Quantum phase diagram of a frustrated spin-1/2 system on a Trellis Ladder

TL;DR

This paper maps the quantum phase diagram of a frustrated spin-1/2 trellis ladder model using DMRG and spin wave analysis, revealing various short-range magnetic phases and applying findings to CaV$_2$O$_5$.

Contribution

It introduces a comprehensive phase diagram for a complex frustrated spin system and compares DMRG with spin wave results, highlighting the dominance of the rung interaction $J_3$.

Findings

Identification of short-range stripe collinear phase at small $J_2$.

Discovery of non-collinear quasi-long range phase at large $J_2/J_3$.

Prediction of magnetic specific heat variation with external field.

Abstract

We study an isotropic Heisenberg spin-1/2 model on a trellis ladder which is composed of two $J_1-J_2$ zigzag ladders interacting through anti-ferromagnetic rung couplings $J_3$. The $J_1$ and $J_2$ are ferromagnetic zigzag spin interaction between two legs and anti-ferromagnetic interaction along each leg of a zigzag ladder. A quantum phase diagram of this model is constructed using the density matrix renormalization group (DMRG) method and linearized spin wave analysis. In small $J_2$ limit a short range stripe collinear phase is found in the presence of $J_3$, whereas, in the large $J_2/J_3$ limit non-collinear quasi-long range phase is found. The system shows a short range non-collinear state in large $J_3$ limit. The short range order phase is the dominant feature of this phase diagram. We also show that the results obtained by DMRG and linearized spin wave analysis show similar phase boundary between stripe collinear and non-collinear short range phases, and the collinear phase region shrinks with increasing $J_3$. We apply this model to understand the magnetic properties of CaV$_2$O$_5$ and also fit the experimental data of susceptibility and magnetization. The variation of magnetic specific heat capacity as function of external magnetic field is also predicted. We note that $J_3$ is a dominant interaction in this system, whereas $J_1$ and $J_2$ are approximately half of $J_3$.

Equations

$$\mathrm{H}$$

$$C(r)\propto \exp \left(\frac{-r}{\xi }\right).$$

$$C(r)\propto \exp \left(\frac{-r}{\xi }\right)\sin \left(\theta r+c\right).$$

$$C(r)\propto r^{-\kappa }\sin \left(\theta r+c\right).$$

$$H=(2J_1+2J_2-J_3)N{s}^2+\sum _{k}s[(2J_1(\cos k-1)$$

$$+2J_2(\cos 2k-1)+J_3)(a_k^{+}{a}_k+b_k^{+}{b}_k)$$

$$+J_3(a_k^{+}{b}_{-k}^{+}+a_k{b}_{-k})].$$

$$a_k=u{c}_k-v{d}_k^{+},$$

$$b_{-k}^{+}=-v{c}_k+u{d}_k^{+},$$

$$H=(2J_1+2J_2-J_3)N{s}^2$$

$$+\sum _k{\omega }_k(c_k^{+}{c}_k+d_k^{+}{d}_k+1),$$

$$ϵ=(J_1+J_2-\frac{J_3}{2})S(S+1)$$

$$+\sum _k\frac{s}{2\pi }{\int }_0^{\pi }\sqrt{J_k^2-J_3^2}dk$$

$$S_{Aj}^z=s-a_j^{+}{a}_j,$$

$$S_{Aj}^{+}=\sqrt{(2s-a_j^{+}{a}_j)}{a}_j,$$

$$S_{Aj}^{-}=a_j^{+}\sqrt{(2s-a_j^{+}{a}_j)},$$

$$S_{Bj}^z=-s+b_j^{+}{b}_j,$$

$$S_{Bj}^{+}=a_j^{+}\sqrt{(2s-a_j^{+}{a}_j)},$$

$$S_{Bj}^{-}=\sqrt{(2s-a_j^{+}{a}_j)}{a}_j,$$

$$S_{Aj}^z=s-a_j^{+}{a}_j,$$

$$S_{Aj}^{+}=\sqrt{2s}{a}_j,$$

$$S_{Aj}^{-}=\sqrt{2s}{a}_j^{+},$$

$$S_{Bj}^z=s-b_j^{+}{b}_j,$$

$$S_{Bj}^{+}=\sqrt{2s}{b}_j^{+},$$

$$S_{Bj}^{-}=\sqrt{2s}{b}_j.$$

$$H=(2J_1+2J_2-J_3)N{s}^2+\sum _{j}s[[J_1(a_j^{+}{a}_{j+1}+b_j^{+}{b}_{j+1})$$

$$+J_2(a_j^{+}{a}_{j+2}+b_j^{+}{b}_{j+2})+J_3{a}_j{b}_j+h.c.]$$

$$-(J_1+J_2)(a_j^{+}{a}_j+b_j^{+}{b}_j)-J_1(a_{j+1}^{+}{a}_{j+1}+b_{j+1}^{+}{b}_{j+1})$$

$$+J_2(a_{j+2}^{+}{a}_{j+2}+b_{j+2}^{+}{b}_{j+2})+J_3(a_j^{+}{a}_j+b_j^{+}{b}_j)].$$

$$a_j=\sum _k\exp (-ikj)a_k,$$

$$a_j^{+}=\sum _k\exp (ikj)a_k^{+}.$$

Full text

I Introduction

In the last couple of decades frustrated low dimensional quantum magnets have been intensively explored in search of various exotic phases like spin fluid with quasi-long-range order (QLRO)  Okamoto and Nomura (1992); Haldane (1982); White and Affleck (1996); Chitra et al. (1995); Kumar et al. (2015); Soos et al. (2016), spin dimer with short-range order (SRO)  Majumdar and Ghosh (1969a); Shastry and Sutherland (1981); Haldane (1982); White et al. (1994); White and Affleck (1996), vector chiral  Chubukov (1991); Furukawa et al. (2012), multipolar phases  Zhitomirsky and Tsunetsugu (2010); Chubukov (1991); Furukawa et al. (2012); Parvej and Kumar (2017) etc. These phases arise in presence of some specific types of spin exchange interactions which may enhance the quantum fluctuations in low-dimensional frustrated systems like one dimensional (1D) spin chains realized in materials, LiCuVO4  Mourigal et al. (2012), Li2CuZrO4  Drechsler et al. (2007), Li2CuSbO4  Dutton et al. (2012a), (N2H5)CuCl3 Maeshima et al. (2003) etc., and quasi-1D spin ladders manifested in form of SrCu2O3 Sandvik et al. (1996), (VO)2P2O7 Johnston et al. (1987); Dagotto and Rice (1996) etc. Frustrated twisted ladders are also realized in materials like Ba3Cu3In4O12 and Ba3Cu3Sc4O12  Dutton et al. (2012b); Kumar et al. (2013); Volkova et al. (2012); Koteswararao et al. (2012). Majority of the 1D frustrated magnetic systems mentioned above are modelled by simple $J_{1}$-$J_{2}$ chain  Majumdar and Ghosh (1969a, b); Shastry and Sutherland (1981); Tonegawa and Harada (1987); Kuboki and Fukuyama (1987); Hamada et al. (1988); Okamoto and Nomura (1992); Chitra et al. (1995); Bursill et al. (1995); White and Affleck (1996); Allen and Sénéchal (1997); Itoi and Qin (2001); Hase et al. (2004); Lu et al. (2006); Drechsler et al. (2007); Kumar et al. (2015); Soos et al. (2016). This model can explain the gapless spin fluid  Okamoto and Nomura (1992); Chitra et al. (1995), gapped dimer  Majumdar and Ghosh (1969a); White and Affleck (1996), gapped non-collinear  Chitra et al. (1995); White and Affleck (1996); Kumar et al. (2015); Soos et al. (2016) and decoupled phases  Kumar et al. (2010a).

In fact many of these 1D systems like LiCuVO4 Mourigal et al. (2012), Li2CuZrO4 Drechsler et al. (2007) show three dimensional ordering at low temperature; therefore, interchain couplings are considered to understand the interesting physics below the three dimensional ordering temperature. However, there are materials with effective spin interactions confined to quasi-1D ladder like structure e.g., SrCu2O3 Sandvik et al. (1996), (VO)2P2O7  Johnston et al. (1987); Dagotto and Rice (1996), CaV2O5, MgV2O5  Korotin et al. (1999, 2000) etc. These systems have antiferromagnetic (AFM) spin exchange interactions along both legs and rungs, and there is also a weak interaction between two adjacent ladders. The ground state (gs) of these systems is a gapped SRO phase  White et al. (1994). The coexisting of spin gap and long range magnetic order in the ladder compound LaCuO2.5 is explained considering interladder coupling  Normand and Rice (1996); Troyer et al. (1997).

The 1D $J_{1}-J_{2}$ system, in large $J_{2}$ limit, is called zigzag ladder Chitra et al. (1995), where two chains are coupled through zigzag bonds, for example LiCuV$O_{4}$  Mourigal et al. (2012). The isolated ladders like zigzag and normal ladders have been extensively studied White and Affleck (1996); White et al. (1994); Almeida et al. (2007); Vekua et al. (2003); Agrapidis et al. (2017); however, the effect of interladder coupling on these ladders is rarely studied. Networks of the coupled zigzag ladders can form a trellis lattice like structure as shown in Fig. 1. The trellis lattice is composed of a number of zigzag ladders coupled through normal rung bonds; alternatively, we can assume coupled normal ladders interacting through zigzag like bond interactions. In this lattice spin exchange interaction strengths $J_{2}$ and $J_{3}$ are along leg and rung of a normal ladder, respectively, and $J_{1}$ is zigzag bond interaction strength between two ladders as shown in Fig. 1.

In this paper we consider only two coupled zigzag ladders and call it trellis ladder because of its geometry. We also impose periodic boundary condition along the width to mimic the trellis lattice. In various interaction limits, two coupled zigzag ladders can behave like a two-leg honeycomb ladder as considered in ref. Luo et al. (2018), where both $J_{1}$ and $J_{2}$ are AFM, but $J_{3}$ can be either ferromagnetic (FM) or AFM. This system shows two types of Haldane phase for the FM $J_{3}$, and columnar dimer and rung singlet phases in presence of the AFM $J_{3}$. Normand et al. have considered a similar coupled ladders with all three AFM $J_{1}$, $J_{2}$ and $J_{3}$ interactions Normand et al. (1997). For large $J_{2}/J_{1}$ they have noticed dimerized gs, whereas non-collinear (NC) long-range order (LRO) for large $J_{3}$ ($J^{\prime}_{2}$). They have found Néel LRO phase in the small $J_{2}<0.4$ limit. Zinke et al. have shown the effect of interchain coupling on NC gs of $J_{1}-J_{2}$ model Zinke et al. (2009), in a two dimensional geometry. The effect of interladder coupling on spin gap and magnon dispersion is calculated using perturbation theory by Miyahara et al. Miyahara et al. (1998). They also try to model the magnetic susceptibility of SrCu2O3 and CaV2O5 using quantum Monte Carlo and mean field type scaling methods Miyahara et al. (1998). However, the system with FM $J_{1}$, and AFM $J_{2}$ and $J_{3}$ has not been studied in the ladder geometry. In this paper, we consider a spin-1/2 trellis ladder structure, which is composed of two zigzag ladders with FM $J_{1}$ and AFM $J_{2}$, and they are coupled by AFM $J_{3}$ as shown in Fig. 1. Our main focus of this paper is to construct the quantum phase diagram (QPD) and also understand the effect of rung interaction $J_{3}$ on the various exotic phases of zigzag ladder Kumar et al. (2015). We notice that in small $J_{2}/|J_{1}|$ limit, gs has collinear striped (CS) SRO on each zigzag ladder; however, spins on one zigzag ladder are aligned antiferromagnetically with respect to the spins on the other zigzag ladder Maiti and Kumar (2019). The NC spin order sets in for moderate value of $J_{2}$. The presence of QLRO in NC regime at small $J_{3}/J_{2}$ limit is a striking effect of the $J_{3}$. In large $J_{3}$ limit, rung dimer is the dominant gs.

This paper is divided into four sections. In section II the model Hamiltonian and numerical method are explained. The numerical results are given in section III. Linear spin wave analysis and experimental data fitting of CaV_{2}$$O_{5} are given in section IV and section V, respectively. All the results are discussed and summarized in Section VI.

II Model Hamiltonian and Numerical Method

A four-legged spin-1/2 ladder made of two coupled zigzag ladders is considered as shown in Fig. 1. The exchange interactions between spins along the legs and rungs are AFM in nature. The diagonal exchange interactions $J_{1}$ in a zigzag ladder are FM. We can write an isotropic Heisenberg spin-1/2 model Hamiltonian for the trellis ladder system as

[TABLE]

where $l=1,2$ are the zigzag ladder indices. $\vec{S}_{l,i}$ is the spin operator at site $i$ on zigzag ladder $l$. We consider $J_{1}=-1$, and $J_{2}$ and $J_{3}$ are variable AFM exchange interaction strengths. We use periodic boundary condition along the rungs, whereas it is open along the legs of the system.

We use density matrix renormalization group (DMRG) method to handle the large degrees of freedom in our system. This method is a state of art numerical technique for 1D or quasi-1D system, and it is based on the systematic truncation of irrelevant degrees of freedom  White (1992); Hallberg (2006); Schollwöck (2005). We use recently developed DMRG method where four new sites are added at every DMRG step Kumar et al. (2010b). This method while constructing superblock, avoids the old-old operator multiplication which leads to the generation of large number of non-zero but small matrix elements in superblock Hamiltonian. The number of eigenvectors $m$ corresponding to the largest eigenvalues of the density matrix, is kept for the renormalization of operators and Hamiltonian of the system block. We have kept $m$ up to 400 to restrict the truncation error less than $10^{-10}$. We have used system sizes up to $N=300$ to minimize the finite size effect.

III Results

We first present an outline of the QPD which is constructed based on various quantities like correlation function $C(r)$, pitch angle $\theta$ and bond order $C(r=1)$. The detailed numerical and analytical calculations are discussed in the following subsections. For $J_{1}=0$, this system is composed of two isolated normal ladders, and two isolated zigzag ladders for $J_{3}=0$. In $J_{1}=0$ limit, gs shows the formation of singlet dimers along the rungs on the normal ladder White et al. (1994). On the other hand, for $J_{3}=0$ the system shows various phases arising due to the presence of frustration in each zigzag ladder, at different exchange coupling limits. For $J_{2}/|J_{1}|<0.25$, the gs of an isolated zigzag ladder has ferromagnetically ordered spins and gapless excitations. In the intermediate parameter regime, $0.25<J_{2}/|J_{1}|<0.67$, NC order arises in this system with a small finite spin gap Chitra et al. (1995); Soos et al. (2016); Kumar et al. (2015); Kumar and Soos (2013). The system behaves like decoupled AFM chains exhibiting QLRO in spin-spin correlation and gapless excitations in $J_{2}/|J_{1}|>0.67$ limit  Soos et al. (2016). We notice that if two zigzag ladders start interacting with each other through rung coupling $J_{3}$, it immediately opens a spin gap in the system. The spin gap in CS(SRO) phase has been explicitly studied in ref. Maiti and Kumar (2019). In section IV we discuss the linear spin wave analysis of this model. At the end, we apply this model to fit magnetic susceptibility and magnetization of CaV_{2}$$O_{5} in large $J_{3}$ limit. We also predict the specific heat curve at high temperature which can be verified experimentally.

III.1 Quantum Phase Diagram

The quantum phase diagram of Hamiltonian in Eq. 1 is shown in $J_{2}/|J_{1}|$ and $J_{3}/|J_{1}|$ parameter space, and we focus mainly on the phases in presence of $J_{3}$. The resulting phase diagram in Fig. 2 shows two distinct phases: the CS(SRO) and NC spin order. In small $J_{3}$ and $J_{2}<0.25$ limit, individual zigzag ladder retains the FM arrangement of spins; however, the spins on two different zigzag ladders are aligned antiparallelly with respect to each other. Therefore, the gs of the whole system has effective multiplicity $S^{z}=0$. The spin-spin correlation decays exponentially along each zigzag chain. This phase can be called as CS (SRO) phase. As we increase $J_{3}$, the correlation length $\xi$ decreases. The details of this phase have been discussed already in the ref. Maiti and Kumar (2019). At higher $J_{3}$ value, even for $J_{2}<0.25$, NC phase emerges but with small amplitude and $\xi$ in spin spin correlation. For $J_{2}>0.25$, spiral arrangement of spins becomes more prominent for lower $J_{3}$. In the NC regime, C(r) is either QLRO (decay following power law) called as NC(QLRO), or SRO (exponentially decaying) called as NC(SRO) for the small or large $J_{3}$, respectively. The $\theta$ vanishes at the boundary between CS(SRO) and NC(SRO) phases. In Fig. 2, color gradient represents $\theta$ distribution in the parameter space. The red solid line with circles represents the boundary between CS(SRO) and NC(SRO) phases in the gs. The region above the green dotted line with square symbols represents the SRO phase where spin correlation length is confined to its neighbor i.e., $\xi\leq 1$. In the large $J_{3}$ limit, the correlation strength along rung dominates, and it tends to form singlet dimers along the rungs. The dimer phase is characterized by large energy gap, and the spin correlation is confined within the nearest neighbors ($\xi\leq 1$). Interestingly, for large $J_{2}/J_{3}$ limit the gs is in unique NC(QLRO) phase. To best of our knowledge, QLRO phase exits with pitch angle $\theta=\pi$ or $\frac{\pi}{2}$  Kumar et al. (2015); Soos et al. (2016), whereas this system shows QLRO even with $\theta<\frac{\pi}{2}$. NC(QLRO) phase lies below the blue dashed line with diamond symbols. The phase boundary between NC(SRO) and NC(QLRO) phases has large errorbar due to the inability to distinguish between the power law and exponential nature of $C(r)$ in this parameter regime. To verify these different phases $C(r)$, $\theta$, $\xi$ and $C(r=1)$ are studied in detail in the following subsections.

III.2 Spin-spin correlation C(r)

We calculate the longitudinal spin-spin correlation $C(r)=<S^{z}_{0}S^{z}_{r}>$, where $S^{z}_{0}$ and $S^{z}_{r}$ are the $z$-component of the spin operators at the reference site [math] chosen at the middle of a zigzag chain and the site $r$ at a distance $r$ from [math]th spin, respectively. In Fig. 1, the distance $r$ is shown along the same zigzag ladder with bold numerics with respect to the reference site [math], whereas, normal numerics represent distances on the other zigzag leg. We note that in $J_{2}/|J_{1}|<\frac{1}{4}$ limit, all the spins are aligned parallelly on individual zigzag ladder and have short range longitudinal correlation for finite $J_{3}$. $C(r)$ follows an exponential behavior as shown in the Fig. 3(a) for $J_{2}=0.1$, and $J_{3}=0.1,0.3,0.5,0.7$ and $0.9$. We notice that each zigzag ladder shows collinear arrangement of spins as $C(r)>0$, but it decays exponentially with $r$ i.e.,

[TABLE]

The fitting curve represents an exponential function with correlation length $\xi$. Fig. 3(b) shows the $C(r)$ of the same reference spin with the spins on the other zigzag leg. The negative values suggest anti-parallel arrangement of spins relative to the reference spin leg. This behavior of $C(r)$ confirms the stripe nature of spin arrangement on each zigzag ladder. Therefore, we call it striped phase. On further increase in $J_{2}$, $C(r)$ starts to oscillate at higher $J_{3}$ even at the limit $J_{2}<0.25$. For $J_{2}=0.23$, C(r) is shown in Fig. 3(c) for the same set of $J_{3}$ values. We note that NC(SRO) arises for $J_{3}\geq 0.3$. While C(r) for $J_{3}=0.1$ is fitted by Eq. 2, C(r) for other $J_{3}$ can be fitted with the equation below,

[TABLE]

The NC order can be easily noticed at lower $J_{3}$ for $J_{2}>0.25$. For $J_{2}=0.3$, C(r) is shown in Fig. 3(d) and fitted by Eq. 3. We note that $\xi$ decreases with $J_{3}$. For moderate $J_{2}$, the NC phase follows SRO behavior, whereas it shows QLRO in the gs for higher $J_{2}>0.45$ but for small $J_{3}$. The transition between NC(SRO) to NC(QLRO) seems continuous, and hence it is difficult to find an accurate phase boundary. In QLRO regime C(r) is fitted with sinusoidal power law function written as

[TABLE]

In Fig. 4(a) and (b), $C(r)$ are plotted for $J_{2}=0.5$ and $0.7$, respectively, with $J_{3}=0.1,0.5$ and 1.0. For $J_{2}=0.5$, and $J_{3}=0.1$, C(r) fits with power law in Eq. 4 where $\kappa\approx 1$, whereas C(r) follows exponential decay at $J_{3}=0.5$ and 1.0 with $\xi=2.29$ and 1.56, respectively. For $J_{2}=0.7$, and $J_{3}=0.1$ and 0.5, C(r) decays algebraically with $\kappa=1.15$ and 1.37, respectively, but exponentially for $J_{3}=1.0$ with $\xi=1.99$. We notice that the width of the NC(QLRO) region increases with $J_{2}$.

III.3 Pitch angle $\theta$

In the NC phase we calculate pitch angle $\theta$ from the fitting parameter in Eq. 3 and Eq. 4. $\theta/\pi$ is plotted as a function of $J_{2}$ for various values of $J_{3}$, as shown in Fig. 5. $\theta/\pi$ versus $J_{2}$ curves are fitted with function $\theta/\pi=a[1-e^{{b(J_{2}-J^{c}_{2})}}]$, where $a$, $b$ and $J^{c}_{2}$ are the fitting parameters. $J^{c}_{2}$ is the phase boundary point between CS(SRO) and NC(SRO) phases for a given $J_{2}$ and $J_{3}$. $\theta$ increases from 0 to $\pi/2$ with $J_{2}$. The $\theta$ $\approx$ $\pi/2$ region is confined to high $J_{2}/J_{3}$ limit. The variation of $\theta$ is represented by color gradient in the phase diagram in Fig. 2.

III.4 Correlation length $\xi$

The correlation length $\xi$ extracted from fitting Eq. 2 and Eq. 3 is a measure of correlation length in CS(SRO) and NC(SRO) phase, respectively. The nature of $\xi$ in CS(SRO) phase is discussed in ref. Maiti and Kumar (2019). In NC(SRO) regime $\xi$ are plotted as function of $J_{2}$ for $J_{3}=0.5,0.6,0.7,0.8,0.9$ in Fig. 6. The correlation length can be fitted by $\xi=c+dJ_{2}$, where $c$ and $d$ are the fitting parameters. We note that $\xi$ increases with $J_{2}/J_{3}$. Higher value of $J_{2}$ needs more strength in $J_{3}$ to keep the same correlation length in NC(SRO) phase. Surprisingly, this behavior is completely opposite in the case of CS(SRO) phase, where higher $J_{2}$ requires lower $J_{3}$ to

sustain the same correlation length  Maiti and Kumar (2019). When $\xi\leq 1$, dominant correlation strengths become confined within the three nearest neighbors among which the rung bond correlation is dominant over other two bond strengths. In fact $\xi\leq 1$ represents the correlation length within nearest neighbor distance; as per our convention of distance, both $r=1$ and $r=2$ are the nearest neighbors to the reference spin. In this limit, the system behaves like a collection of singlet rung dimers. The varying strength of nearest neighbor bond correlations depending on $J_{2}$ and $J_{3}$ are discussed in the next subsection.

III.5 Nearest neighbor bond correlation $C(r=1)$

It is quite interesting to see the relative strength of nearest neighbor $C(r=1)$ or longitudinal bond order in the parameter space. The magnitude of $C(r=1)$ along the rung $|C^{R}|$, along the leg $|C^{L}|$, and along the zigzag leg $C^{D}$ are shown for $J_{2}/|J_{1}|=0.1,0.4$, and $0.7$ in Fig. 7(a), (c) and (d), respectively. The bonds along three directions are shown in the schematic Fig. 7(b). We notice that $C^{D}$ and $C^{L}$ have positive values for CS(SRO) phase whereas, $C^{L}$ becomes negative for NC phase. In the NC phase $|C^{L}|$ is dominant for small $J_{3}$, but $|C^{R}|$ dominates for $J_{3}>0.08$ and $0.38$ for $J_{2}$ = 0.4 and 0.7,respectively. The effect of $J_{3}$ on $C^{D}$ is weak, and also the magnitude of $C^{D}$ is small. Therefore, we can safely conclude that major contributions of energy come from $|C^{R}|$ and $|C^{L}|$ in NC phase. $|C^{R}|$ increases exponentially with $J_{3}$ and saturates to a value which is nearly equal to 0.25.

IV Linear spin wave analysis

In the CS(SRO) phase spins on the same zigzag ladder are arranged ferromagnetically, whereas spins from different zigzag ladders are arranged antiferromagnetically to each other. We perform the linear spin wave analysis of the Hamiltonian for this phase. We use the Holstein-Primakoff transformation to the Hamiltonian in Eq. 1. The details of the calculation are given in appendix A.

The Hamiltonian can be written in terms of bosonic operators $a_{j}$,$b_{j}$,$a^{+}_{j}$ and $b^{+}_{j}$, where $a_{j}$/$a^{+}_{j}$ and $b_{j}$/$b^{+}_{j}$ correspond to spin up and spin down operators or spins on leg $l=1$ and $l=2$, respectively. We consider only up to quadratic terms. After Fourier transformation, the resultant Hamiltonian can be written as

[TABLE]

The above Hamiltonian can be transformed to diagonal form using the Bogoliubov transformation i.e.,

[TABLE]

where $u^{2}-v^{2}=1$, $u^{2}+v^{2}=\frac{J_{k}}{\sqrt{J^{2}_{k}-J^{2}_{3}}}$ and $2uv=\frac{J_{3}}{\sqrt{J^{2}_{k}-J^{2}_{3}}}$, and $J_{k}=2J_{1}(\cos{k}-1)+2J_{2}(\cos{2k}-1)+J_{3}$. Applying Bogoliubov transformation, we get

[TABLE]

where $\omega_{k}=S(\sqrt{J^{2}_{k}-J^{2}_{3}})$.

The gs energy per bond is given by

[TABLE]

The $\epsilon$ can be minimized using $\frac{d\omega_{k}}{dk}=0$ and we find these conditions; $\cos{k}=\frac{-J_{1}}{4J_{2}}$ and $\cos{k}=\frac{-J_{1}}{4J_{2}}\pm\frac{\sqrt{(J_{1}+4J_{2})^{2}-4J_{2}J_{3}}}{4J_{2}}$. The second condition $J_{3}\leq\frac{(J_{1}+4J_{2})^{2}}{4J_{2}}$ for any real value of $\cos{k}$, gives the phase boundary between CS(SRO) and NC(SRO) phases. This boundary is similar to that found by DMRG calculation.

V Fitting experimental data of CaV2O5

There are many vanadate compounds like CaV2O5, MgV2O5, NaV2O5 etc., which are suspected to behave effectively like two leg ladders coupled by zigzag bonds forming trellis lattice like structure. Among these materials the interladder coupling ($J_{1}$) in CaV2O5 is expected to be ferromagnetic. The LDA+U calculations performed by Korotin et al. Korotin et al. (2000) give an estimation of the $J_{1}$ ,$J_{2}$, and $J_{3}$ exchange interaction strengths as $-28K$, $122K$ and $608K$, respectively. In this compound, $V^{4+}$ ions have one electron in d-orbital and behave like spin-1/2 ions. The experimental magnetic susceptibility $\chi(T)$ is taken from sample 1 and magnetization $M(H)$ is taken from sample 2 of ref. Johnston et al. which are represented by circles in Fig. 8(a) and (b), respectively. The dimer model fitting of susceptibility data deviates significantly from the experimentally observed data. The experimental data is shown as circle and dimer fit is shown by red dashed line in Fig. 8(a). The model Hamiltonian in Eq. 1 is used with $J_{1}=-272K$, $J_{2}=272K$ and $J_{3}=612K$ to fit the experimental data of $M(H)$ and $\chi(T)$. The fitting curve of $\chi(T)$ shown by black solid curve is in excellent agreement with experimental data for $T>160K$. As shown in Fig. 8(b) $M-H$ curve fitted with dimer model shown by red dashed line is quite off at high $H$, whereas our model gives excellent fitting, as shown by the black solid line at $T=200K$. We predict $M-H$ curve at other three different $T=100,300$ and $500K$. We notice the enhancement of $M$ as a function of $T$, which is quite unusual. This behavior of $M-H$ curve can be understood in terms of large singlet-triplet gap. A moderate temperature enhances the possibility to reach higher magnetic state for a given field $H$.

We also predict the magnitude of specific heat $C_{v}$ as a function of $T$ for four values of magnetic field $H=0,10,20,50$ Tesla as shown in Fig.9. The $C_{v}$ has broad peak at $T\approx 235K$. The effect of magnetic field $H$ is small. The $C_{v}$ decreases with $H$, but the suppression of $C_{v}$ is visible only near the peak. Initially CaV2O5 was assumed to be only a dimer system with singlet-triplet energy gap   $660K$  Onoda and Nishiguchi (1996). We use the model Hamiltonian in Eq. 1, and our fittings of $\chi(T)$ and $M(H)$ with same model parameters suggest that $J_{1}$ and $J_{2}$ are only 1/2 of $J_{3}$. It is found that our predicted values of $J_{1}$ and $J_{2}$ are significantly different from the predicted values in ref. Korotin et al. (2000), whereas the value of $J_{3}$ is similar with their calculated value by LDA+U method.

VI Discussion and Conclusions

In this paper we have studied the isotropic Heisenberg spin-1/2 model, given in Eq. 1, on the trellis ladder. The QPD of this model is constructed. The phase boundaries of the QPD are calculated based on the correlation function $C(r)$, pitch angle $\theta$ and correlation length $\xi$ using the DMRG method. Our linear spin wave analysis of this model predicts phase boundary of CS(SRO) and NC(SRO) phases, and it is quite consistent with our DMRG results. We also use this model to fit $\chi-T$ and $M-H$ data of CaV2O5, and understand the temperature $T$ dependence of $M-H$ curves and magnetic field $H$ dependence of $C_{v}-T$ curves.

In fact our lattice system can also be mapped to a two coupled $J_{1}-J_{2}$ Heisenberg spin-1/2 chains. Zinke et al. studied the effect of interchain coupling $J_{3}$ on non-collinear phase in a coupled 2D array of $J_{1}-J_{2}$ spin chains using the coupled cluster theory  Zinke et al. (2009). They showed that the collinear to non-collinear transition point $J^{c}_{2}$ increases with $J_{3}$. However, our model shows that the critical value $J^{c}_{2}$ decreases with $J_{3}$. This inconsistency may be because of the confined geometry or ladder structure in our case. The $J^{c}_{2}$ value at phase boundary of CS(SRO) and NC(SRO) phases decreases with $J_{3}$, and it can also be shown by linear spin wave analysis. As shown in Fig. 5, the variation of $\theta$ for $J_{2}>0.3$ decreases with $J_{3}$ and this trend is consistent with literature  Zinke et al. (2009), and this may happen because of the deconfinement of quasi-particle along rung of the model. In Fig. 2 of QPD the majority of the parameter space is SRO phase which is basically gapped spin liquid phase Mila (2000); Dagotto and Rice (1996); however, for small value of $J_{3}/J_{2}$, an incommensurate (QLRO) phase appears, which is quite unique in this ladder system. The $J_{1}-J_{2}$ spin-1/2 zigzag model in similar parameter space shows either incommensurate (SRO) or decoupled phase  Kumar et al. (2015); White and Affleck (1996). The QLRO in the system may be induced because of dominant effective anti-ferromagnetic interaction along the leg.

We apply this model to understand the magnetic properties of the CaV2O5, and have reliable fitting of the experimental data Johnston et al. . We apply a criterion of simultaneous fitting of both experimental $\chi-T$ and $M-H$ curves. Our best fit suggests that $J_{2}/|J_{1}|$ is close to 1, and $J_{1}$ is approximately $-272K$. For a given $H$, $M(H)$ for this system increases with $T$, whereas in general magnetization decreases with increasing temperature. We notice that in a highly gapped system, higher $T$ allows the system to access the higher magnetic states easily; therefore, it is much easier to magnetize this system at moderate temperature for a given $H$. Our calculated singlet-triplet energy gap is $~{}459K$, whereas dimer model predicts it as   $660K$. The Knight shift and spin-lattice relaxation measurements for CaV2O5 show energy gaps are $464K$ and $616K$, respectively  Iwase et al. (1996). Our predicted energy gap is closer to the Knight shift measurement. The modelling of $\chi(T)$ of CaV2O5 was done by Miyahara et al. using QMC method, and they showed that small $J_{1}$ does not effect the magnetic $\chi(T)$, as shown in Fig.6 of ref. Miyahara et al. (1998). They estimated the value of $J_{1}=45K$, $J_{2}=67K$ and $J_{3}=672K$. Johnston et al. treated this system as collection of dimers, and extracted the value of $J_{3}=667K$ with small $J_{1}$ and $J_{2}$  Johnston et al. . Korotin et al. also calculated the value of $J_{1}=-28K$, $J_{2}=122K$ and $J_{3}=608K$; however, their calculation also assumes other types of interactions  Korotin et al. (2000). Our simultaneous fitting of experimental $\chi-T$ and $M-H$ data also suggests it as dominant dimer with $J_{3}=612K$, but -$J_{1}$ and $J_{2}$ are only about half in magnitude of the $J_{3}$.

In summary, we study the QPD of model Hamiltonian in Eq. 1 on the trellis ladder. We show that $J_{3}$ plays an important role to localize the system. This system shows interesting CS(SRO) and NC(QLRO) which is rare in ladder like structures. This model Hamiltonian is used to fit the experimental magnetic properties of CaV2O5 and we also show that the interaction $J_{1}$ and $J_{2}$ are much larger than earlier predicted values, and $J_{1}$ is ferromagnetic in nature. In many zigzag ladder systems like LiCuVO4 Mourigal et al. (2012), Li2CuZrO4 Drechsler et al. (2007), Li2CuSbO4  Dutton et al. (2012a) etc., where three dimensional ordering occurs at low $T$, this model can be applied to understand the effect of interladder coupling in the system. We have also predicted the $M-H$ and $C_{v}-T$ curves which can be verified experimentally.

Acknowledgements MK thanks Prof. Z. G. Soos for useful discussions and also thanks DST for a Ramanujan Fellowship SR/S2/RJN-69/2012 and DST India for funding computation facility through SNB/MK/14-15/137. Debasmita thanks Dayasindhu Dey for help in linearised spin wave theory calculations.

VII Appendix A

For up spins the Holstein-Primakoff transformations take the form

[TABLE]

For the down spin

[TABLE]

We use the linear approximation at classical limit

[TABLE]

for spin up, and for spin down

[TABLE]

In terms of bosonic operators, the Hamiltonian takes the form upto quadratic order as

[TABLE]

Fourier transforms of the bosonic operators are,

[TABLE]