Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in

t-J

chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to

J_{\perp}=0.6

, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin-spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behavior persists upon moderate hole doping

x \alt 0.75

. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.

DOI: 10.48550/arxiv.cond-mat/0504306

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Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in

t-J

J_{\perp}=0.6

x \alt 0.75

. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.

We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-

\frac{1}{2}

Luttinger liquid coupled by a Kondo exchange interaction

{J}_{K}

to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling

{J}_{1}

and a next-nearest-neighbor (frustrating) exchange interaction

{J}_{2}

. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling

{J}_{K}

with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios

{J}_{K}/{J}_{1}

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

, and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.

We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling

α

. This bond-anisotropic parameter continuously interpolates among the Ising-like limit

α=0

, the isotropic point

α=1

, and the XY-dominated regime

α\gg 1

. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge

c \simeq 1

, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe

c \simeq 1/2

, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point

α= 1

{J}_{d}

(along the diagonals). An extensive comparison of the renormalization group and quantum Monte Carlo results for parameters where the simulations by the latter method are possible shows a very good agreement between the two methods. It is found, by analyzing ground state energies and spin-spin correlation functions, that there is a transition between two ordered magnetic states. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\lesssim}0.5$, the ground state displays a N\'eel order. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\gtrsim}0.5$, a collinear magnetic ground state in which interchain spin correlations are ferromagnetic becomes stable. In the vicinity of the transition point, ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\approx}0.5$, the ground state is disordered. But, the nature of this disordered ground state is unclear. While the numerical data seem to show that the chains are disconnected, the possibility of a genuine disordered two-dimensional state, hidden by finite size effects, cannot be excluded.