Thermodynamic properties of the $S=1/2$ twisted triangular spin tube
Takuya Ito Chihiro Iino Naokazu Shibata








 ito@cmpt.phys.tohoku.ac.jp



 Department of Physics, Tohoku University, Sendai 980-8578, Japan
 





             


Thermodynamic properties of the twisted three-leg spin tube under magnetic field are studied by the finite-T density-matrix renormalization group method. The specific heat, spin, and chiral susceptibilities of the infinite system are calculated for both the original and its low-energy effective models. The  obtained results show that the presence of the chirality is observed as a clear peak in the specific heat at low temperature and the contribution of the chirality dominates the low-temperature part of the specific heat as the exchange coupling along the spin tube decreases. The peak structures in the specific heat, spin, and chiral susceptibilities are strongly modified near the quantum phase transition where the critical behaviors of the spin and chirality correlations change. These results confirm that the chirality plays a major role in characteristic low-energy behaviors of the frustrated spin systems.




75.10.Jm, 75.10.Pq, 75.40.-s

                             

                              
Thermodynamic properties of the S=1/2 twisted triangular spin tube
    Naokazu Shibata
    December 30, 2023
==================================================================








§ INTRODUCTION


Geometrical frustration as a source of diverse and complex behaviors of electrons is an interesting research subject of condensed matter physics<cit.>.
In quantum spin systems the chiral degrees of freedom at low temperatures play an important role in complicated behaviors<cit.>. The antiferromagnetic Heisenberg models composed of weakly coupled triangular units are typical systems in which three S=1/2 spins in each triangular unit form a fourfold ground-state multiplet consisting of spin and chiral degrees of freedom<cit.>. The appearance of chiral degrees of freedom at low temperature is reflected in the characteristic behaviors of thermodynamic quantities and its critical correlation finally causes various quantum phase transitions at zero temperature.

The twisted three-leg spin tube shown in Fig. <ref> is one of the simplest one-dimensional systems whose low-energy states are described by both spin and chirality of the triangular unit<cit.>. The degeneracy in each triangular unit is lifted by the exchange coupling along the spin tube and many quantum states including chirality liquid, spin liquid, dimer state, and spin-gap state are realized depending on the exchange coupling and the magnetic field as shown in Fig. <ref><cit.>.

In this paper we study thermal properties of the twisted three-leg spin tube whose low-energy states are described by spin and chiral degrees of freedom.
We calculate the partition function of the system by using the finite-T density-matrix renormalization group (DMRG) method <cit.> and directly determine the specific heat, spin, and chiral susceptibilities using the eigenvectors of the maximum eigenvalue of the quantum transfer matrix<cit.>. The obtained results for the infinite system show that the presence of the chirality is observed as a peak structure in the specific heat and the contribution from the chirality becomes dominant at low temperature as the exchange coupling along the spin tube decreases. The peak structures in the specific heat and the susceptibilities at low temperatures are strongly modified near the quantum phase transitions where the critical behaviors in the ground state change. 










§ MODEL AND METHOD

The Hamiltonian of the twisted three-leg spin tube is defined as

    H    =     J_⊥∑_j=1∑_i=1^3S_i,j·S_i+1,j - h_z∑_j=1S^z_ Tj
       +    J_∥∑_j=1∑_i=1^3(S_i,j·S_i,j+1 + S_i,j·S_i+1,j+1),

where S_i,j is the S=1/2 spin operator at rung i and leg j,
and J_⊥ (J_∥) is the antiferromagnetic exchange coupling of
intra (inter) triangles. The second term represents the Zeeman energy under the external magnetic field h_z. The total spin of the jth unit triangle is defined by
S_ Tj = ∑_i=1^3 S_i,j.
Here we use the periodic boundary conditions for
the perpendicular direction to the tube axis, and we regard J_⊥ as energy unit throughout this paper.

To calculate thermodynamic quantities of the Hamiltonian
we use the finite-T DMRG method<cit.>.
This method enables us to iteratively expand the quantum transfer matrix<cit.> in the imaginary time direction and obtain the partition function of the infinite system from the maximum eigenvalue of the quantum transfer matrix within a desired accuracy controlled by the number of keeping states, m, in the calculation.
We treat each unit triangle of the spin tube as a single site and keep up to 300 basis states (m=300) in each block of the quantum transfer matrix.
The accuracy of the results depends on the number of keeping states m and the Trotter number N_ T of the quantum transfer matrix used in the calculation<cit.>. In the present calculation the error coming from finite m is in a range of O(10^-5) to O(10^-12) and the error from the finite Trotter number is removed by using an extrapolation of N_ T in a range of N_ T=50 – 200. Hereafter we set k_B as 1.





§ EFFECTIVE HAMILTONIAN

We start from the decoupling limit J_∥ = 0, where the unit triangles perpendicular to the tube axis are independent of each other.
The eigenstates in this limit are obtained by diagonalizing an 8× 8 Hamiltonian of three S=1/2 spins in the unit triangle and they are shown to form two fourfold multiplets in the absence of a magnetic field as in Fig. <ref>(a). The higher multiplet has total spin S_ Tj=3/2 and the lower one is composed of S_ Tj=1/2 states with chiral degrees of freedom, which are represented by pseudospin τ^z=± 1/2. The degeneracy in each multiplet is partially lifted by Zeeman energy and only chiral degrees of freedom remain in the ground state under magnetic field until a large Zeeman energy fully polarizes the spins.

The exchange interaction along the spin tube J_∥ generates matrix elements between the eigenstates of unit triangles, and the effective Hamiltonian acting on the chiral degrees of freedom is derived within the first-order perturbation with respect to J_∥ as

    H_XY =-J_∥/3∑ _j=1 (τ ^+_jτ ^-_j+1 +  H.c.),

where τ^+_j(τ^-_j) is the chirality pseudospin-1/2 raising (lowering) operator<cit.>.
In the presence of a finite spin excitation gap in the magnetization plateau phase, the spin excitations are suppressed and a chirality liquid state is realized at low temperature. Thus the low-energy thermodynamic properties in the magnetization plateau phase are expected to be characterized by the chirality effective Hamiltonian H_XY. This is confirmed by comparing the specific heat of both the original and effective Hamiltonians shown in Fig. <ref>(b), where we find almost identical results below T/J_∥∼ 1, which means the effective Hamiltonian of chiral degrees of freedom reproduces the original specific heat at low temperatures.

With the increase of the magnetic field, the energy gap of the spin excitation decreases and finally vanishes at the transition to the spin and chirality liquid state. This change in the ground state is due to the large Zeeman shift of the S_ Tj^z=3/2 state that modifies the effective Hamiltonian for the chiral degrees of freedom as

    H_ t `-J = ∑ _j=1∑ _σ = ↑ ,↓[ -t(c^†_σ jc_σ j+1   +    H.c.) 
    - J(τ ^+_jτ ^-_j+1 +  H.c.) 
       +    Vn_jn_j+1 + μ n_j],

where c^†_σ j (c_σ j) and n_j are hard-core boson creation (annihilation) operator with chirality pseudospin-1/2 and its number operator, respectively<cit.>. The coefficient of each term is given as t=1/2J_∥, J=1/3J_∥, V=2/3J_∥, and μ = -2J_∥+(h_z - 3/2J_⊥).
This effective Hamiltonian corresponds to the extended t-J model with XY exchange coupling. The field-induced phase transition from the plateau phase to the spin and chirality liquid phase then corresponds to the metal-insulator transition in the t-J model<cit.>.
The equivalence between the effective t-J model and the original spin tube in the low-energy region is shown in Fig. <ref>(d), where we find that the specific heat of the original spin tube is well reproduced by the effective Hamiltonian at low temperatures. This agreement between the two models is seen even for J_∥/J_⊥=0.2 as shown in Fig. <ref>(c).



§ RESULTS



 §.§ Chirality liquid state

To clarify the thermodynamic behaviors of chirality as effective degrees of freedom of the unit triangle, we first investigate the specific heat C of the twisted three-leg spin tube in the region of the plateau phase (P_ ch in Fig. <ref>).
The results for J_∥/J_⊥=0.05 under various magnetic field are presented in Fig. <ref>(a), where we find at least two peaks, one of whose position at lower temperature is independent of the magnetic field. The presence of the magnetic-field-independent peak at low temperature is consistent with the expectation that the chirality has the smallest energy scale given by the effective Hamiltonian of Eq. (<ref>). The other peaks at higher temperature are magnetic field dependent and their positions correspond to the excitation energies to excited states of different S_ Tj^z and S_ Tj shown in Fig. <ref>(a).
The specific heat C_ XY of the effective Hamiltonian consisting of only chirality operators in Eq. (<ref>) is represented by the dashed line in the same figure whose low-temperature asymptotic behavior is consistent with the prediction of conformal field theory (CFT) (C=π T/2J_∥),<cit.> as 
shown in the inset of Fig.4 (a). The almost identical results between C and C_ XY
at low temperatures show that the low-energy properties are described by chiral degrees of freedom. This coincidence of the specific heat also indicates that the energy scale of chirality is well separated from that of spin excitations as is shown in the temperature dependence of the entropy in Fig. <ref>(b), where the plateau of S∼ln2 coming from the chiral degree of freedom appears in the intermediate temperature range between the lowest and the second lowest peaks of the specific heat around h_z/J_⊥=0.8, while different plateaus of ln 4 and ln 3 appear near h_z/J_⊥=0.2 and h_z/J_⊥=1.4 that correspond to the lower and upper boundary of the plateau phase, where different local spin states with S_ Tj^z=-1/2 and 3/2 are included in the ground state, respectively. The presence of the large spin excitation gap around h_z/J_⊥=0.8 is also confirmed in the temperature dependence of the spin susceptibility shown in Fig. <ref>(c), where we find vanishing spin susceptibility below the intermediate-temperature region T/J_⊥∼ 0.1. From these results we conclude that the presence of the chirality in the spin tube is observed in the specific heat at low temperature.










 §.§ Spin and chirality liquid state

We next examine the interplay between the spin and chirality in the TLL2 phase at higher magnetic field, where both spin and chiral excitations are described by gapless collective excitations. Since the Zeeman energy enhances the probability of the spin-polarized S_ Tj^z=3/2 states (holes in the effective t-J model), its effect on the chiral excitations is expected. Here we study the effective Hamiltonian Eq. (<ref>) of the spin tube to see how the specific heat of chirality is modified with the increase of the magnetic field.

At h_z/J_⊥∼ 1.49, the ground state is in the plateau phase (P_ch) and we see a single peak at low temperature in Fig. <ref>(a), which continuously connects to the specific heat of the chirality XY model, Eq. (<ref>), at lower magnetic field shown in Fig. <ref>(a). After the phase transition at h_z/J_⊥=1.5, the number of spin-polarized S_ Tj=3/2 states starts to increase with the magnetic field. This increase of the spin-polarized states enhances a gapless spin excitation at finite temperature and a small increase of the specific heat appears at low temperatures T/J_⊥∼ 0.002. Further increase in magnetic field modifies the peak structure of chiral excitations and it finally falls into decay toward lower temperature. The peak structure of chiral excitations finally seems to vanish at h_z/J_⊥=1.65, which corresponds to the transition field to the fully spin-polarized state. This field dependence of the specific heat is clearly seen in Fig. <ref>, which shows a contour map of C around the TLL2 phase.

To understand the magnetic field dependence of the specific heat,
we next investigate the spin and chiral susceptibilities.
Figures <ref>(b) and <ref>(c) show the temperature dependence of the spin and chiral susceptibilities, χ_s and χ_c. We find that the low-temperature chiral susceptibility monotonically increases with the increase of magnetic field up to the transition field of h_z/J_⊥=1.65. This enhancement of the chiral susceptibility is due to the increase in the number of spin-polarized S_ Tj=3/2 states because the effective interaction between the chirality operators monotonically decreases with the increase of the number of spin-polarized S_ Tj=3/2 states and this decrease of the interaction results in the enhancement of the chiral susceptibility.
A similar sharp increase appears in the spin susceptibility at the transition to the TLL2 phase at h_z/J_⊥=1.5 at low temperature. This increase of the spin susceptibility originates from the appearance of low-energy spin excitations of the spin-polarized S_ Tj=3/2 states. The increase in the number of spin-polarized S_ Tj=3/2 states first enhances the possibility of the exchange process and the energy of the spin excitations increases, which suppresses the spin susceptibility at low temperature. However, further increase of the magnetic field suppresses the exchange processes between S_ Tj^z=3/2 and 1/2 states and the energy scale of the spin excitations decreases, which enhances the spin susceptibility near the phase transition to the fully spin-polarized states.
Thus the behaviors of the spin and chirality are significantly modified by the presence of the spin-polarized S_ Tj=3/2 state, whose number is controlled by the magnetic field.



 §.§ Up-up-down state


We finally study how the exchange coupling between the neighboring triangular units modifies the thermodynamic behaviors of the spin tube.
We start from the chirality liquid state in the magnetization plateau phase to see the effect of interactions between the chiral degrees of freedom. The J_∥ dependence of the specific heat of the triangular spin tube is shown in Fig. <ref>(a). In the weak-coupling region of J_∥/J_⊥ < 0.25, we find a clear peak at low temperature where magnetic susceptibility is suppressed. The released entropy associated with this peak formation is close to ln 2 per unit triangle as shown in Fig. <ref>(b), which shows this peak in C is coming from chiral degrees of freedom. With increasing J_∥/J_⊥, the peak structure in C shifts to higher temperature and merges with the main peak at higher temperature, whose origin is the spin degrees of freedom of the original spins in triangular units. The lower peak is deformed into a shoulder structure at around J_∥/J_⊥=0.25 and then
a small peak is formed again. This peak shift and transient behavior of the specific heat implies a reconstruction of the excitation spectrum at J_∥/J_⊥∼ 0.25, which corresponds to the phase transition from the P_ch phase to the up-up-down (UUD) phase in Fig. <ref>.
In particular the vanishing specific heat at low temperatures means the disappearance of low-energy chiral excitations characterizing the chirality liquid phase. Because the spin excitation gap is always present in the magnetization plateau phase as is seen in a vanishing low-temperature spin susceptibility in Fig. <ref>(c), the low-energy thermodynamic properties are scaled by finite energy gaps of both the spin and chiral excitations. This change in the low-energy chiral excitation is consistent with the presence of the phase transition from chirality liquid to UUD phase in the ground state<cit.>.






§ SUMMARY


In this paper we have studied thermal properties of the twisted three-leg spin tube by using the finite-T DMRG method. The obtained specific heat in the magnetization plateau state shows a clear peak structure independent of the magnetic field, indicating the presence of the chiral degrees of freedom at low temperature. With the increase in magnetic field, spin-polarized local states with no chirality are included in the ground state with the phase transition to the TLL2 phase. The spin susceptibility sharply increases at the transition at low temperature and then decreases with the increase in the number of spin polarized local states up to around half the number of unit triangles. The chiral susceptibility gradually increases with the increase in the number of spin-polarized states and shows diverging behavior in the limit of low temperature at the transition to the fully spin-polarized state.
We have also studied the effect of interactions J_∥ between the triangular unit in the spin tube. With the increase of the exchange interaction between the spins in neighboring triangular units, the temperature dependence of the specific heat shows that the energy scale of the chiral excitations increases and the excitation gap finally appears with vanishing chiral susceptibility at low temperatures, consistent with the transition to the UUD ordered phase in the ground state. These results confirm that the chirality is an important physical quantity characterizing low-energy thermal properties of the twisted spin tube as a highly frustrated quantum spin system.


This work was supported by JSPS (JP) Grant No. 26400344.



apsrev4-1





