Possibility of Two-component Tomonaga-Luttinger Liquid and Magnetization Cusp in Frustrated Integer-spin Antiferromagnetic Tubes with Uniform Field

Using field theory techniques, we study uniform-field effects for integer-spin antiferromagnetic (AF) tubes (cylinder type spin systems) in the weak interchain-coupling regime. When the number of chains is odd in a tube, the AF interchain coupling yields the frustration. In the case without external fields, a nonlinear sigma model (NLSM) method shows that the lowest excitations in the frustrated tubes are doubly degenerate in addition to the magnon triplet. This implies that a sufficiently strong uniform field induces a two-component Tomonaga-Luttinger liquid (TLL), due to the condensation of the doubly degenerate lowest magnons. However, if multi magnon bands are condensed simultaneously, it is generally possible that interactions among the resulting massless modes are relevant and they break the TLL phase. Combining some field theory arguments with the NLSM, we predict that the two-component TLL critical state survives despite of the interactions. Furthermore, we predict that when the uniform field is so strong that the second lowest magnons are also condensed, the two-component TLL is destroyed and a new one-component TLL emerges. This quantum phase transition may be detected as a magnetization cusp. In one-dimensional (1D) gapped spin systems with spin rotation symmetry, generally a uniform magnetic field, which makes the rotation symmetry reduce to a U(1) one, removes the gap and induces a magnoncondensed critical phase regarded as a TomonagaLuttinger liquid (TLL) [1] with central charge c = 1. Zamolodchikov’s c theorem [2] actually supports that a one-component TLL state with c = 1 tends to occur in U(1) symmetric 1D systems. However, it is known that frustration effects often break such conventional scenarios. In this short paper, we focus on the field-induced critical phases in frustrated integer-spin tubes, and discuss the possibility of the existence of an unconventional two-component TLL state with c = 1 + 1 and a quantum phase transition. Here, the tube means a ladder with a periodic boundary condition along the interchain (rung) direction. For more details, refer to Ref. [3]. The N-leg spin tube Hamiltonian is given by Ĥ = N ∑ n=1 ∑ j [J~Sn, j ·~Sn, j+1 + J⊥~Sn, j ·~Sn+1, j −HS n, j], (1) where ~Sn, j is the integer-spin operator on site j in the nth chain, J(> 0) and J⊥ are the exchange couplings along the chain and rung directions (~SN+1, j = ~S1, j), respectively, and H is the external uniform field. When the leg number N is odd and the rung coupling is antiferromagnetic (AF), i.e., J⊥ > 0, the system has the frustration along the rung. Our interest is in such frustrated tubes. For N-leg integer-spin systems, we can construct an extended Sénéchal’s nonlinear sigma model (NLSM) method [4], which first maps each chain in the N-leg system to a NLSM, and then treats the rung coupling terms perturbatively [3]. Although this method would be efficient especially in the weak rung-coupling regime, we believe that the results are valid even in an intermediate coupling regime (|J⊥| ∼ J), at least qualitatively. The method shows that the rung coupling induces the hybridization among the massive magnon bands in neighboring chains, and as a result, the magnon-band splitting occurs with a finite mass gap surviving. In tubes, each resultant band has a wave number k for the rung direction. Furthermore, it also shows that the lowest magnon excitations in frustrated tubes possess a two-fold degeneracy in addition to the triple one of the spin-1 magnon triplet. This extra degeneracy is guaranteed by the πrotation symmetry with respect to the center axis of the cross section of the tube. Note that the symmetry is absent in ladders. Here, recall that (as mentioned above) a sufficiently strong field H yields the condensation of Sz = 1 magnons. Therefore, one can immediately expect that a two-component TLL state with c = 1 + 1 appears as a result of the condensation of the lowest doubly degenerate magnons, in the frustrated tubes with a strong field H. However, such a critical phase may actually be broken down by the interactions between the two massless modes in the condensed state. To discuss the interaction effects, we use some field theory approaches in addition to the NLSM. Applying the Ginzburg-Landau (GL) analysis for the field-induced critical phase in the spin-1 AF chain [5] to our NLSM theory in frustrated tubes, we obtain the following lowenergy Lagrangian for the two-component TLL expected above: LTLL = ∑q=±p K 2v [(∂tθq) 2 − v(∂xθq)]. The indices q = ±p represent the rung-direction wave number of each massless mode in the two-component TLL, K is the TLL parameter [6], and v is the velocity of the massless modes. The bosonic field θk(x) is associated with the spin operator ~Sn, j as follows: S̃ 0,uni ∼ ∑ q=±p [a∂xφq/ √ π + M +Cu cos( √ 4πφq + 2πMx/a)], S̃ q,stag ∼C1se √ πθq [1 +C2s sin( √ 4πφq + 2πMx/a)], (2) where S̃α k, j = S̃ α k,uni +(−1) jS̃α k,stag is the “Fourier transformation” of the original spin Sα n, j, the field φq is the dual of θq, a is the lattice spacing (x = j×a), M is proportional to the magnetization per site, and Cu,1s,2s are the nonuniversal constants. The final terms in the right-hand side in Eqs. (2) are expected from the study of 2-leg spin-1/2 ladder [7]. Note that we have never taken into account the interactions between the two TLLs, i.e., fields (θp,φp) and (θ−p,φ−p), yet. In the bosonization or conformal field theory plus renormalization group picture [1], the relevant interaction terms are always represented by a product of some vertex operators, e.g., e±i √ 4πφq and e±i √ πθq [8]. Relying on the symmetries [9] in frustrated tubes, we can restrict possible interaction forms in the low-energy theory as follows. From S q,stag in Eq. (2), the U(1) rotation around z axis corresponds to the shift θq → θq + const. It means that the low-energy theory must not have any vertex operators including eiC1θq or eiC2(θp+θ−p) (C1,2 is a multiple of √ π). Moreover, Eq. (2) tells us that the one-site translation along the chain corresponds to φq → φq + √ πM and θq → θq + √ π . We hence see the absence of all the vertex terms with eiC1φq or eiC2(φp+φ−p), except for the case where M is specific commensurate values [9]. The remaining possible terms are only eiC1(θp−θ−p) and eiC1(φp−φ−p). Using the translation symmetry along the rung and the π-rotation one, we can predict that these vertex terms are also prohibited in the low-energy theory. Thus, we can conclude that a two-component TLL emerges when the lowest doubly degenerate magnons are condensed in the frustrated tubes. (The above argument also predicts the existence of a c = 1 state in non-frustrated systems.) H