We consider the ground-state phase diagram of a one-dimensional spin-1/2 XXZ chain with a spatially modulated Dzyaloshinskii-Moriya interaction in the presence of an alternating magnetic field applied along the z[over ̂] axis. The model is studied using the continuum-limit bosonization approach and the finite system exact numerical technique. In the absence of a magnetic field, the ground-state phase diagram of the model includes, besides the ferromagnetic and gapless Luttinger-liquid phases, two gapped phases: the composite (C1) phase characterized by the coexistence of long-range-ordered (LRO) alternating dimerization and spin chirality patterns, and the composite (C2) phase characterized by, in addition to the coexisting spin dimerization and alternating chirality patterns, the presence of LRO antiferromagnetic order. In the case of mentioned composite gapped phases, and in the case of a uniform magnetic field, the commensurate-incommensurate type quantum phase transitions from a gapful phase into a gapless phase have been identified and described using the bosonization treatment and finite chain exact diagonalization studies. The upper critical magnetic field corresponding to the transition into a fully polarized state has been also determined. It has been shown that the very presence of a staggered component of the magnetic field vapes the composite (C1) in favor of the composite gapped (C2) phase.
We consider the spin-1/2 Heisenberg chain with alternating spin exchange in the presence of additional modulation of exchange on odd bonds with period 3. We study the ground state magnetic phase diagram of this hexamer spin chain in the limit of very strong antiferromagnetic (AF) exchange on odd bonds using the numerical Lanczos method and bosonization approach. In the limit of strong magnetic field commensurate with the dominating AF exchange, the model is mapped onto an effective XXZ Heisenberg chain in the presence of uniform and spatially modulated fields, which is studied using the standard continuum-limit bosonization approach. In the absence of additional hexamer modulation, the model undergoes a quantum phase transition from a gapped phase into the only one gapless Lüttinger liquid (LL) phase by increasing the magnetic field. In the presence of hexamer modulation, two new gapped phases are identified in the ground state at magnetization equal to [Formula: see text] and [Formula: see text] of the saturation value. These phases reveal themselves also in the magnetization curve as plateaus at corresponding values of magnetization. As a result, the magnetic phase diagram of the hexamer chain shows seven different quantum phases, four gapped and three gapless, and the system is characterized by six critical fields which mark quantum phase transitions between the ordered gapped and the LL gapless phases.
We study the interplay of electric and magnetic order in the one-dimensional Heisenberg spin-
21 XXZ chain with large Ising anisotropy in the presence of the Dzyaloshinskii-Moriya (DM) interaction and with longitudinal and transverse magnetic fields, interpreting the DM interaction as a coupling between the local electric polarization and an external electric field. We obtain the ground state phase diagram using the density matrix renormalization group method and compute various ground state quantities like the magnetization, staggered magnetization, electric polarization and spin correlation functions, etc. In the presence of both longitudinal and transverse magnetic fields, there are three different phases corresponding to a gapped N\'eel phase with antiferromagnetic (AF) order, gapped saturated phase, and a critical incommensurate gapless phase. The external electric field modifies the phase boundaries but does not lead to any new phases. Both external magnetic fields and electric fields can be used to tune between the phases. We also show that the transverse magnetic field induces a vector chiral order in the N\'eel phase (even in the absence of an electric field) which can be interpreted as an electric polarization in a direction parallel to the AF order.
We study the phase diagram of the spin-1/2 XXZ Heisenberg chain in a staggered magnetic field. An exact solution is obtained along the phase boundary, where the exchange anisotropy and the staggered-field strength satisfy an exact analytical relation, yielding a highly degenerate manifold of ground states with off-diagonal long-range order. We demonstrate that these exact ground states lie on the boundary between the ferromagnetic phase and a gapless Luttinger liquid phase. Upon further decreasing the anisotropy, the gapless phase eventually gives way to a gapped quantum-disordered phase. Large-scale density-matrix renormalization group calculations confirm this phase structure and indicate that the transition across the exact boundary is of first order. Our results provide an exact benchmark for understanding quantum phase transitions driven by staggered magnetic fields in one-dimensional quantum spin systems.
The ground-state phase diagram of a spin
S=21XXZ Heisenberg chain with spatially modulated Dzyaloshinskii-Moriya interaction $\mathcal{H}={\ensuremath{\sum}}_{n}{J\left({S}_{n}^{x}{S}_{n+1}^{x}+{S}_{n}^{y}{S}_{n+1}^{y}\right)+{J}_{z}{S}_{n}^{z}{S}_{n+1}^{z}+[{D}_{0}+{(\ensuremath{-}1)}^{n}{D}_{1}]({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x})}$ is studied using the continuum-limit bosonization approach and extensive density-matrix renormalization group computations. It is shown that the effective continuum-limit bosonized theory of the model is given by the double-frequency sine-Gordon model (DSG) where the frequencies, i.e., the scaling dimensions of the two competing cosine perturbation terms, depend on the effective anisotropy parameter ${\ensuremath{\gamma}}^{*}={J}_{z}/\sqrt{{J}^{2}+{D}_{0}^{2}+{D}_{1}^{2}}$. Exploring the ground-state properties of the DSG model we show that the zero-temperature phase diagram contains the following four phases: (i) the ferromagnetic phase at ${\ensuremath{\gamma}}^{*}\ensuremath{\le}\ensuremath{-}1$; (ii) the gapless Luttinger-liquid (LL) phase at $\ensuremath{-}1<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C1}}^{*}=\ensuremath{-}1/\sqrt{2}$; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern $\ensuremath{\epsilon}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}({\mathbf{S}}_{n}{\mathbf{S}}_{n+1})$ with the LRO alternating spin chirality pattern $\ensuremath{\kappa}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}\left({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x}\right)$ at ${\ensuremath{\gamma}}_{\text{C1}}^{*}<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C2}}^{*}$; and (iv) at ${\ensuremath{\gamma}}^{*}>{\ensuremath{\gamma}}_{\text{C2}}^{*}>1$ the gapped composite (C2) phase characterized in addition to the coexisting spin dimerization and alternating chirality patterns, by the presence of LRO antiferromagnetic order. The transition from the LL to the C1 phase at ${\ensuremath{\gamma}}_{\text{C1}}^{*}$ belongs to the Berezinskii-Kosterlitz-Thouless universality class, while the transition at ${\ensuremath{\gamma}}^{*}={\ensuremath{\gamma}}_{\text{C2}}^{*}$ from C1 to C2 phase is of the Ising type.
A frustrated spin-
1/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling
J2>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of
J1/J2 and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D
+) phase with the same sign of the
x,y- and
z-component dimer order parameters, two VC dimer (VCD
+/VCD${}_{\ensuremath{-}}$) phases with the sign of the
z-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D
+ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb
2Cu
2Mo
3O
12 is discussed.
We have studied the phase diagram of the one dimensional XXZ model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the quantum renormalization group (QRG) approach to get the stable fixed points, critical point and the scaling of coupling constants. This model has three phases, ferromagnetic, spin-fluid and Néel phases which are separated by a critical line which depends on the DM coupling constant. We have shown that the staggered magnetization is the order parameter of the system and investigated the influence of DM interaction on the chiral ordering as a helical magnetic order.