Spin-1 Antiferromagnetic Heisenberg Chain Research Articles

Enhancing quantum utility: Simulating large-scale quantum spin chains on superconducting quantum computers

We present the quantum simulation of the frustrated quantum spin-12 antiferromagnetic Heisenberg spin chain with competing nearest-neighbor (J1) and next-nearest-neighbor (J2) exchange interactions in the real superconducting quantum computer with qubits ranging up to 100. In particular, we implement the Hamiltonian with the next-nearest neighbor exchange interaction in conjunction with the nearest-neighbor interaction on IBM's superconducting quantum computer and carry out the time evolution of the spin chain by employing the first-order Trotterization. Furthermore, our implementation of the second-order Trotterization for the isotropic Heisenberg spin chain, involving only nearest-neighbor exchange interaction, enables precise measurement of the expectation values of staggered magnetization observable across a range of up to 100 qubits. Notably, in both cases, our approach results in a constant circuit depth in each Trotter step, independent of the number of qubits. Our demonstration of the accurate measurement of expectation values for the large-scale quantum system using superconducting quantum computers designates the quantum utility of these devices for investigating various properties of many-body quantum systems. This will be a stepping stone to achieving the quantum advantage over classical ones in simulating quantum systems before the fault tolerance quantum era. Published by the American Physical Society 2024

State space geometry of the spin-1 antiferromagnetic Heisenberg chain

We study the phase diagram of the spin-1 antiferromagnetic Heisenberg chain with uniaxial anisotropy and applied magnetic field in terms of the genuine multipartite entanglement as witnessed by the mean quantum Fisher information density. By considering the manifold generated by the staggered magnetization operator, we connect the state space curvature in the vicinity of the ground state of the Heisenberg chain to the genuine multipartite entanglement. Our analysis demonstrates that the quantum critical points and symmetry protected topological (SPT) phase exhibit large state space curvature, while the separable phases are completely flat, offering insight into the physical interpretation of state space curvature. We further show that the entanglement in the SPT phase is enhanced by the presence of uniaxial anisotropy and undiminished in the presence of uniform magnetic fields. The magnon condensate phase induced by large fields is shown to emanate from the Gaussian critical point and exhibits massive multipartite entanglement over a robust region of the parameter space.

Even-odd effects in the J1−J2 SU(N) Heisenberg spin chain

The zero-temperature phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin chain is investigated by means of complementary field theory and numerical approaches for general $N$. A fully gapped $\mathrm{SU}(N)$ valence bond solid made of $N$ sites is formed above a critical value of ${J}_{2}/{J}_{1}$ for all $N$. We find that the extension of this $N$-merized phase for larger values of ${J}_{2}$ strongly depends on the parity of $N$. For even $N$, the phase smoothly interpolates to the large ${J}_{2}$ regime where the model can be viewed as a zigzag $\mathrm{SU}(N)$ two-leg spin ladder. The phase exhibits both a $N$-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the $N$-merized phase with odd $N$ has only a finite extent with no incommensuration. A gapless phase in the $\mathrm{SU}{(N)}_{1}$ universality class is stabilized for larger ${J}_{2}$ that stems from the existence of a massless renormalization group flow from $\mathrm{SU}{(N)}_{2}$ to $\mathrm{SU}{(N)}_{1}$ conformal field theories when $N$ is odd.

Duality, criticality, anomaly, and topology in quantum spin-1 chains

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation ${U}_{\mathrm{KT}}$, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that ${U}_{\mathrm{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under ${U}_{\mathrm{KT}}$) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when $H$ is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model $H+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}$ is exactly equivalent to a gapless spin-$1/2$ XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.

Identifying phase transition point of <i>J</i><sub>1</sub>-<i>J</i><sub>2</sub> antiferromagnetic Heisenberg spin chain by machine learning

Studying quantum phase transitions through order parameters is a traditional method, but studying phase transitions by machine learning is a brand new field. The ability of machine learning to classify, identify, or interpret massive data sets may provide physicists with similar analyses of the exponentially large data sets embodied in the Hilbert space of quantum many-body system. In this work, we propose a method of using unsupervised learning algorithm of the Gaussian mixture model to classify the state vectors of the <i>J</i><sub>1</sub>-<i>J</i><sub>2</sub> antiferromagnetic Heisenberg spin chain system, then the supervised learning algorithm of the convolutional neural network is used to identify the classification point given by the unsupervised learning algorithm, and the cross-validation method is adopted to verify the learning effect. Using this method, we study the <i>J</i><sub>1</sub>-<i>J</i><sub>2</sub> Heisenberg spin chain system with chain length <i>N</i> = 8, 10, 12, 16 and obtain the same conclusion. The first order phase transition point of <i>J</i><sub>1</sub>-<i>J</i><sub>2</sub> antiferromagnetic Heisenberg spin chain system can be accurately found from the ground state vector, but the infinite order phase transition point cannot be found from the ground state vector. The first order and the infinite order phase transition point can be found from the first excited state vector, which indirectly shows that the first excited state may contain more information than the ground state of <i>J</i><sub>1</sub>-<i>J</i><sub>2</sub> antiferromagnetic Heisenberg spin chain system. The visualization of the state vector shows the reliability of the machine learning algorithm, which can extract the feature information from the state vector. The result reveals that the machine learning techniques can directly find some possible phase transition points from a large set of state vectorwithout prior knowledge of the energy or locality conditions of the Hamiltonian, which may assists us in studying unknown systems. Supervised learning can verify the phase transition points given by unsupervised learning, thereby indicating that we can discover some useful information about unknown systems only through machine learning techniques. Machine learning techniques can be a basic research tool in strong quantum-correlated systems, and it can be adapted to more complex systems, which can help us dig up hidden information.

Physics of integer-spin antiferromagnetic chains: Haldane gaps and edge states

The antiferromagnetic Heisenberg spin chain with integer spin has short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling $J$ of the Heisenberg chain. We discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We next study the robustness of the edge states of the spin one chain by studying by the DMRG algorithm a spin one ladder. We show that the peculiar hidden topological order of the spin one chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin one chain.

Review of one-dimensional few-body systems in ultracold atomic physics

We review some recent theoretical and experimental developments of one-dimensional few-body problems in ultracold atomic system. The experiments have so far realized the deterministic loading of few atoms in the ground state of a potential well, the observation of tunneling dynamics out of the metastable trap controlled by a magnetic gradient for a repulsively or attractively interacting system, the preparation of two fermionic atoms in an isolated double-well potential with a full control over the quantum state of the system, the formation of a Fermi sea by studying quasi-one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions, and the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These achievements make the ultracold atoms an ideal platform to study many-body physics in a bottom-up approach, i.e., one starts from the fundamental building block of the system and observes the emergence of many-body effects by adding atoms one by one into the system. Corresponding theoretical models have been developed to explain the experimental data, to tackle the crossover boundary between few and many particles, and even explore the solvability and integrability of the models, especially the energy spectrum of interacting few atoms such as two atoms in a harmonic trap, two heteronuclear atoms of unequal mass in a ring trap, and two atoms in a <inline-formula><tex-math id="M6000">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20181993_M6000.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20181993_M6000.png"/></alternatives></inline-formula>-barrier split double well potential. After a brief review of Bethe-Ansatz method, a theory for the tunneling of one atom out of a trap containing two interacting cold atoms is developed based on the calculation of the quasiparticle wave function, and the tunneling dynamics of two atoms starting from the NOON state is explored from the exactly solved model of <inline-formula><tex-math id="M6001">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20181993_M6001.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20181993_M6001.png"/></alternatives></inline-formula>-barrier split double well based on a Bethe ansatz type hypothesis of the wave functions. It was shown that the spectroscopy and spin dynamics for strongly interacting few atoms of spin-1/2 and spin-1 can be described by effective spin chain Hamiltonians, which serves as a useful and efficient tool to study the quantum magnetism with clod atoms.

Antiferromagnetic Heisenberg Spin Chain of a Few Cold Atoms in a One-Dimensional Trap.

We report on the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These chains are stabilized by strong repulsive interactions between the two spin components without the need for an external periodic potential. We independently characterize the spin configuration of the chains by measuring the spin orientation of the outermost particle in the trap and by projecting the spatial wave function of one spin component on single-particle trap levels. Our results are in good agreement with a spin-chain model for fermionized particles and with numerically exact diagonalizations of the full few-fermion system.

Quantum phase transition in dimerised spin-1/2 chains

Quantum phase transition in dimerised antiferromagnetic Heisenberg spin chain has been studied. A staircase structure in the variation of concurrence within strongly coupled pairs with that of external magnetic field has been observed indicating multiple critical (or critical like) points. Emergence of entanglement due to external magnetic field or magnetic entanglement is observed for weakly coupled spin pairs too in the same dimer chain. Though closed dimerised isotropic XXX Heisenberg chains with different dimer strengths were mainly explored, analogous studies on open chains as well as closed anisotropic (XX interaction) chains with tilted external magnetic field have also been studied.

Suppression of the impurity-induced local magnetism by the opening of a spin pseudogap in Ni-dopedSr2CuO3

The $S=1/2$ antiferromagnetic Heisenberg spin chain compound ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ doped with $1%$ and $2%$ of Ni impurities has been studied by means of $^{63}\mathrm{Cu}$ nuclear magnetic resonance. A strong decrease of the spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$ at low temperatures points toward a spin gap, while a stretching exponent $\ensuremath{\lambda}<1$ and a frequency dependence of ${T}_{1}^{\ensuremath{-}1}$ indicate that this spin gap varies spatially and should rather be characterized as a spin pseudogap. The magnitude of the spin pseudogap scales with doping level. Our results therefore evidence the finite-size character of this phenomenon. Moreover, an unusual narrowing of the low-temperature NMR lines reveals the suppression of the impurity-induced staggered paramagnetic response with increasing doping level.

Magnetocaloric effect in the spin-1/2 chain with competing interactions

We study the magnetocaloric effect for the integrable antiferromagnetic Heisenberg spin chain with competing interactions. We computed the Grüneisen parameter, which is closely related to the magnetocaloric effect, for the quantum spin chain in the thermodynamic limit. This is obtained by means of the solution of a set of non-linear integral equations, which describes the thermodynamics of the model. We also provide results for the entropy S and the isentropes in the (H, T) plane.

Magnetization and phase transition induced by circularly polarized laser in quantum magnets

We theoretically predict a nonequilibrium phase transition in quantum spin systems induced by a laser, which provides a purely quantum-mechanical way of coherently controlling magnetization. Namely, when a circularly polarized laser is applied to a spin system, the magnetic component of a laser is shown to induce a magnetization normal to the plane of polarization, leading to an ultrafast phase transition. We first demonstrate this phenomenon numerically for an $S=1$ antiferromagnetic Heisenberg spin chain, where a new state emerges with magnetization perpendicular to the polarization plane of the laser in place of the topologically ordered Haldane state. We then elucidate its physical mechanism by mapping the system to an effective static model. The theory also indicates that the phenomenon should occur in general quantum spin systems with a magnetic anisotropy. The required laser frequency is in the terahertz range, with the required intensity being within a prospective experimental feasibility.

Sign structure and ground-state properties for aspin−St−Jchain

The antiferromagnetic Heisenberg spin chain of odd spin $S$ is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration $\ensuremath{\delta}$ and hole hopping energy $t$ are considered, the spin chain is replaced by a $\text{spin}\ensuremath{-}S$ $t\ensuremath{-}J$ chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The first result of this paper is that, for the model considered here, the ${\mathbb{Z}}_{2}$ sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the second result of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the $t\ensuremath{-}J$ chain for arbitrary $S$ and $\ensuremath{\delta}$. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The third result of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small $t/J$ limit. The large $t/J$ phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless.

Spin gap in the single spin-12chain cuprateSr1.9Ca0.1CuO3

We report $^{63}$Cu nuclear magnetic resonance and muon spin rotation measurements on the S=1/2 antiferromagnetic Heisenberg spin chain compound Sr$_{1.9}$Ca$_{0.1}$CuO$_3$. An exponentially decreasing spin-lattice relaxation rate 1/T$_1$ indicates the opening of a spin gap. This behavior is very similar to what has been observed for the cognate zigzag spin chain compound Sr$_{0.9}$Ca$_{0.1}$CuO$_2$, and confirms that the occurrence of a spin gap upon Ca doping is independent of the interchain exchange coupling $J'$. Our results therefore generally prove the appearance of a spin gap in an antiferromagnetic Heisenberg spin chain induced by a local bond disorder of the intrachain exchange coupling $J$. A low temperature upturn of 1/T$_1$ evidences growing magnetic correlations. However, zero field muon spin rotation measurements down to 1.5 K confirm the absence of magnetic order in this compound which is most likely suppressed by the opening of the spin gap.

Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z -axis anisotropy λ and magnetic field B . To demonstrate its accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first seven lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z -axis anisotropy and the magnetic field, the 2- and 4-spin correlation functions are calculated, and the results are compared with those obtained by bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined by Roomany–Wyld renormalization group (RG) finite size scaling. The results are in good agreement with those obtained by the level-spectroscopy method.

Ground-state phase diagram of a spin-12frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases

The ground-state phase diagram of a spin-1/2 XXZ chain with competing ferromagnetic nearest-neighbor (J_1<0) and antiferromagnetic second-neighbor (J_2>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range -4<J_1/J_2<0 has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and Neel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and Neel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.

Modeling spin interactions in carbon peapods using a hybrid density functional theory

The $\text{Sc}@{\text{C}}_{82}$ endohedral fullerenes inside a single-wall semiconducting or metallic nanotube form a well-defined chain of antiferromagnetically coupled spins. Using hybrid density functional theory (DFT), we find that the spin resides mainly on the fullerene cage, whether or not the fullerenes are in a nanotube. The spin interactions decay exponentially with fullerene separation and the system can be described by a simple antiferromagnetic Heisenberg spin chain. Energy parameters for a generalized Hubbard-Anderson model are deduced from the DFT calculations and yield a second-order Heisenberg exchange energy, which is in good agreement with total-energy calculations for parallel and antiparallel spin configurations. Within the accuracy of the calculations, neither semiconducting nor metallic nanotubes affect the interactions between the fullerene electron spins.

Extended String Order Parameter in the (S=1, 1/2) Mixed Spin Chain

We have found a set of order parameters that characterize the plateau state of the ( S =1, 1/2) spin chain system. The order parameters are regarded as extensions of the string order parameter of the S =1 antiferromagnetic Heisenberg spin chain. The extension is made by mapping the S z =1/2 ( S z =-1/2) state of an S =1/2 spin onto the S z =0 ( S z =-1) state of an S =1 spin. Numerical data indicate that, when single-ion anisotropy D is introduced, all the order parameters decrease continuously and vanish in the large- D region. Our estimate of the critical value D c for the phase transition agrees well with the phase boundary determined by a previous level spectroscopy analysis. We present estimates of three critical exponents that are associated with the order parameters. A scaling relation between the exponents holds well in the present system.

Exchange coupling in a one-dimensional Wigner crystal

We consider a long quantum wire at low electron densities. In this strong interaction regime a Wigner crystal may form, in which electrons comprise an antiferromagnetic Heisenberg spin chain. The coupling constant J is exponentially small, as it originates from tunneling of two neighboring electrons through the segregating potential barrier. We study this exponential dependence, properly accounting for the many-body effects and the finite width of the wire.

Exchange coupling in a one-dimensional Wigner crystal

We consider a long quantum wire at low electron densities. In this strong-interaction regime a Wigner crystal may form, in which electrons comprise an antiferromagnetic Heisenberg spin chain. The coupling constant $J$ is exponentially small, as it originates from tunneling of two neighboring electrons through the segregating potential barrier. We study this exponential dependence, properly accounting for the many-body effects and the finite width of the wire.