FIELD-INDUCED TOMONAGA–LUTTINGER LIQUID OF A QUASI-ONE-DIMENSIONAL S = 1 ANTIFERROMAGNET

Final Reading

Brief Review

July 9, 2007 10:39 WSPC/147-MPLB 01362

Modern Physics Letters B, Vol. 21, No. 16 (2007) 965–976c© World Scientific Publishing Company

FIELD-INDUCED TOMONAGA LUTTINGER LIQUID OF A

QUASI-ONE-DIMENSIONAL S = 1 ANTIFERROMAGNET

M. HAGIWARA∗, H. TSUJII†, C. R. ROTUNDU‡,‖, B. ANDRAKA‡,Y. TAKANO‡, T. SUZUKI§ and S. SUGA¶

∗Center for Quantum Science and Technology under Extreme Conditions (KYOKUGEN),Osaka University, 1–3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

[email protected]†Department of Physics, Kanazawa University,Kakuma-machi, Kanazawa 920-1192, Japan

‡Department of Physics, University of Florida, P. O. Box 118440,Gainesville, Florida 32611-8440, USA

§Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwa, Chiba 277-8581, Japan

¶Department of Applied Physics, Osaka University,Suita, Osaka 565-0871, Japan

Received 15 June 2007

We review the results of specific-heat experiments on the S = 1 quasi-one-dimensional(quasi-1D) bond-alternating antiferromagnet Ni(C9H24N4)(NO2)ClO4, alias NTENP.At low temperatures above the transition temperature of a field-induced long-rangeorder, the magnetic specific heat (Cmag) of this compound becomes proportional totemperature (T ), when a magnetic field along the spin chains exceeds the critical fieldHc at which the energy gap vanishes. The ratio Cmag/T , which increases as the magneticfield approaches Hc from above, is in good quantitative agreement with a prediction ofconformal field theory combined with the field-dependent velocity of the excitationscalculated by the Lanczos method. This result is the first conclusive evidence for aTomonaga-Luttinger liquid in a gapped quasi-1D antiferromagnet.

Keywords: Tomonaga–Luttinger liquid; one-dimensional S = 1 antiferromagnet; specificheat.

1. Introduction

The concept of the Tomonaga–Luttinger liquid (TLL),1, 2 which was introduced by

Haldane3 in the early 1980s, encompasses a large class of one-dimensional (1D)

quantum liquids. The striking feature of the TLL is a power-law singularity in the

long-range behavior of various correlation functions with anomalous critical expo-

nents. Low-energy excitations are collective, in contrast with the Fermi liquid, whose

low-energy excitations are quasiparticles. Although the specific heat is proportional

‖Present address: Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA.

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966 M. Hagiwara et al.

to temperature (T ), similar to that of the Fermi liquid, the linearity arises from the

dispersion of the collective excitations that is linear in momentum k. According to

conformal field theory,4, 5 the coefficient γsp of the T -linear specific heat is given

by RπkB/3~v, where R is the gas constant, kB the Boltzmann constant, and v the

velocity of the excitations.

Recent advances in sophisticated microfabrication techniques have enabled re-

searchers to create 1D metals such as an edge state in a fractional quantum Hall

system,6 a quantum wire,7 and a metallic single-walled carbon nanotube (SWNT).8

As evidence for a TLL state, power-law behavior has been reported in the tem-

perature dependences of various quantities, including the tunneling conductance

through a point contact fed by 1D quantum Hall edge states,6 conductance in a 1D

wire,7 and photoemission intensity of a SWNT near the Fermi energy.8

In 1D metals, the velocities of the collective modes for the spin and charge de-

grees of freedom take different values, resulting in spin-charge separation. This sep-

aration manifests itself, among others, in specific heat: γsp becomes RπkB(1/3~vs +

1/3~vc), where vs and vc are the velocities of the spin and charge excitations, re-

spectively. To our knowledge, however, no thermodynamic signature of a TLL has

been observed in 1D metals, which are available in too small quantities for specific-

heat measurements in many cases. Moreover, the electronic specific heat of a TLL is

qualitatively indistinguishable from that of a Fermi liquid, both being proportional

to temperature.

The concept of the TLL is not limited to 1D metals; it also applies to 1D

antiferromagnets. In a TLL in an antiferromagnet, the gapless point k0 of the

linear dispersion moves with the magnetic field9 and is related to the magnetization.

Calorimetry is an excellent tool for the search of a TLL in antiferromagnets, since no

other state of these materials is known to have a specific heat proportional to T . The

low-energy physics of 1D spins has been studied theoretically for a particularly long

time in the uniform S = 1/2 Heisenberg antiferromagnet. It was therefore natural

to search for a TLL in antiferromagnetic S = 1/2 linear-chain compounds. The S =

1/2 quasi-1D antiferromagnets copper benzoate and Cu(C4H4N2)(NO3)2 were the

first to exhibit a linear T dependence of specific heat due to k-linear dispersion.10, 11

The magnetic excitations in these materials are inherently gapless (albeit only in

zero magnetic field in the case of copper benzoate), since the antiferromagnetic

bonds between the Cu2+ ions are uniform.

A TLL should occur also in gapped 1D antiferromagnets such as S = 1 chains,

S = 1/2 two-leg ladders, and S = 1/2 chains with alternating bonds. In such sys-

tems placed in an increasing magnetic field, one of the Zeeman-split triplet branches

goes down in energy, until the energy gap collapses at a critical magnetic field Hc.

At this quantum critical point, a transition takes place from a gapped spin liquid

to a TLL,12, 13 provided that the system possesses a U(1) symmetry.

In fact, the results of experiments on four gapped quasi-1D antiferromagnets

had been interpreted in terms of a TLL prior to our work, but these did not


Field-Induced TLL of a Quasi-1D S = 1 Antiferromagnet 967

provide strong evidence for it. A TLL was invoked to explain the divergence of

the NMR spin-lattice relaxation rate (1/T1)14 and an anomalous low-temperature

hump in the magnetic specific heat (Cmag)15–17 of the spin-1/2 ladder compound

Cu2(C5H12N2)2Cl4, alias CuHpCl. Neutron results and a careful examination of the

crystal structure suggested,18 however, that CuHpCl is three-dimensional rather

than quasi-1D. A similar interpretation of the NMR relaxation rate19 and the mag-

netic specific heat20 was put forward for the spin-1/2 alternating-bond chain com-

pound pentafluorophenyl nitronyl nitroxide (F5PNN). This interpretation implies

that the TLL does not fully develop in F5PNN before it is destroyed by a three-

dimensional long-range order (LRO). Moreover, 1/T1 is dominated by an anomaly

at the onset of the LRO. An increase of 1/T1 with decreasing temperature was also

observed in the S = 1 Heisenberg antiferromagnet (CH3)4NNi(NO2)3 at magnetic

fields above Hc.21 But the experiment used polycrystals, making the interpretation

of the data ambiguous for this highly anisotropic material.

The experimental evidence for a TLL was also indirect and weak in the

fourth gapped quasi-1D antiferromagnet Ni2(C5H14N2)2N3(PF6), abbreviated as

NDMAP. For this S = 1 Haldane-gap material, it was proposed22 that the critical

exponent of the TLL explains the shape of the H–T phase diagram of the field-

induced LRO22, 23 in the magnetic fields above Hc for H‖chain. The principal axis of

the NiN6 octahedra which determines the crystal-field anisotropy in this compound

deviates from the chain direction by 15◦, with the sign of the angle alternating from

chain to chain. Because of this unfortunate geometry, no magnetic-field direction

strictly satisfies a U(1) symmetry for all the chains simultaneously. This lack of ax-

ial symmetry probably explains why clear evidence for a TLL is absent in NDMAP.

Including NDMAP, none of the four compounds exhibits the crucial hallmark of a

TLL — a specific heat proportional to T .

The material we have studied, Ni(C4H24N4)NO2(ClO4), alias NTENP,24 is an

S = 1 bond-alternating-chain antiferromagnet. Similar to NDMAP, this material

exhibits an LRO above Hc,25, 26 at which the energy gap between the singlet ground

state and the first excited triplets27 closes. The compound has a good U(1) sym-

metry, since its chains are nearly parallel to the principal axes of the NiN6−nOn

octahedra, where n permutes semi-randomly among 0, 1, and 2. In this review, we

examine an unambiguous TLL behavior recently observed in the specific heat of

NTENP above Hc. The main results were published in Ref. 26, following a prelim-

inary report25 for magnetic fields perpendicular to the chain direction.

2. Crystallographic and Magnetic Properties of NTENP

NTENP has a triclinic crystal structure (space group P 1) with lattice constants

a = 10.747(1) A, b = 9.413(2) A, c = 8.789(2) A, and angles α = 95.52(2)◦,

β = 108.98(3)◦ and γ = 106.83(3)◦ at room temperature.24 Bridged by nitrito

groups, the Ni2+ ions form chains involving two bond lengths that alternate along

with the permutation of the NiN6−nOn octahedra. The inversion centers within each



chain are located in the nitrito groups, guaranteeing that the magnetic moments

can have no staggered components which break a U(1) symmetry. The chains, which

run along the a axis, are well separated by ClO−4 anions, thus having a good 1D

nature.

The model spin Hamiltonian for NTENP is

H =∑

i

[JS2i−1 · S2i + δJS2i · S2i+1 − µBSigH

+ D(Szi )2 + E{(Sx

i )2 − (Sy

i )2}], (1)

where J is the larger of the two exchange constants resulting from the alternation of

the Ni2+–Ni2+ bond lengths, δ the ratio of these exchange constants, g the g tensor

of Ni2+, µB the Bohr magneton, and D and E the axial and orthorhombic single-

ion anisotropy constants, respectively. Assuming E = 0, the magnetic susceptibility

gives the parameters J/kB = 54.2 K, δ = 0.45, D/J = 0.25, and g‖ = 2.14 for the

Hamiltonian. Here, g‖ is the component of g in the chain direction.27 The analysis

of the excitations observed by inelastic neutron scattering in the magnetic fields

shows that E is at most 5% of D.28 This high degree of symmetry is why we have

searched for a TLL in NTENP.

The large alternation of the exchange constants along the chain causes the

ground state of NTENP to be a singlet-dimer state rather than a Haldane spin

liquid. The evidence for this ground state comes from the magnetic susceptibility,

magnetization, and ESR experiments on Zn-doped NTENP.27 Introduced into a

singlet dimer of S = 1 spins, a nonmagnetic impurity such as Zn creates a localized,

unpaired S = 1 spin. The effect of doping is in strong contrast with that for a

Haldane spin liquid, in which an impurity generates two S = 1/2 spins.29 The

experiments of Ref. 27 indeed observed impurity-induced S = 1 spins. Similar to

those of a Haldane compound, however, the low-lying excitations of NTENP are

gapped triplets.

The critical fields Hc parallel and perpendicular to the chain direction are 9.3 T

and 12.4 T, respectively.27 These values are from magnetization measurements of a

single-crystal sample at 1.3 K, as shown in Fig. 1. The critical field is taken to be at

the local maximum of the field derivative of magnetization for each field direction.

3. Experimental Details

We prepared single crystals of hydrogenous and deuterated NTENP for specific-

heat measurements as described in Ref. 24. Measurements in magnetic fields ap-

plied along the chain direction were performed at the National High Magnetic

Field Laboratory (NHMFL) in Tallahassee, Florida. A relaxation calorimetry was

used for these measurements, with a deuterated sample weighing about 5 mg, in

magnetic fields provided by a 20 T superconducting magnet.30 Separately, a hy-

drogenous sample of a similar mass was also studied. The minimum temperature of

the experiments was 150 mK. Measurements in magnetic fields normal to the chain



Fig. 1. Magnetization of a single crystal of NTENP at 1.3 K for the magnetic fields appliedparallel and perpendicular to the chains.27 The arrows show the critical fields which are taken tobe at the local maximum of the field derivative of magnetization.

direction were made at KYOKUGEN, Osaka University. An adiabatic method was

used on a hydrogenous sample weighing about 1 g in magnetic fields up to 16 T at

temperatures down to about 200 mK.

The lattice contribution to the specific heat was determined to be 3.31 ×

T 3 mJ/K mol from the data at temperatures between 0.2 K and 1.1 K in zero

field, where the singlet ground state with a large energy gap of about 10 K made

the magnetic component of the specific heat negligible. This lattice contribution

has been subtracted from the raw data. For the hydrogenous samples, we have also

subtracted the nuclear contribution of the hydrogen atoms Cnuc given by Eq. (2)

from the data.

Cnuc = nR(∆/T )2 exp(∆/T )

(1 + exp(∆/T ))2, (2)

where n = 24 is the number of hydrogen atoms per molecular unit, and ∆ =

gnµnH/kB. Here, gn and µn are the g factor of the hydrogen nucleus (the proton)

and the nuclear magneton, respectively.

4. Results and Discussion

Figures 2(a) and 2(b) display the magnetic specific heat for deuterated and hy-

drogenous NTENP samples in magnetic fields parallel and perpendicular to the

chains, respectively. Although a hydrogenous sample was also studied for H‖chain,

we present only the deuterated-sample data for this field direction, since they are



Fig. 2. Magnetic specific heat of (a) a deuterated sample for the magnetic fields applied parallelto the chains and (b) a hydrogenous sample for the magnetic fields applied perpendicular to thechains. From Ref. 26.

more extensive and free of uncertainties due to subtraction of the nuclear contribu-

tion. The hydrogenous-sample data were very similar. For both field directions, a

sharp peak signals the LRO above the critical field Hc. Both the peak height and

its position (ordering temperature Tc) increase very rapidly with increasing field for

H⊥chain, whereas the increase of Tc is less pronounced for H‖chain. Moreover, the

peak height for H‖chain is nearly constant, even decreasing slightly with increasing

field H > 12 T.

For the purpose of the present review, the crucial difference between the results

for the two field directions is in the temperature region above Tc. Here, the spe-

cific heat is linear in temperature for H‖chain when H ≥ 13 T, whereas no such



0

200

400

600

800

1,000

0 1 2 3 4

13 T14 T16 T18 T20 T

Cm

ag (

mJ/

K m

ol)

H||chain

T (K)

Fig. 3. Magnetic specific heat of a deuterated sample for the magnetic fields applied parallel tothe chains. For clarity, the data are offset by 50 mJ/K mol at each field increment. The solidstraight lines encompass the T -linear regions above the Tc of the LRO and the arrows indicatethe deviation points between the data and the straight lines.

linearity is observed for H⊥chain. Qualitatively, the T -linear specific heat indicates

a TLL state. The absence of linearity for H⊥chain adds strong support to this

identification, since the field applied in this direction breaks the U(1) symmetry

of the spin Hamiltonian and makes the system belong to the 2D Ising universality

class.31–33

Figure 3 highlights the linear T dependence by setting off the data by

50 mJ/Kmol at each field increment and by drawing a straight line at each field.

The linear behavior extends progressively to a higher temperature as the magnetic

field increases. In the figure, the arrow at each field marks the upper end of the linear

region. Since this is a crossover point, its position contains a degree of ambiguity.

These points are plotted along with Tc in the H–T phase diagram of Fig. 4, which

clearly indicates the stabilization of the TLL by the magnetic field, as expected by

theory.34

As indicated by the nearly field-independent slopes of the straight lines in Fig. 3,

the value of the T -linear specific heat depends only weakly on the magnetic field.

The ratio γsp = Cmag/T in the TLL region rises slightly as the magnetic field

decreases toward Hc ∼ 9.3 T. As we have pointed out in the Introduction, γsp for

a TLL is RπkB/3~v, which depends only on velocity v of the low-lying excitations.

Using the Lanczos method to diagonalize the Hamiltonian for a chain of up to 20

spins, we have calculated the dispersion curve and have extracted v as a function

of the ground-state magnetization. The results are sufficiently independent of the

system size N for N = 16, 18, and 20. From these results and the experimental



0.0 0.5 1.0 1.5 2.0 2.5 3.08

10

12

14

16

18

20

22

T (K)

H (

T)

H||chain

TLL

LRO Specific heatNMR

Fig. 4. Magnetic field versus temperature phase diagram of NTENP for the magnetic fieldsparallel to the chains, adopted from Ref. 26. The regions marked LRO and TLL are the long-range-ordered and Tomonaga–Luttinger-liquid phases, respectively. The solid circles indicate the peakpositions of the specific heat for the deuterated samples and the open circles show the positionsof the arrows in Fig. 3. The open square is the position of the maximum NMR relaxation rate at12 T from Ref. 42. The solid line is the best fit of the data to the expression Tc = A|H − Hc|α

with Hc = 9.17(1) T and α = 0.264(2). The broken line is a visual guide.

magnetization curve,27 we obtain γsp as a function of the magnetic field as shown

in Fig. 5. The calculated value at about 20 T is lower than the experimental one

by about 20%. Unfortunately, no calculated γsp exists at fields between 16 T and

20 T where no corresponding magnetization value is accessible to N = 16–20. At

fields below about 16 T, however, the agreement between the calculation and the

experimental data is excellent. The increase of γsp with decreasing H toward Hc,

captured by both experiment and calculation, is associated with the divergence of

the density of states at the band edge, which is commonly seen in 1D quantum

many-body systems.35 The overall quantitative agreement between the experiment

and calculation is conclusive evidence for a TLL.

The transition temperature of LRO plotted in Fig. 4 is for H‖chain. The

LRO can be regarded as a Bose–Einstein condensation (BEC) of the triplets.36, 37

According to the BEC theory,38 the phase boundary in the H–T phase diagram

obeys the power law Tc ∝ |H − Hc|α. The solid line is the best fit of the data up

to 13 T to this expression with Hc = 9.17(1) T and α = 0.264(2). The prelimi-

nary value α = 0.334 reported in Ref. 25 was larger, with systematically higher

Tc, probably due to a slight deviation of the magnetic fields from the chain direc-

tion. A Hartree–Fock approximation37 gives α = 2/3, which has been supported by

experiments on TlCuCl3,39 BaCuSi2O6,

40 and NiCl2-4SC(NH2)2.41 The question

of whether the smaller α for NTENP contradicts the theoretical value cannot be



0

50

100

150

200

12 14 16 18 20 22 24

Calculation

Experiment

sp (

mJ/

K2 m

ol)

H (T)

Fig. 5. Field dependence of γsp = Cmag/T . The solid circles are experimental values taken fromFig. 3, and the open circles are calculations given by RπkB/3~v where R and v are the gas constantand the velocity of the low-lying excitations, respectively. From Ref. 26.

addressed until measurements are extended to temperatures closer to the T = 0

limit. The Tc value42 obtained by NMR at 12 T is in good agreement with the

present results as shown in the figure. Similar to the NMR results for F5PNN,19

however, the relaxation rate 1/T1 of Ref. 38 cannot be interpreted unambiguously

in terms of a TLL, because an anomaly at Tc dominates the data.

Finally, let us return to the field dependence of the height of the specific-heat

peak at Tc. As Tc of a second-order LRO transition increases with increasing field,

like in NTENP, the specific-heat peak usually becomes higher and broader, be-

cause the width of the critical region increases with increasing Tc. This behavior

is observed for H⊥chain, but not for H‖chain, as pointed out earlier. The unusual

field dependence of the specific-heat peak height for H‖chain is probably a result

of strong correlations in the TLL. In a TLL, the correlation function decays only

algebraically, with a well developed short-range order (SRO). Figure 6 shows the

entropy obtained by integrating the data in Fig. 3, contrasting its temperature

and field dependence for H‖chain with that for H⊥chain. For H‖chain, we see a

substantial entropy drop in the temperature region between 4 K and Tc due to

the development of an SRO. As the increasing magnetic field makes the TLL re-

gion wider, the entropy drop becomes larger, leaving less entropy to be expended

at Tc and thereby making the specific-heat peak smaller. This field dependence of

entropy in the TLL region is in strong contrast with its behavior in H⊥chain. In

this field configuration, the increasing field simply competes more strongly with the

antiferromagnetic interactions as usual, causing the entropy to remain higher until

the critical region is reached near Tc.



Fig. 6. Entropy for H‖chain (top) and H⊥chain (bottom). The entropy drop between 4 K andTc is larger for H‖chain than for H⊥chain.

5. Conclusions

The specific heat of NTENP for H‖chain above Hc is proportional to temperature

in a region immediately above Tc of the field-induced long-range order, with the

field dependence of the ratio Cmag/T in good agreement with the prediction of

conformal field theory. This finding, which is the first conclusive evidence for a

Tomonaga–Luttinger liquid (TLL) in a gapped quasi-1D antiferromagnet, illustrates

the ubiquity of TLLs in one dimension. Obviously, our work is just a beginning, to

be followed by more experiments that explore the interesting physics of the TLL in

NTENP. For instance, renewed efforts are called for to observe a k-linear dispersion

with an incommensurate k0 with inelastic neutron scattering. Another potentially

rewarding route of investigation will be an NMR to look for, and characterize, a

power-law increase of the spin-lattice relaxation rate with decreasing temperature.



The NMR experiment of Ref. 38 was done only at 12 T, for which specific heat

does not yet show a linear T dependence. An experiment at higher fields is clearly

needed to study a power law of 1/T1 uncorrupted by the anomaly at Tc. Lastly,

the search must continue for the existence of more TLLs in new gapped quasi-1D

antiferromagnets with various zero-field ground states.

Acknowledgments

We thank N. Tateiwa and T. C. Kobayashi for the measurements at KYOKU-

GEN, A. Zheludev and D. L. Maslov for useful discussions, and G. E. Jones,

T. P. Murphy, and E. C. Palm for assistance. This work was supported in part

by the Grant-in-Aid for Scientific Research on Priority Areas “High Field Spin

Science in 100 T” (No. 451) from the Japanese Ministry of Education, Culture,

Sports, Science and Technology, by the DOE under grant No. DE-FG02-99ER45748,

and by the NHMFL In-House Research Program. The NHMFL is supported by NSF

Cooperative Agreement DMR-0084173 and by the State of Florida.

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FIELD-INDUCED TOMONAGA–LUTTINGER LIQUID OF A QUASI-ONE-DIMENSIONAL S = 1 ANTIFERROMAGNET

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