Heisenberg Antiferromagnetic Ladder Research Articles

Dimerization and spin decoupling in a two-leg Heisenberg ladder with frustrated trimer rungs

We study the antiferromagnetic spin-half Heisenberg ladder in the presence of an additional frustrating rung spin that is motivated and relevant also for the description of real two-dimensional materials such as the two-dimensional trimer magnet ${\mathrm{Ba}}_{4}{\mathrm{Ir}}_{3}{\mathrm{O}}_{10}$. We study the zero-temperature phase diagram, where we combine numerical and analytical methods into an overall consistent description. All numerical simulations are also accompanied by studies of the dynamical spin structure factor obtained via the density matrix renormalization group. Overall, we find in the regime of strong rung coupling a gapped dimerized phase related to competing symmetry sectors in Hilbert space that ultimately results in frustration-driven spin-Peierls transition. In the weak rung-coupling regime, the system is uniform yet shows a gapped spinon continuum together with a sharp coherent low-energy branch that renders the system critical overall. In either case, the additional rung spin quickly gets sidelined and nearly decouples once its bare coupling to the ladder drops somewhat below the direct Heisenberg coupling of the legs.

Inter-layer modified -wave superconductivity in an effectively doped spin-1 ladder

We construct a four-leg spin-1/2 t–J type model to simulate a doped two-leg spin-1 antiferromagnetic Heisenberg ladder. Employing renormalized mean-field theory with simple Gutzwiller factors, we obtain three degenerate superconducting states with different pairing symmetry. Through improving the Gutzwiller factors, we find that the state C with inter-layer modified -wave pairing has the lowest energy in a large doping range. Besides, we use the density matrix renormalization group method to solve the model. The negative binding energy reveals the pairing tendency, and the pair–pair correlation functions exhibit a slowly decaying behavior on certain types of bonds. From the pair correlations, we confirm the inter-layer modified -wave superconducting state as the ground state of the model.

Magnetic-Order Crossover in Coupled Spin Ladders.

We report a novel crossover behavior in the long-range-ordered phase of a prototypical spin-1/2 Heisenberg antiferromagnetic ladder compound (C_{7}H_{10}N)_{2}CuBr_{4}. The staggered order was previously evidenced from a continuous and symmetric splitting of ^{14}N NMR spectral lines on lowering the temperature below T_{c}≃330 mK, with a saturation towards ≃150 mK. Unexpectedly, the split lines begin to further separate away below T^{*}∼100 mK, while the linewidth and the line shape remain completely invariable. This crossover behavior is further corroborated by the NMR relaxation rate T_{1}^{-1} measurements. A very strong suppression reflecting the ordering, T_{1}^{-1}∼T^{5.5}, observed above T^{*}, is replaced by T_{1}^{-1}∼T below T^{*}. These original NMR features are indicative of the unconventional nature of the crossover, which may arise from a unique arrangement of the ladders into a spatially anisotropic and frustrated coupling network.

Dichotomy between Attractive and Repulsive Tomonaga-Luttinger Liquids in Spin Ladders.

We present a direct NMR method to determine whether the interactions in a Tomonaga-Luttinger liquid (TLL) state of a spin-1/2 Heisenberg antiferromagnetic ladder are attractive or repulsive. For the strong-leg spin ladder compound (C_{7}H_{10}N)_{2}CuBr_{4} we find that the isothermal magnetic field dependence of the NMR relaxation rate T_{1}^{-1}(H) displays a concave curve between the two critical fields bounding the TLL regime. This is in sharp contrast to the convex curve previously reported for a strong-rung ladder, (C_{5}H_{12}N)_{2}CuBr_{4}. We show that the concavity and the convexity of T_{1}^{-1}(H), which is a fingerprint of spin fluctuations, directly reflect the attractive and repulsive fermionic interactions in the TLL, respectively. The interaction sign is alternatively determined from an indirect method combining bulk magnetization and specific heat data.

Electron spin resonance modes in a strong-leg ladder in the Tomonaga-Luttinger liquid phase

Magnetic excitations in the strong-leg quantum spin ladder compound (C$_7$H$_{10}$N)$_2$CuBr$_4$ (known as DIMPY) in the field-induced Tomonaga-Luttinger spin liquid phase are studied by means of high-field electron spin resonance (ESR) spectroscopy. The presence of a gapped ESR mode with unusual non-linear frequency-field dependence is revealed experimentally. Using a combination of analytic and exact diagonalization methods, we compute the dynamical structure factor and identify this mode with longitudinal excitations in the antisymmetric channel. We argue that these excitations constitute a fingerprint of the spin dynamics in a strong-leg spin-1/2 Heisenberg antiferromagnetic ladder and owe its ESR observability to the uniform Dzyaloshinskii-Moriya interaction.

Scattering particles in quantum spin chains

A variational approach for constructing an effective particle description of the low-energy physics of one-dimensional quantum spin chains is presented. Based on the matrix product state formalism, we compute the one- and two-particle excitations as eigenstates of the full microscopic Hamiltonian. We interpret the excitations as particles on a strongly-correlated background with non-trivial dispersion relations, spectral weights and two-particle S matrices. Based on this information, we show how to describe a finite density of excitations as an interacting gas of bosons, using its approximate integrability at low densities. We apply our framework to the Heisenberg antiferromagnetic ladder: we compute the elementary excitation spectrum and the magnon-magnon S matrix, study the formation of bound states and determine both static and dynamic properties of the magnetized ladder.

Quantum phase transition and magnetic plateau in three-leg antiferromagnetic Heisenberg spin ladder with unequal J1–J2–J1 legs

Magnetic properties of spin-1/2 antiferromagnetic three-leg Heisenberg ladders, where antiferromagnetic interactions in legs are J1, J2 and J1 respectively and in the rungs are J⊥, have been investigated by bond-mean field method. As J⊥ changes, magnetization curves show different behavior. For J⊥=0.5, there are cusps in magnetization curves, while for J⊥=3.0, the 1/3 magnetization plateau appears, which can be explained by energy spectra. Furthermore, for J⊥=3.0 the 1/3 magnetization plateaus will become wider or narrow down with J2 changing. In addition, the mean-field bond parameters and the concurrences, which confirm the phase transitions, are also studied.

Attractive Tomonaga-Luttinger liquid in a quantum spin ladder.

We present NMR measurements of a strong-leg spin-1/2 Heisenberg antiferromagnetic ladder compound (C7H10N)2CuBr4 under magnetic fields up to 15 T in the temperature range from 1.2 K down to 50 mK. From the splitting of NMR lines, we determine the phase boundary and the order parameter of the low-temperature (three-dimensional) long-range-ordered phase. In the Tomonaga-Luttinger regime above the ordered phase, NMR relaxation reflects characteristic power-law decay of spin correlation functions as 1/T1∝T(1/2K-1), which allows us to determine the interaction parameter K as a function of field. We find that field-dependent K varies within the 1<K<2 range, which signifies attractive interaction between the spinless fermions in the Tomonaga-Luttinger liquid.

Field-induced multiple-point quantum criticality in the three-leg Heisenberg ladder

Field-induced quantum (zero temperature) criticality described using parameters that are not associated with a symmetry-breaking long-range order is found in the spin-$\frac{1}{2}$ antiferromagnetic three-leg Heisenberg ladder. These parameters represent the spin bond order, which is a consequence of the probability for spins on adjacent sites to be bound in a singletlike arrangement. They underwent two phase transitions, one at the lower critical field ${h}_{c1}$ and the other at the upper critical field ${h}_{c2}$. The field dependence of these bond parameters and of the magnetization is calculated for several values of the rung coupling in all regimes. This yields the coupling field phase diagram. It is found that the rung coupling must exceed a threshold value for the plateau, at one third of the saturation magnetization, to appear. The results obtained here within the bond mean-field theory compare well with existing numerical data.

Static and Dynamic Properties of Antiferromagnetic Heisenberg Ladders: Fermionic versus Bosonic Approaches

In terms of spinless fermions via the Jordan-Wigner transformation along a snake-like path and spin waves modified so as to restore the sublattice symmetry, we investigate static and dynamic properties of two-leg antiferromagnetic Heisenberg ladders. The specific heat is finely reproduced by the spinless fermions, whereas the magnetic susceptibility is well described by the modified spin waves. The nuclear spin-lattice relaxation rate is discussed in detail with particular emphasis on its novel field dependence.

Ground-state phase diagram of a spin-12frustrated three-leg antiferromagnetic Heisenberg ladder

Using the density matrix renormalization group method we have studied the ground-state phase diagram of a spin-1/2 frustrated three-leg antiferromagnetic Heisenberg ladder, and identified two phases corresponding to the spin-3/2 chain and the standard spin ladder. Varying the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ leads to a quantum phase transition from one ground state to the other, but does not alter the gapless nature of the excitation spectra. On the phase boundary, the degenerate ground states form a chiral spin liquid with a central charge $c=2.$ The critical value of ${J}_{\ifmmode\times\else\texttimes\fi{}}$ is insensitive to the number of legs in the ladder, suggesting that a quantum phase transition may occur at ${J}_{\ifmmode\times\else\texttimes\fi{}}=0.6$ for the isotropic spin-1/2 antiferromagnetic Heisenberg model on the square lattice.

Fermionic Description of Spin-Gap States of Antiferromagnetic Heisenberg Ladders in a Magnetic Field

Employing the Jordan-Wigner transformation on a unique path and then making a mean-field treatment of the fermionic Hamiltonian, we semiquantitatively describe the spin-gap states of Heisenberg ladders in a field. The appearance of magnetization plateaux is clarified as a function of the number of legs.

Yang-Lee Circle Theorem for an Antiferromagnetic Heisenberg Ladder

The Yang-Lee zeros of an antiferromagnetic Heisenberg ladder model aredetermined. It is found that if J4⩽0 Yang-Lee zeros are locatedon the unit circle and on the negative real axis in the complex activityplane. In particular, if J4⩽0 and 2J2⩾J4, Yang-Lee zerosare located on the unit circle and the Yang-Lee circle theorem is valid. IfJ4>0, Yang-Lee zeros are located on some complicated curves.

Magnetic properties of a molecular-based spin-ladder system:(5IAP)2CuBr4⋅2H2O

We have synthesized and characterized a spin-1/2 Heisenberg antiferromagnetic ladder: bis 5-iodo-2-aminopyridinium tetrabromocuprate (II) dihydrate. X-ray diffraction studies show the structure of the compound to consist of well isolated stacked ladders and the interaction between the ${\mathrm{Cu}}^{2+}$ atoms to be due to direct $\mathrm{Br}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}\mathrm{Br}$ contacts. Magnetic susceptibility and magnetization studies show the compound to be in the strong-coupling limit, with the interaction along the rungs ${(J}^{\ensuremath{'}}\ensuremath{\approx}13\mathrm{K})$ much greater than the interaction along the rails $(J\ensuremath{\approx}1\mathrm{K}).$ Magnetic critical fields are observed near 8.3 and 10.4 T, respectively, establishing the existence of the energy gap.

Low-energy properties of spin-1/2 two-leg antiferromagnetic Heisenberg ladders with ferromagnetic diagonal coupling

Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.

Magnetic ordering in La 2Cu 2O 5

Muon spin relaxation (μSR) measurements were carried out on La 2Cu 2O 5, which is a candidate for the spin 1 2 4-leg Heisenberg antiferromagnetic ladder system. At room temperature, the μSR spectrum can be fitted with the static Gaussian Kubo–Toyabe function. This spectrum originates from static nuclear dipole fields of copper. At ∼140 K , a sudden decrease of the initial asymmetry was observed. This suggests that a long-range magnetic ordering of the Cu 2+ spins occurs due to nonnegligible interladder interactions.

Dynamical spin correlations in the Heisenberg ladder under a magnetic field and correlation functions in the SO(5) ladder

The zero-temperature dynamical spin-spin correlation functions are calculated for the spin-$\frac{1}{2}$ two-leg antiferromagnetic Heisenberg ladder in a magnetic field above the lower critical field ${H}_{c1}$. The dynamical structure factors are calculated, which exhibit both massless modes and massive excitations. These modes appear in different sectors characterized by the parity in the rung direction and by the momentum in the direction of the chains. The structure factors have power-law singularities at the lower edges of their support. The results are also applicable to the spin-1 Heisenberg chain. The implications are briefly discussed for the various correlation functions and the $\ensuremath{\pi}$ resonance in the SO(5) symmetric ladder model.

Thermodynamic properties of the spin-1/2 antiferromagnetic ladder under magnetic field

Specific heat ($C_V$) measurements in the spin-1/2 Cu$_2$(C$_2$H$_{12}$N$_2$)$_2$Cl$_4$ system under a magnetic field up to $H=8.25 T$ are reported and compared to the results of numerical calculations based on the 2-leg antiferromagnetic Heisenberg ladder. While the temperature dependences of both the susceptibility and the low field specific heat are accurately reproduced by this model, deviations are observed below the critical field $H_{C1}$ at which the spin gap closes. In this Quantum High Field phase, the contribution of the low-energy quantum fluctuations are stronger than in the Heisenberg ladder model. We argue that this enhancement can be attributed to dynamical lattice fluctuations. Finally, we show that such a Heisenberg ladder, for $H>H_{C1}$, is unstable, when coupled to the 3D lattice, against a lattice distortion. These results provide an alternative explanation for the observed low temperature ($T_C\sim 0.5K$ -- $0.8K$) phase (previously interpreted as a 3D magnetic ordering) as a new type of incommensurate gapped state.

Correlations and Néel Order of Randomly Diluted Quantum Spin Ladders

We present a Monte Carlo study of the correlation length $\xi$ of randomly diluted antiferromagnetic Heisenberg ladders, composed of two spin--1/2 chains. For weak and intermediate inter--chain couplings, $J_{\bot}/J \leq 1$, we find an enhancement of correlations that is strongest for a fraction $z^* \approx J_{\bot}/(8J)$ of dilutants. We are able to access the experimentally relevant low--temperature regime, $T/J \approx 1/500$, and find that the recently inferred N\'eel temperature of $Sr(Cu_{1-z}Zn_z)_2O_3$ corresponds to a curve of constant correlation length $\xi \approx 18$ of the single diluted ladder with $J_{\bot}/J \alt 1/2$. The primary reason for the N\'eel ordering is argued to be a strong enhancement of two--dimensional correlations due to a Cu--Sr--Cu exchange coupling of $\approx 10 meV$ in the stacking direction of the ladders.

Spin gap of the antiferromagnetic Heisenberg ladder: self-consistent mean-field theory

The properties including the spin gap and group-state energy of the spin- 1 2 antiferromagnetic Heisenberg ladder are investigated by using the self-consistent mean-field theory. We get the result that the spin gap is nonzero for all interchain couplings. Our numerical results agree well with the recent theoretical and experimental results.