The one-dimensional Ising model with a transverse field
TL;DR: In this paper, 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.
About: This article is published in Annals of Physics.The article was published on 1970-03-01. It has received 1266 citations till now. The article focuses on the topics: Square-lattice Ising model & Ising model.
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TL;DR: In this article, the properties of entanglement in many-body systems are reviewed and both bipartite and multipartite entanglements are considered, and the zero and finite temperature properties of entangled states in interacting spin, fermion and boson model systems are discussed.
Abstract: Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.
3,096 citations
TL;DR: It is demonstrated, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point, which connects the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point.
Abstract: Classical phase transitions occur when a physical system reaches a state below a critical temperature characterized by macroscopic order. Quantum phase transitions occur at absolute zero; they are induced by the change of an external parameter or coupling constant, and are driven by quantum fluctuations. Examples include transitions in quantum Hall systems, localization in Si-MOSFETs (metal oxide silicon field-effect transistors; ref. 4) and the superconductor-insulator transition in two-dimensional systems. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart. This phenomenon, known as entanglement, is the resource that enables quantum computation and communication. The role of entanglement at a phase transition is not captured by statistical mechanics-a complete classification of the critical many-body state requires the introduction of concepts from quantum information theory. Here we connect the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point. We demonstrate, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point.
1,571 citations
Additional excerpts
...At the phase transition the correlation length ξ diverges as ξ ∼v λ - λ c v - ν with ν e1 (ref...
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TL;DR: In this article, the authors review recent developments in the physics of ultracold atomic and molecular gases in optical lattices and show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics.
Abstract: We review recent developments in the physics of ultracold atomic and molecular gases in optical lattices. Such systems are nearly perfect realisations of various kinds of Hubbard models, and as such may very well serve to mimic condensed matter phenomena. We show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics. After a short presentation of the models and the methods of treatment of such systems, we discuss in detail, which challenges of condensed matter physics can be addressed with (i) disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv) lattice gases in “artificial” magnetic fields, and, last but not least, (v) quantum information processing in lattice gases. For completeness, also some recent progress related to the above topics with trapped cold gases will be discussed. Motto: There are more things in heaven and earth, Horatio, Than are dreamt of in your...
1,535 citations
TL;DR: In this paper, the authors studied the entanglement in the transverse Ising model, a special case of the one-dimensional infinite-lattice anisotropic XY model, which exhibits a quantum phase transition.
Abstract: What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example of such a system is the one-dimensional infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest-neighbor entanglement (though not the nearest-neighbor entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the remainder of the lattice.
1,274 citations
TL;DR: In this article, the spin correlation functions of the one-dimensional $XY$ model were studied in the presence of a constant magnetic field and it was shown that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.
Abstract: The spin-correlation functions of the one-dimensional $XY$ model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.
679 citations
References
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IBM1
TL;DR: In this article, two genuinely quantum models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found.
3,382 citations
TL;DR: In this article, the low frequency collective modes of protons are calculated for ferrroelectric crystals of the KH 2 PO 4 type taking into account the tunneling frequency and a simplified interaction between proton sites.
550 citations
TL;DR: In this article, the statistical mechanics of an infinite one-dimensional classical lattice gas were studied and it was shown that for a large class of interactions, such a system has no phase transition and the equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations.
Abstract: We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be aK-system.
497 citations
TL;DR: In this article, the authors considered the Ising model on a half-plane of infinite extent and studied some of the consequences connected with the presence of the boundary, including the thermodynamic properties associated with the boundary.
Abstract: We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field $\mathfrak{H}$. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of $\mathfrak{H}$ and proportional to the total number of lattice sites, while the other depends on $\mathfrak{H}$ and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature ${T}_{c}$. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For $T={T}_{c}$, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small $\mathfrak{H}$. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at $T={T}_{c}$, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for $Tl{T}_{c}$, may be analytically continued beyond the point $\mathfrak{H}=0$. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.
334 citations