots, N-1.

This yields H = \sum_k \begin{pmatrix} c_k^\dagger & c_{-k} \end{pmatrix} \begin{pmatrix} \epsilon & \delta_k \\ \delta_k & -\epsilon \end{pmatrix} \begin{pmatrix} c_k \\ c_{-k}^\dagger \end{pmatrix} + \text{constant}, where \epsilon = h - J \cos k and \delta_k = -J \sin k. The Bogoliubov transformation then decouples the modes via new quasiparticle operators \gamma_k = u_k c_k + v_k c_{-k}^\dagger, with u_k^2 + v_k^2 = 1 and coefficients chosen to satisfy u_k^2 = \frac{1}{2} (1 + \frac{\epsilon}{\Lambda_k}), v_k^2 = \frac{1}{2} (1 - \frac{\epsilon}{\Lambda_k}), where \Lambda_k = \sqrt{\epsilon^2 + \delta_k^2}. This results in a diagonal form H = \sum_k E_k (\gamma_k^\dagger \gamma_k - 1/2), with the quasiparticle dispersion relation

E_k = 2 \sqrt{(h - J \cos k)^2 + (J \sin k)^2} = 2J \sqrt{1 + \left(\frac{h}{J}\right)^2 - 2 \frac{h}{J} \cos k}.

The excitation gap closes at the critical point h = J, specifically at k = 0.^[3] This fermionic representation allows computation of the exact ground-state energy E_0 = - \frac{1}{2\pi} \int_0^\pi dk \, E_k and correlation functions, such as the spontaneous longitudinal magnetization \langle |\sigma^z_i| \rangle = \left(1 - \left(\frac{h}{J}\right)^2\right)^{1/8} for h < J, via Wick's theorem on the fermionic vacuum. The transverse magnetization \langle \sigma^x_i \rangle is given by an elliptic integral but approximates to \frac{h}{\sqrt{h^2 + J^2}} for h \gg J. The solution highlights the model's quantum criticality, with the spectrum determining critical exponents like the correlation length exponent \nu = 1.^[3]

Kramers-Wannier Duality

The Kramers-Wannier duality provides a powerful symmetry in the one-dimensional transverse-field Ising model, mapping the Hamiltonian to an equivalent form that interchanges the roles of the interaction strength J and the transverse field h. This self-duality reveals the equivalence between the low-field ferromagnetic regime and the high-field paramagnetic regime without requiring the full diagonalization of the model. Pierre Pfeuty further elaborated on this in 1970.^[3] The duality transformation redefines the spin operators in terms of dual variables residing on the bonds of the original lattice. Specifically, the dual operators are given by

\tau^x_j = \sigma^z_j \sigma^z_{j+1}, \quad \tau^z_j = \prod_{i \leq j} \sigma^x_i,

where \sigma^\alpha_j (\alpha = x, z) are the Pauli operators at site j. Under this mapping, the original Hamiltonian H = -J \sum_j \sigma^z_j \sigma^z_{j+1} - h \sum_j \sigma^x_j transforms into the dual form \tilde{H} = -h \sum_j \tau^z_j \tau^z_{j+1} - J \sum_j \tau^x_j, which has the same structure but with J and h swapped.^[15] At the self-dual point where h = J, the model is invariant under this transformation, identifying this coupling as the critical point that separates the two phases. The exact location of this quantum critical point was confirmed through the fermionic solution of the model.^[3] The duality implies a deep connection between the phases: the ordered phase (h < J), characterized by spontaneous magnetization along the z-direction, maps to the disordered phase (h > J), where the transverse magnetization dominates; conversely, the disorder parameter (string order) in one description becomes the order parameter in the dual. This mapping underscores the absence of a finite-temperature transition in one dimension while highlighting the quantum phase transition at zero temperature.^[15] This quantum duality extends the original Kramers-Wannier duality formulated for the two-dimensional classical Ising model, leveraging the isomorphism between the 1D quantum chain and the anisotropic 2D classical lattice.^[16]

Generalizations

Higher Dimensions

The transverse-field Ising model in higher dimensions d > 1 extends the one-dimensional formulation to lattices with increased connectivity, such as the d-dimensional hypercubic lattice, where nearest-neighbor interactions lead to richer phase behavior without exact solvability. The Hamiltonian is

H = -J \sum_{\langle i,j \rangle} \sigma_i^z \sigma_j^z - h \sum_i \sigma_i^x,

with J > 0 favoring ferromagnetic alignment along the z-direction and h introducing quantum fluctuations via the transverse field.^[17] At zero temperature, the model exhibits a quantum phase transition at a critical field strength h_c that increases with dimension d due to enhanced ordering tendencies from more neighbors. For h < h_c, the ground state is in a magnetically ordered phase characterized by spontaneous breaking of the \mathbb{Z}_2 symmetry and long-range correlations in the z-component of spins. For h > h_c, the system enters a quantum disordered phase with gapped excitations and exponentially decaying correlations. In dimensions d \geq 2, the ordered phase further supports finite-temperature transitions to a classically disordered state, governed by thermal fluctuations, while the disordered phase remains paramagnetic at all finite temperatures.^[17] Mean-field theory approximates the critical behavior by treating spin interactions in a self-consistent manner, yielding h_c \approx z J where z = 2d is the lattice coordination number. This approach predicts classical critical exponents, including \beta = 1/2 for the magnetization discontinuity and \nu = 1/2 for the divergence of the correlation length, which become accurate above the upper critical dimension d = 3.^[17] Absent exact solutions for generic d > 1, the model is analyzed using numerical and approximate techniques. Quantum Monte Carlo methods, leveraging the world-line formalism and absence of the sign problem on bipartite lattices, enable efficient simulations of ground-state properties and dynamics up to large system sizes in two and three dimensions. Tensor network approaches, such as infinite projected entangled pair states (iPEPS), provide variational approximations for two-dimensional ground states and entanglement properties near criticality. While exact solvability holds for specialized two-dimensional lattices or limits, the standard hypercubic cases rely on perturbative expansions and renormalization group analyses for deeper insights. The zero-temperature quantum critical point maps to the universality class of the (d+1)-dimensional classical Ising model through the quantum-to-classical correspondence, where imaginary time introduces an extra dimension; this implies dynamic exponent z=1 and Ising-like hyperscaling in d < 3.^[17]

Other Variants

The transverse-field Ising model with long-range interactions features couplings that decay as a power law, J(r) \sim 1/r^{1+\sigma}, where \sigma > 0 controls the range.^[18] For \sigma < 1, the quantum critical point exhibits mean-field behavior due to the dominance of long-range effects, while for $1 < \sigma < 2, non-classical exponents emerge, altering the universality class from the short-range case.^[19] These models are analytically tractable in one dimension via extensions of the Jordan-Wigner transformation and numerically studied using quantum Monte Carlo methods.^[20] Variants with anisotropic or staggered transverse fields introduce spatial inhomogeneity, breaking translation symmetry and leading to distinct quantum phase transitions.^[21] In the presence of staggered Dzyaloshinskii-Moriya interactions, the model displays quantum criticality belonging to the Ising universality class but with modified entanglement scaling and correlation lengths.^[22] Such configurations can induce chiral ordering or gapless excitations, shifting the system away from standard ferromagnetic or paramagnetic phases.^[23] The quantum impurity version of the transverse-field Ising model describes a single spin coupled to a fermionic bath via Ising interactions, often mapped to Kondo-like problems.^[24] This setup is solvable using the numerical renormalization group (NRG) method, revealing a quantum phase transition between screened and unscreened impurity states analogous to Kondo physics.^[25] The transverse field acts as a competing energy scale, driving critical behavior with non-Fermi-liquid properties near the transition.^[26] Post-2000 developments have extended the model to open quantum systems incorporating dissipation, such as local dephasing or spontaneous emission, which stabilize mixed steady states and induce dissipative phase transitions.^[27] For instance, boundary-dissipated chains exhibit exact solutions for their density matrices, revealing competition between coherent transverse fields and incoherent losses.^[28] Connections to quantum annealing leverage the model as a benchmark for adiabatic optimization, where time-dependent transverse fields simulate ground-state searches in glassy systems.^[29] In holographic contexts, the critical transverse-field Ising model maps to AdS/CFT dualities, with Gibbs states encoding bulk geometries via quantum error-correcting codes.^[30] Experimental realizations of the transverse-field Ising model have been achieved using trapped ions, where laser-induced couplings simulate spin interactions and magnetic fields.^[31] Systems with up to 300 ions demonstrate quantum phase transitions and advanced Hamiltonian learning in long-range variants as of 2025.^[32] Superconducting qubit arrays, such as those in transmon-based processors, implement the model through flux-tunable Josephson junctions, enabling simulations of dynamical phase transitions on scales of 53 qubits.^[33]

References

[1]
None
Below is a merged summary of the Transverse-Field Ising Model (TFIM) that consolidates all information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a combination of narrative text and a table to organize key information efficiently. The response retains all unique details mentioned across the summaries, avoiding redundancy where possible.
[2]
https://doi.org/10.1103/RevModPhys.77.259
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[PDF] The One-Dimensional king Model with a Transverse Field The Ising ...
The one-dimensional Ising model with a transverse field is solved exactly by trans- forming the set of Pauli operators to a new set of Fermi operators.
[4]
[PDF] Tutorial 3: The Transverse Ising Model
The transverse Ising model is a quantum spin system where each spin is a qubit, with a quantum state of an assembly of qubits along a one-dimensional chain.
[5]
Ising model in a transverse field. I. Basic theory - IOPscience
A brief review is first made of systems for which the spin-1/2 Ising model in a transverse field provides a useful description.
[6]
Collective motions of hydrogen bonds - ScienceDirect.com
Volume 1, Issue 6, November 1963, Pages 132-137. Solid State Communications. Collective motions of hydrogen bonds. Author links open overlay panel P.G. de ...
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The one-dimensional Ising model with a transverse field
The one-dimensional Ising model with a transverse field is solved exactly by transforming the set of Pauli operators to a new set of Fermi operators.
[8]
[PDF] Beitrag zum Verständnis der magnetischen Erscheinungen in festen ...
XXI, 1920. Lenz, Magnetische Erscheinungen in festen Körpern. sich je nach Art des stofflichen Feldes um eine jeweils bestimmte Wechselwirkung zwischen.
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Beitrag zur Theorie des Ferromagnetismus | Zeitschrift für Physik A ...
Volume 31, pages 253–258, (1925); Cite this article. Download PDF · Zeitschrift für Physik. Beitrag zur Theorie des Ferromagnetismus. Download PDF. Ernst Ising.
[10]
Crystal Statistics. I. A Two-Dimensional Model with an Order ...
65, 117 – Published 1 February, 1944 ... Abstract. The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed ...
[11]
[PDF] Quantum Ising Models - Arun Paramekanti - University of Toronto
Dec 9, 2018 · The transverse-field Ising model, parameterized as in Equation 1 has this quantum phase ... Lieb, T. D. Schultz, and D. C. Mattis. Two ...
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[13]
[PDF] Statistics of the Two-Dimensional Ferromagnet. Part I
H. A. KRAMERS AND G. H. WANNIER. Because of (32) Eq. (34) tells us that if the energy is continuous at the Curie point it must have the value. (36) which is a ...
[14]
https://doi.org/10.1103/AnnPhys.16.407
[15]
Quantum-critical properties of the long-range transverse-field Ising ...
Jun 23, 2021 · The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series ...
[16]
PRX Quantum 4, 010302 (2023) - Probing Critical Behavior of Long ...
Jan 5, 2023 · Here we report the quantum simulation of a long-range transverse-field Ising model using up to 61 ions and probe the critical behavior of its quantum phase ...Missing: criticality | Show results with:criticality
[17]
[PDF] Quantum-critical properties of the one-dimensional long-range ...
The zero-temperature criticality of the long-range transverse-field Ising model with al- gebraically decaying ferromagnetic and antiferromagnetic Ising ...
[18]
Quantum criticality in spin-1/2 anisotropic XY model with staggered ...
The quantum criticality and entanglement properties are investigated in the one-dimensional spin-1/2 transverse field Ising model with staggered Dzyaloshinskii ...
[19]
Characterizing quantum criticality in the transverse field Ising model ...
Aug 7, 2025 · Characterizing quantum criticality in the transverse field Ising model with staggered Dzyaloshinsky-Moriya interaction. December 2023; Journal ...
[20]
Solitonic excitations in the Ising anisotropic chain under large ...
Dec 29, 2021 · This transition is in the universality class of the transverse field Ising model [19] as identified from the dual-field double-sine Gordon model
[21]
Quantum phase transition of Ising-coupled Kondo impurities
Jun 18, 2004 · We investigate a model of two Kondo impurities coupled via an Ising interaction. Exploiting the mapping to a generalized single-impurity ...
[22]
Numerical renormalization group method for quantum impurity ...
Apr 2, 2008 · As a specific example, consider the Kondo model, which describes a magnetic impurity with spin S ⃗ coupled to the electrons of a conduction band ...
[23]
Quantum critical Kondo destruction in the Bose-Fermi Kondo model ...
Jul 15, 2013 · For couplings that localize the impurity spin in the absence of a transverse field, increasing such a field eventually causes a continuous, ...
[24]
Exact solution of the boundary-dissipated transverse field Ising model
In this work, we shall present an exact solution to a transverse field Ising ... Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain ...
[25]
[PDF] Phase diagram of the dissipative quantum Ising model on a square ...
Dec 14, 2018 · The competition between interactions and dissipative processes in a quantum many-body system can drive phase transitions of different order.
[26]
Error measurements for a quantum annealer using the one ... - Nature
Jun 27, 2022 · To harness the power of quantum annealing, machines must be constructed to faithfully implement the relevant transverse field Ising Hamiltonian ...<|control11|><|separator|>
[27]
[1611.07528] Towards holography via quantum source-channel codes
Nov 22, 2016 · As an example, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they exhibit non-trivial ...
[28]
Quantum simulation of the transverse Ising model with trapped ions
Oct 14, 2011 · The Ising model with a transverse field is described by the following Hamiltonian: Equation (1). where σ(i)α, α = x,y,z, are the Pauli ...
[29]
Simulating groundstate and dynamical quantum phase transitions ...
Oct 10, 2022 · We optimise a translationally invariant, sequential quantum circuit on a superconducting quantum device to simulate the groundstate of the quantum Ising model.