Bosonization and density-matrix renormalization group studies of the Fulde-Ferrell-Larkin-Ovchinnikov phase and irrational magnetization plateaus in coupled chains

We review the properties of two coupled fermionic chains, or ladders, under a magnetic field parallel to the lattice plane. Results are computed by complementary analytical (bosonization) and numerical (density-matrix renormalization group) methods, which allows a systematic comparison. Limiting cases such as coupled-band and coupled-chain regimes are discussed. We particularly focus on the evolution of the superconducting correlations under increasing field and on the presence of irrational magnetization plateaus. We found the existence of large doping-dependent magnetization plateaus in the weakly interacting and strong-coupling limits and in the nontrivial case of isotropic couplings. We report on the existence of extended Fulde-Ferrell-Larkin-Ovchinnikov phases [Phys. Rev. 135, A550 (1964); Sov. Phys. JETP 20, 762 (1965);] within the isotropic $t\text{−}J$ and Hubbard models, deduced from the evolution of different observables under magnetic field. Emphasis is put on the variety of superconducting order parameters present at high magnetic field. We have also computed the evolution of the Luttinger exponent corresponding to the ungapped spin mode appearing at finite magnetization. In the coupled-chain regime, the possibility of having polarized triplet pairing under high field is predicted by bosonization.

Figure 1

Splitting of the band dispersion in a noninteracting doped ladder ($\mu$ is the chemical potential) due to Zeeman effect at low magnetic field $H$.

Figure 2

Band evolution corresponding to Fig. 3.

Figure 3

Magnetization curves for the free system with $t_{\Vert }=0.2$ and $t_{\perp }=1.0$ showing a plateau at $m=\delta$ of width $2(t_{\perp }-2t_{\Vert })$.

Figure 4

(Color online) Plateaus in the strong-coupling limit. We chose $J_{\perp }=2.5$, $J_{\Vert }=0.3$, and $t_{\Vert }=t_{\perp }=1.0$. The $m=\delta$ plateau is an effect of interactions and the ground state in this plateau has unpaired holes. Still, holes are paired for magnetizations with $m⩽m_c<\delta$.

Figure 5

(Color online) Plateau phase diagram of a two-leg ladder for $t_{\perp }=J_{\perp }=1.0$ and $t_{\Vert }=J_{\Vert }=0.2$. Doping immediately destroys the $m=0$ plateau present at half-filling, while a large doping-dependent $m=\delta$ plateau appears for a small critical field.

Figure 6

(Color online) Local densities of holes and spins in the ground state of the $m=\delta$ phase computed on a ladder with $L=48$ and 6 holes, $t_{\perp }=t_{\Vert }=1.0$, and $J_{\perp }=0.5$. As $J_{\Vert }$ increases from 0 to $J_{\Vert }$ hole pairs form in the isotropic limit.

Figure 7

Schematic representation of bosonized bands in the coupled-band models.

Figure 8

(Color online) Absolute value of superconducting correlations for zero and a finite magnetization in an isotropic doped ladder with $\delta =0.063$. Notations are “Rg” for rung, “Lg” for leg, “$\sim 0$” for numerically irrelevant signal, “$\mathit{exp}$” for exponentially decaying correlations, and “$\pi m$” and “$2k_F$” for the wave vectors. Note that the $S^z=1$ “$2k_F$” oscillations are smoothed out by taking the absolute value. $M=1000$ states were kept with discarded weight of order ${10}^{-6}$.

Figure 9

(Color online) Same correlations as Fig. 8 but using the Hubbard model with $\delta =0.063$ and $U∕t=8.0$. $M=1600$ states were kept with discarded weight of order ${10}^{-6}$.

Figure 10

(Color online) Verification of the $q=\pi m$ relation of FFLO-like pairing at finite magnetization in an isotropic ladder with $J∕t=0.5$.

Figure 11

(Color online) Convergence with the number of kept states $M$ for a fixed length $L=64$. The plotted superconducting correlation is the singlet one and parameters are $m=0.094$, $\delta =0.063$, and $J∕t=0.5$ which corresponds to the phase above the $m=\delta$ plateau and below the superconducting upper critical field $H_c$.

Figure 12

(Color online) Convergence with size $L$ for the same parameters as in Fig. 11, $M$ being fixed.

Figure 13

(Color online) Extrapolation (a) of fitted exponent as a function of discarded weight $w$ and (b) of the inverse of the number of kept states $M$ for different lengths $L$. Extrapolated results from (b) are tentatively extrapolated vs $L$ in (c). Large error bars occur when $L$ is too small (for $L=32$) and because $M$ is too small (for $L=128$). Parameters are given in the caption of Fig. 11.

Figure 14

(Color online) Normalized four-point spin correlations $T\left(x\right)=⟨S_2^{+}\left(x\right)S_1^{+}\left(x\right)S_2^{-}\left(0\right)S_1^{-}\left(0\right)⟩$ for various magnetizations in the isotropic $t\text{−}J$ ladder with $J∕t=0.75$ and $\delta =0.063$. Their decay exponent is $2K_{s+}^{-1}$ which gives access to $K_{s+}\left(H\right)$. Correlations were computed on a system with $L=64$ and $M=1200$ states kept.

Figure 15

(Color online) $K_{s+}(H∕{\Delta }_s)$ from DMRG computations on a system with parameters of Fig. 14.

Figure 16

(Color online) Magnetization curves for the isotropic $t\text{−}J$, Hubbard, and noninteracting systems. Irrational plateaus are clearly visible at $m=\delta$. $H_c$ represents the superconducting critical field. The energy scale $J$ is defined as $4t^2∕U$ in the Hubbard model. The noninteracting system curve has been rescaled $(H\to H∕4)$ for clarity.

Figure 17

Phase diagram for the isotropic two-leg $t\text{−}J$ ladder in the $(H,\delta )$ plane for $J∕t=0.35$. The large FFLO phase is delimited by the upper limit of the $m=0$ plateau and the critical field $H_c$. Note also that in this part of the diagram, the $m=\delta$ phase is not expected to be a superconducting phase but rather metallic.

Figure 18

(Color online) (a) Two-particle gap and (b) pairing energy for isotropic ladders as a function of magnetization. An anomaly in the two-particle gap at $m=\delta$ is clearly visible, while the pairing energy remains finite up to the superconducting-metallic state transition. $m_c$ represents the magnetization corresponding to the superconducting critical field $H_c$.

Figure 19

(Color online) $H_c$, $H_P$, and $m=\delta$ plateau on an isotropic two-leg $t\text{−}J$ ladder doped with two holes as a function of $J∕t$ (this corresponds to the $\delta \to 0$ limit of the phase diagram of Fig. 17). The curve with squares corresponds to the upper limit of the $m=0$ plateau and the lower limit of the $m=\delta$ plateau, while the curve with circles to the upper limit of the $m=\delta$ plateau. The FFLO phase is delimited by the curve with circles and the curve with triangles. Results are extrapolated from DMRG results on systems with $L=32,48,64$. For small $J∕t$, a Nagaoka phase is found but because of strong finite-size effects, we do not discuss this part of the diagram. Lines are guides for the eyes.