arXiv:1911.05270 · Physical review. B./Physical review. B

Variety of order-by-disorder phases in the asymmetric J1−J2 zigzag ladder: From the delta chain to the J1−J2 chain

Tomoki Yamaguchi, Stefan‐Ludwig Drechsler, Y. Ohta, Satoshi Nishimoto

Published Mar 9, 2020• 21 citations

Abstract

We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin-$\frac{1}{2}$ antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; ${J}_{1}<0$) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and ${J}_{2}/|{J}_{1}|$. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.

DOI: 10.1103/physrevb.101.104407

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Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$-$J_2$ Chain

Cliò Efthimia Agrapidis, Stefan-Ludwig Drechsler, Jeroen van den Brink +1 more

arXiv:1806.08662SciPost PhysicsFeb 6, 201924 citations

Melting of a frustration-induced dimer crystal and incommensurability in the J1 - J2 two-leg ladder

Arthur Lavarélo, Guillaume Roux, Nicolas Laflorencie

arXiv:1107.5498Physical Review BOct 4, 201142 citations

Spin-1/2 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic (F) legs and alternate F and AF rungs

Chatterjee, Monalisa, Manoranjan Kumar, Z. G. Soos

arXiv:2403.05238Mar 8, 20240 citations

Phase diagram of the quantum spin- 1 2 Heisenberg- Γ model on a frustrated zigzag chain

Hidehiro Saito, Chisa Hotta

arXiv:2404.10615Physical review. B./Physical review. BJul 8, 202415 citations

Anomalous fractional quantization in the kagomelike Heisenberg ladder: Emergence of the effective spin-1 chain

Tomoki Yamaguchi, Y. Ohta, Satoshi Nishimoto

arXiv:2103.01455Physical review. B./Physical review. BMay 10, 20213 citations

Magnetic phase diagram of the spin- 12 antiferromagnetic zigzag ladder

Toshiya Hikihara, Tsutomu Momoi, Akira Furusaki +1 more

arXiv:1004.0622Physical Review BJun 28, 201063 citations

arXiv:1911.05270 · Physical review. B./Physical review. B

Variety of order-by-disorder phases in the asymmetric J1−J2 zigzag ladder: From the delta chain to the J1−J2 chain

Tomoki Yamaguchi, Stefan‐Ludwig Drechsler, Y. Ohta, Satoshi Nishimoto

Published Mar 9, 2020• 21 citations

arXiv HTML PDF

Abstract

We study an asymmetric

{J}_{1}\ensuremath{-}{J}_{2}

zigzag ladder consisting of two different spin-

\frac{1}{2}

antiferromagnetic (AFM;

{J}_{2}, \ensuremath{\gamma}{J}_{2}>0

) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM;

{J}_{1}<0

) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of

\ensuremath{\gamma}

and

{J}_{2}/|{J}_{1}|

. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.

DOI: 10.1103/physrevb.101.104407

Related papers

Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$ - $J_2$ ChainCoexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$-$J_2$ Chain

Cliò Efthimia Agrapidis, Stefan-Ludwig Drechsler, Jeroen van den Brink +1 more

Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.

arXiv:1806.08662SciPost PhysicsFeb 6, 201924 citations

Melting of a frustration-induced dimer crystal and incommensurability in the J1 - J2 two-leg ladder

Arthur Lavarélo, Guillaume Roux, Nicolas Laflorencie

The phase diagram of an antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs is determined using numerical methods (exact diagonalization and density-matrix renormalization group) supplemented by strong-coupling and mean-field analysis. Interestingly, this model displays remarkable features, bridging the physics of the ${J}_{1}$ - ${J}_{2}$ chain and of the unfrustrated ladder. The phase diagram as a function of the transverse coupling ${J}_{\ensuremath{\perp}}$ and the frustration ${J}_{2}$ exhibits an Ising transition between a columnar phase of dimers and the usual rung-singlet phase of two-leg ladders. The transition is driven by resonating valence bond fluctuations in the singlet sector while the triplet spin gap remains finite across the transition. In addition, frustration brings incommensurability in the real-space spin-correlation functions, the onset of which evolves smoothly from the ${J}_{1}$ - ${J}_{2}$ chain value to zero in the large-${J}_{\ensuremath{\perp}}$ limit. The onset of incommensurability in the spin structure factor and in the dispersion relation is also analyzed. The physics of the frustrated rung-singlet phase is well understood using perturbative expansions and mean-field theories in the large-${J}_{\ensuremath{\perp}}$ limit. Last, we discuss the effect of the nontrivial magnon dispersion relation on the thermodynamical properties of the system. The relation of this model and its physics to experimental observations on compounds that are currently investigated, such as BiCu ${}_{2}$ PO ${}_{6}$ , is eventually addressed.The phase diagram of an antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs is determined using numerical methods (exact diagonalization and density-matrix renormalization group) supplemented by strong-coupling and mean-field analysis. Interestingly, this model displays remarkable features, bridging the physics of the ${J}_{1}$-${J}_{2}$ chain and of the unfrustrated ladder. The phase diagram as a function of the transverse coupling ${J}_{\ensuremath{\perp}}$ and the frustration ${J}_{2}$ exhibits an Ising transition between a columnar phase of dimers and the usual rung-singlet phase of two-leg ladders. The transition is driven by resonating valence bond fluctuations in the singlet sector while the triplet spin gap remains finite across the transition. In addition, frustration brings incommensurability in the real-space spin-correlation functions, the onset of which evolves smoothly from the ${J}_{1}$-${J}_{2}$ chain value to zero in the large-${J}_{\ensuremath{\perp}}$ limit. The onset of incommensurability in the spin structure factor and in the dispersion relation is also analyzed. The physics of the frustrated rung-singlet phase is well understood using perturbative expansions and mean-field theories in the large-${J}_{\ensuremath{\perp}}$ limit. Last, we discuss the effect of the nontrivial magnon dispersion relation on the thermodynamical properties of the system. The relation of this model and its physics to experimental observations on compounds that are currently investigated, such as BiCu${}_{2}$PO${}_{6}$, is eventually addressed.

arXiv:1107.5498Physical Review BOct 4, 201142 citations

Spin-1/2 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic (F) legs and alternate F and AF rungs

Chatterjee, Monalisa, Manoranjan Kumar, Z. G. Soos

The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$ . The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$ . Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$ , $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at $J_L = J_F/2 \leq J_A$ , that is reversed on interchanging $-J_F$ and $J_A$ . Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$ .The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$. The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$, $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at $J_L = J_F/2 \leq J_A$, that is reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$.

arXiv:2403.05238Mar 8, 20240 citations

Phase diagram of the quantum spin- 1 2 Heisenberg- Γ model on a frustrated zigzag chain

Hidehiro Saito, Chisa Hotta

We investigate the quantum spin-1/2 zigzag chain with frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg interactions, incorporating additional off-diagonal exchange interactions known as the $\mathrm{\ensuremath{\Gamma}}$ term, both with and without an applied magnetic field. Based on the density-matrix renormalization group calculation, we map out the ground-state phase diagram that shows a variety of magnetic and nonmagnetic phases including multicritical points and several exactly solvable points. Upon introducing a finite $\mathrm{\ensuremath{\Gamma}}$ term, we observe the persistent dimer-singlet state of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, sustaining a nonzero spin gap, while also hosting a gapless nonmagnetic excitation that manifests in the substantial zero-energy peak in the nematic dynamical structure factor. This gapless peak-mode, remaining almost as a fluctuation to the ground state, induces a dilute but robust concentration of nematicity on top of singlets on dimers, which we call the nematic singlet-dimer phase. When the whole nematic excited mode condenses and replaces the singlet, the nematic-dimer phase transforms into the Ising-type ferromagnetic or antiferromagnetic long-range orders. The $\mathrm{\ensuremath{\Gamma}}$ term spontaneously selects magnetic easy axes, and their orientations dictate the type of magnetic order under geometric frustration effects as predicted by Landau's mean-field theory. These theoretical findings provide insights into the exotic low-temperature phase observed in ${\mathrm{YbCuS}}_{2}$ , characterized by gapless excitations and seemingly nonmagnetic behavior.We investigate the quantum spin-1/2 zigzag chain with frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg interactions, incorporating additional off-diagonal exchange interactions known as the $\mathrm{\ensuremath{\Gamma}}$ term, both with and without an applied magnetic field. Based on the density-matrix renormalization group calculation, we map out the ground-state phase diagram that shows a variety of magnetic and nonmagnetic phases including multicritical points and several exactly solvable points. Upon introducing a finite $\mathrm{\ensuremath{\Gamma}}$ term, we observe the persistent dimer-singlet state of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, sustaining a nonzero spin gap, while also hosting a gapless nonmagnetic excitation that manifests in the substantial zero-energy peak in the nematic dynamical structure factor. This gapless peak-mode, remaining almost as a fluctuation to the ground state, induces a dilute but robust concentration of nematicity on top of singlets on dimers, which we call the nematic singlet-dimer phase. When the whole nematic excited mode condenses and replaces the singlet, the nematic-dimer phase transforms into the Ising-type ferromagnetic or antiferromagnetic long-range orders. The $\mathrm{\ensuremath{\Gamma}}$ term spontaneously selects magnetic easy axes, and their orientations dictate the type of magnetic order under geometric frustration effects as predicted by Landau's mean-field theory. These theoretical findings provide insights into the exotic low-temperature phase observed in ${\mathrm{YbCuS}}_{2}$, characterized by gapless excitations and seemingly nonmagnetic behavior.

arXiv:2404.10615Physical review. B./Physical review. BJul 8, 202415 citations

Anomalous fractional quantization in the kagomelike Heisenberg ladder: Emergence of the effective spin-1 chain

Tomoki Yamaguchi, Y. Ohta, Satoshi Nishimoto

We study a kagomelike spin- $\frac{1}{2}$ Heisenberg ladder with competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the density-matrix renormalization group based calculations, we obtain the ground-state phase diagram as a function of the ratio between the FM and AFM exchange interactions. Five different phases exist. Three of them are spin-polarized phases: an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as FR1 and FR2 phases). The spontaneous magnetization per site is $m=\frac{1}{2}, \frac{1}{3}$ , and $\frac{1}{6}$ in the FM, FR1, and FR2 phases, respectively. This can be understood from the fact that an effective spin-1 Heisenberg chain formed by the upper and lower leg spins has a three-step fractional quantization of the magnetization per site as $m=1, \frac{1}{2}$ , and 0. In particular, an anomalous ``intermediate'' state $m=\frac{1}{2}$ of the effective spin-1 chain with the reduced Hilbert space of a spin from three to two dimensional is highly unexpected in the context of conventional spin-1 physics. Thus, surprisingly, the effective spin-1 chain behaves like a spin- $\frac{1}{2}$ chain with SU(2) symmetry. The remaining two phases are spin-singlet phases with translational symmetry breaking in the presence of valence bond formations. One of them is octamer-singlet phase with a spontaneous octamerization long-range order of the system, and the other is period-4 phase characterized by the magnetic superstructure with a period of four structural unit cells. In these spin-singlet phases, we find the coexistence of valence bond structure and gapless chain. Although this may be emerged through the order-by-disorder mechanism, there can be few examples of such a coexistence.We study a kagomelike spin-$\frac{1}{2}$ Heisenberg ladder with competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the density-matrix renormalization group based calculations, we obtain the ground-state phase diagram as a function of the ratio between the FM and AFM exchange interactions. Five different phases exist. Three of them are spin-polarized phases: an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as FR1 and FR2 phases). The spontaneous magnetization per site is $m=\frac{1}{2}, \frac{1}{3}$, and $\frac{1}{6}$ in the FM, FR1, and FR2 phases, respectively. This can be understood from the fact that an effective spin-1 Heisenberg chain formed by the upper and lower leg spins has a three-step fractional quantization of the magnetization per site as $m=1, \frac{1}{2}$, and 0. In particular, an anomalous ``intermediate'' state $m=\frac{1}{2}$ of the effective spin-1 chain with the reduced Hilbert space of a spin from three to two dimensional is highly unexpected in the context of conventional spin-1 physics. Thus, surprisingly, the effective spin-1 chain behaves like a spin-$\frac{1}{2}$ chain with SU(2) symmetry. The remaining two phases are spin-singlet phases with translational symmetry breaking in the presence of valence bond formations. One of them is octamer-singlet phase with a spontaneous octamerization long-range order of the system, and the other is period-4 phase characterized by the magnetic superstructure with a period of four structural unit cells. In these spin-singlet phases, we find the coexistence of valence bond structure and gapless chain. Although this may be emerged through the order-by-disorder mechanism, there can be few examples of such a coexistence.

arXiv:2103.01455Physical review. B./Physical review. BMay 10, 20213 citations

Magnetic phase diagram of the spin- 12 antiferromagnetic zigzag ladder

Toshiya Hikihara, Tsutomu Momoi, Akira Furusaki +1 more

We study the one-dimensional spin- $\frac{1}{2}$ Heisenberg model with antiferromagnetic nearest-neighbor ${J}_{1}$ and next-nearest-neighbor ${J}_{2}$ exchange couplings in magnetic field $h$ . With varying dimensionless parameters ${J}_{2}/{J}_{1}$ and $h/{J}_{1}$ , the ground state of the model exhibits several phases including three gapped phases (dimer, 1/3-magnetization plateau, and fully polarized phases) and four types of gapless Tomonaga-Luttinger liquid (TLL) phases which we dub TLL1, TLL2, spin-density-wave $({\text{SDW}}_{2})$ , and vector chiral phases. From extensive numerical calculations using the density-matrix renormalization-group method, we investigate various (multiple-)spin-correlation functions in detail and determine dominant and subleading correlations in each phase. For the one-component TLLs, i.e., the TLL1, ${\text{SDW}}_{2}$ , and vector chiral phases, we fit the numerically obtained correlation functions to those calculated from effective low-energy theories of TLLs and find good agreement between them. The low-energy theory for each critical TLL phase is thus identified, together with TLL parameters which control the exponents of power-law decaying correlation functions. For the TLL2 phase, we develop an effective low-energy theory of two-component TLL consisting of two free bosons (central charge $c=1+1$ ), which explains numerical results of entanglement entropy and Friedel oscillations of local magnetization. Implications of our results to possible magnetic phase transitions in real quasi-one-dimensional compounds are also discussed.We study the one-dimensional spin-$\frac{1}{2}$ Heisenberg model with antiferromagnetic nearest-neighbor ${J}_{1}$ and next-nearest-neighbor ${J}_{2}$ exchange couplings in magnetic field $h$. With varying dimensionless parameters ${J}_{2}/{J}_{1}$ and $h/{J}_{1}$, the ground state of the model exhibits several phases including three gapped phases (dimer, 1/3-magnetization plateau, and fully polarized phases) and four types of gapless Tomonaga-Luttinger liquid (TLL) phases which we dub TLL1, TLL2, spin-density-wave $({\text{SDW}}_{2})$, and vector chiral phases. From extensive numerical calculations using the density-matrix renormalization-group method, we investigate various (multiple-)spin-correlation functions in detail and determine dominant and subleading correlations in each phase. For the one-component TLLs, i.e., the TLL1, ${\text{SDW}}_{2}$, and vector chiral phases, we fit the numerically obtained correlation functions to those calculated from effective low-energy theories of TLLs and find good agreement between them. The low-energy theory for each critical TLL phase is thus identified, together with TLL parameters which control the exponents of power-law decaying correlation functions. For the TLL2 phase, we develop an effective low-energy theory of two-component TLL consisting of two free bosons (central charge $c=1+1$), which explains numerical results of entanglement entropy and Friedel oscillations of local magnetization. Implications of our results to possible magnetic phase transitions in real quasi-one-dimensional compounds are also discussed.

arXiv:1004.0622Physical Review BJun 28, 201063 citations

We study an asymmetric

{J}_{1}\ensuremath{-}{J}_{2}

zigzag ladder consisting of two different spin-

\frac{1}{2}

antiferromagnetic (AFM;

{J}_{2}, \ensuremath{\gamma}{J}_{2}>0

) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM;

{J}_{1}<0

) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of

\ensuremath{\gamma}

and

{J}_{2}/|{J}_{1}|

Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic

J_1

J_2

Chain

{J}_{1}

{J}_{2}

chain and of the unfrustrated ladder. The phase diagram as a function of the transverse coupling ${J}_{\ensuremath{\perp}}$ and the frustration

{J}_{2}

exhibits an Ising transition between a columnar phase of dimers and the usual rung-singlet phase of two-leg ladders. The transition is driven by resonating valence bond fluctuations in the singlet sector while the triplet spin gap remains finite across the transition. In addition, frustration brings incommensurability in the real-space spin-correlation functions, the onset of which evolves smoothly from the

{J}_{1}

{J}_{2}

chain value to zero in the large-${J}_{\ensuremath{\perp}}$ limit. The onset of incommensurability in the spin structure factor and in the dispersion relation is also analyzed. The physics of the frustrated rung-singlet phase is well understood using perturbative expansions and mean-field theories in the large-${J}_{\ensuremath{\perp}}$ limit. Last, we discuss the effect of the nontrivial magnon dispersion relation on the thermodynamical properties of the system. The relation of this model and its physics to experimental observations on compounds that are currently investigated, such as BiCu

{}_{2}

{}_{6}

, is eventually addressed.

The frustrated ladder with alternate ferromagnetic(F) exchange

-J_F

and AF exchange

J_A

to first neighbors and F exchange

-J_L

to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of

2N

spins-1/2 with periodic boundary conditions. The ground state is a singlet

(S = 0)

and the singlet-triplet gap

\varepsilon_T

is finite for the exchanges considered. Spin-1/2 string correlation functions

g_1(N)

and

g_2(N)

are defined for an even number

N

of consecutive spins in systems with two spins per unit cell; the ladder has string order

g_2(\infty)> 0

and

g_1(\infty) = 0

. The minimum

N^*

g_2(N)

is related to the range of ground-state spin correlations. Convergence to

g_2(\infty)

is from below, and

g_1(N)

decreases exponentially for

N \geq N^*

. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of

J_F

J_L

and

J_A

reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the

J_1-J_2

model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at

J_L = J_F/2 \leq J_A

, that is reversed on interchanging

-J_F

and

J_A

. Inversion symmetry is spontaneously broken in the dimer phase of the

J_1-J_2

model where the Kekulé diagrams are the doubly degenerate ground states at

J_2/J_1 = 1/2

{\mathrm{YbCuS}}_{2}

, characterized by gapless excitations and seemingly nonmagnetic behavior.

We study a kagomelike spin-

\frac{1}{2}

Heisenberg ladder with competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the density-matrix renormalization group based calculations, we obtain the ground-state phase diagram as a function of the ratio between the FM and AFM exchange interactions. Five different phases exist. Three of them are spin-polarized phases: an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as FR1 and FR2 phases). The spontaneous magnetization per site is

m=\frac{1}{2}, \frac{1}{3}

, and

\frac{1}{6}

in the FM, FR1, and FR2 phases, respectively. This can be understood from the fact that an effective spin-1 Heisenberg chain formed by the upper and lower leg spins has a three-step fractional quantization of the magnetization per site as

m=1, \frac{1}{2}

, and 0. In particular, an anomalous ``intermediate'' state

m=\frac{1}{2}

of the effective spin-1 chain with the reduced Hilbert space of a spin from three to two dimensional is highly unexpected in the context of conventional spin-1 physics. Thus, surprisingly, the effective spin-1 chain behaves like a spin-

\frac{1}{2}

chain with SU(2) symmetry. The remaining two phases are spin-singlet phases with translational symmetry breaking in the presence of valence bond formations. One of them is octamer-singlet phase with a spontaneous octamerization long-range order of the system, and the other is period-4 phase characterized by the magnetic superstructure with a period of four structural unit cells. In these spin-singlet phases, we find the coexistence of valence bond structure and gapless chain. Although this may be emerged through the order-by-disorder mechanism, there can be few examples of such a coexistence.

We study the one-dimensional spin-

\frac{1}{2}

Heisenberg model with antiferromagnetic nearest-neighbor

{J}_{1}

and next-nearest-neighbor

{J}_{2}

exchange couplings in magnetic field

h

. With varying dimensionless parameters

{J}_{2}/{J}_{1}

and

h/{J}_{1}

, the ground state of the model exhibits several phases including three gapped phases (dimer, 1/3-magnetization plateau, and fully polarized phases) and four types of gapless Tomonaga-Luttinger liquid (TLL) phases which we dub TLL1, TLL2, spin-density-wave

({\text{SDW}}_{2})

, and vector chiral phases. From extensive numerical calculations using the density-matrix renormalization-group method, we investigate various (multiple-)spin-correlation functions in detail and determine dominant and subleading correlations in each phase. For the one-component TLLs, i.e., the TLL1,

{\text{SDW}}_{2}

, and vector chiral phases, we fit the numerically obtained correlation functions to those calculated from effective low-energy theories of TLLs and find good agreement between them. The low-energy theory for each critical TLL phase is thus identified, together with TLL parameters which control the exponents of power-law decaying correlation functions. For the TLL2 phase, we develop an effective low-energy theory of two-component TLL consisting of two free bosons (central charge

c=1+1

), which explains numerical results of entanglement entropy and Friedel oscillations of local magnetization. Implications of our results to possible magnetic phase transitions in real quasi-one-dimensional compounds are also discussed.