# -Symmetry breaking and violation of axial symmetry in and other insulating spin systems

###### Abstract

We describe the Bose-Einstein condensate of magnetic bosonic quasiparticles in insulating spin systems using a phenomenological standard functional method for = 0. We show that results that are already known from advanced computational techniques immediately follow. The inclusion of a perturbative anisotropy term that violates the axial symmetry allows us to remarkably well explain a number of experimental features of the dimerized spin-1/2 system . Based on an energetic argument we predict a general intrinsic instability of an axially symmetric magnetic condensate towards a violation of this symmetry, which leads to the spontaneous formation of an anisotropy gap in the energy spectrum above the critical field. We, therefore, expect that a true Goldstone mode in insulating spin systems, i.e., a strictly linear energy-dispersion relation down to arbitrarily small excitations energies, cannot be observed in any real material.

###### pacs:

75.10.Hk, 75.45.+j, 75.30.Gw, 73.43.Nq##
I Introduction

The concept of Bose-Einstein condensation (BEC), i.e., the occupation of a single quantum state by a macroscopic number of bosons, has been extended from real bosonic particles to
various types of quasiparticles with integer total spin I. Affleck (1991); Giamarchi and Tsvelik (1998); T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000); Ch. Rüegg, N. Cavadini, A. Furrer, H.-U.G del, K.
Kr mer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch (2003); P. Wachter, B. Bucher and J.
Malar (2004); M. Matsumoto, B. Normand, T. M. Rice and M. Sigrist (2004); J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P.
Jeambrun, J.M.J. Keeling, F. M. Marchetti, M.H. Szymanska, R. Andr , J.L.
Staehli, V. Savona, P.B. Littlewood, B. Deveaud and L.S.
Dang (2006); S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A.
Melkov, A.A. Serga, B. Hillebrands and A.N. Slavin (2006); D. Snoke (2006); R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer and K.
West (2006); V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A.
Melkov and A. N. Slavin (2008). Such discrete magnetic, electronic or lattice excitations are then treated as a set of bosons forming a Bose gas D. Snoke (2006). These quasiparticles usually possess a small effective mass, which permits to study BEC even at room temperature V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A.
Melkov and A. N. Slavin (2008). Both experiment J. Harada (2007); R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer and K.
West (2006) and theory T. Haugset, H. Haugerund and F.
Ravndal (1998); J. O. Andersen (2004) suggest that the occurrence of a BEC in a 3D interacting Bose system has its origin in the spontaneous breaking of the symmetry, thereby leading to a gapless and linear excitation spectrum in
the long-wavelength limit, i.e., to a Goldstone mode J. Goldstone, A. Salam, and S.
Weinberg (1962).

Experimental observations in a number of quantum spin systems can be explained within the theory of BEC, e.g., by a condensation of triplet states in dimerized spin-1/2 systems (hereafter called triplons) such as T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000); Ch. Rüegg, N. Cavadini, A. Furrer, H.-U.G del, K.
Kr mer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch (2003); T. Giamarchi, Ch. Rüegg and
O. Tcherniyshyov (2008). The bosonic character of the these magnetic quasiparticles allows to describe this spin-dimer system as a weakly interacting Bose gas. Inelastic neutron-scattering measurements in the condensate phase of revealed, in accordance with theoretical investigations M. Matsumoto, B. Normand, T. M. Rice and M.
Sigrist (2002), the presence of a seemingly gapless and linear excitation spectrum down to very low excitation energies of the order of 0.75 meV in = 14 T Ch. Rüegg, N. Cavadini, A. Furrer, H.-U.G del, K.
Kr mer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch (2003). This observation has been interpreted as a manifestation of the Goldstone mode.

In the last years various scenarios for the consequences of anisotropy on the properties of a
magnetic BEC have been discussed in detail A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005); S.E. Sebastian, P. Tanedo, P. A. Goddard, S.-C. Lee,
A. Wilson, S. Kim, S. Cox, R. D. McDonald, S. Hill, N. Harrison, C. D.
Batista, and I. R. Fisher (2006). The presence of any kind of anisotropy will, in principle, explicitly break the rotational (i.e, axial) symmetry of the bosonic system J. Sirker, A. Weisse and O.P.
Sushkov (2005). The degree of spontaneous -symmetry breaking then depends on the order of magnitude of the anisotropic terms compared to the energy scale associated with the isotropic interactions
of the Bose gas S.E. Sebastian, P. Tanedo, P. A. Goddard, S.-C. Lee,
A. Wilson, S. Kim, S. Cox, R. D. McDonald, S. Hill, N. Harrison, C. D.
Batista, and I. R. Fisher (2006). The relatively large triplon bandwith in N. Cavadini, G. Heigold, W. Henggeler, A. Furrer,
H.-U. Güdel, K. Krämer and H. Mutka (2001) exceeds the spin-nonconserving terms, such as an intradimer exchange (IE) anisotropy and a Dzyaloshinsky-Moriya (DM) anisotropy, by more than two orders of magnitude A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005). Nonetheless, the question to what extent the existing anisotropies in do affect the magnetic phase diagram, the Goldstone mode and other measurable quantities of is still an issue under investigation V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004); A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); O. Vyaselev, M. Takigawa, A. Vasiliev, A. Oosawa and
H. Tanaka (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005).

By taking a perturbative anisotropy term into account we will consider in the following the influence of an IE-like anisotropy that explicitly violates the axial symmetry, and we will study the consequences on the condensate phase of . The influence of a possible DM-like anisotropy J. Sirker, A. Weisse and O.P.
Sushkov (2005); J. Sirker, A. Weisse and O.P.
Sushkov (2004) is not considered here DM . Based on an energy consideration we will then argue that, as a consequence of an unavoidable magnetoelastic coupling, even an axially symmetric magnetic system is unstable towards a spontaneous violation of this symmetry as soon as the BEC-state is formed.

## Ii Functional method

We describe the condensate at = 0 with a macroscopic wave function, a complex scalar field . Standard functional methods used to describe a dilute Bose gas in the classical limit at = 0 yield an extremal condition for the potential energy per dimer J. O. Andersen (2004), namely

(1) |

where is the chemical potential, is a constant related to a repulsive short-range
interaction T. Haugset, H. Haugerund and F.
Ravndal (1998), and is the complex conjugate of . The minimum value then determines the condensate fraction , here defined as with the number of condensed triplons and the number of dimers.

In a dimerized antiferromagnet, the chemical potential is where is the Bohr magneton, the Landé -factor, and the critical magnetic field beyond which a triplet state is energetically equally favorable as the singlet state. This can be expressed in terms of the energy gap = separating at zero field the and the states, respectively.

In the case of an explicitly violated axial symmetry we may include a perturbation term J. Sirker, A. Weisse and O.P.
Sushkov (2005); I. Affleck (1991) to the potential energy that can arise in a real magnetic system from various sources such as anisotropic intra- and interdimer interaction constants and , respectively. For we have, for example, 5.6 T W. Shiramura, K. Takatsu, H. Tanaka, K. Kamishima, M.
Takahashi, H. Mitamura and T. Goto (1999), = 315 F. Yamada, T. Ono, H. Tanaka, G. Misguich, M. Oshikawa
and T. Sakakibara (2008), and a of the order of 0.01 meV A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005), depending on the orientation of the magnetic field with respect to the crystal lattice A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004). Such an anisotropy term may arise from a pre-existing violated axial symmetry of the system, or from a spontaneous distortion at the magnetic phase transition that we will discuss below.
We, therefore, have to minimize

(2) |

where we assume that itself is at first unchanged by the presence of the small perturbative anisotropy .

We first want to compare the results of this simple formalism with corresponding predictions from advanced Hartree-Fock (HF) computations and with experimental data on . Despite the simple formalism used here, we can later make specific predictions that would otherwise be more difficult to obtain.

##
Iii Results

###
iii.1 Comparison with results from Hartree-Fock calculations

Without any explicit anisotropy we obtain the well-known minimum value = ) = T. Nikuni, M. Oshikawa, A. Oosawa and H. Tanaka (2000). The phase of is not fixed in this case, leading to the landscape sketched in Fig. 1 (left panel, ”mexican-hat potential”). However, any nonzero value for locks the phase of to the imaginary axis (i.e., = and therefore = const.), and leads to an optimum value in a ”Napoleon’s hat potential” (see Fig. 1, right panel). The minimum potential energy becomes which is smaller than in the axially symmteric case. The saddle-point value for on the real axis for is . Note that these energy densities are all expressed per dimer. The corresponding energies per triplon are .

A vanishing is realized when = 0. As a consequence, the
gap field that would be observed in an ideal system with axial symmetry is renormalized to a value = (with = ) above which condensation occurs. Taking a reasonable value for 0.01 meV for A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005) and with =2.06 A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004), we obtain a renormalization of the critical field due to alone by 0.2 T for this particlular magnetic-field direction.

The resulting condensate fraction, where , is in full agreement with Hartree-Fock calculations for spin dimer systems T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000).

In Fig. 2a we show the triplon condensate fraction as deduced from our magnetization -data of R. Dell’Amore, A. Schilling and K. Krämer (2008), see Fig. 2b, that we have already corrected for a small fraction of noncondensed triplons T. Nikuni, M. Oshikawa, A. Oosawa and H. Tanaka (2000). These data have been obtained from the simple relation (with the number of dimers and the total triplon fraction at = 0) T. Nikuni, M. Oshikawa, A. Oosawa and H. Tanaka (2000), without assuming any specific value for . A linear fit to the data in the dilute limit ( 7.5 T J. Sirker, A. Weisse and O.P. Sushkov (2005)) yields 0.003 T and = 311.4 0.5 which is in very good agreement with available literature values J. Sirker, A. Weisse and O.P. Sushkov (2005); Misguich and Oshikawa (2004); F. Yamada, T. Ono, H. Tanaka, G. Misguich, M. Oshikawa and T. Sakakibara (2008). The deviation from the linear behavior at larger magnetic fields can be attributed to the contribution of higher triplet states M. Matsumoto, B. Normand, T. M. Rice and M. Sigrist (2004). Our simple formalism does not include the influence of such triplet states, nor does it allow for a determination of itself, but this latter correction is of the order of a few percents at most in our data R. Dell’Amore, A. Schilling and K. Krämer (2008); T. Nikuni, M. Oshikawa, A. Oosawa and H. Tanaka (2000), as it is typical for a weakly interacting Bose gas, see inset of Fig. 2a.

Focusing further on the effects of an explicit violation of axial symmetry, the condensate fraction at = 0 is changed by 2 in our calculation, see Figs. 2c and d. If we use the fact that the total triplon fraction and take the result = /2 from Ref. T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000) assuming a quadratic triplon dispersion relation, we obtain a shift in that is again in full agreement with the corresponding HF calculations J. Sirker, A. Weisse and O.P.
Sushkov (2005), see again Figs. 2c and d.

We may relate the minimum value at = 0 to the transition temperature if we assume that , the energy gain per triplon upon condensation, is proportional to with a field-dependent proportionality factor of the order of unity. Any nonzero leads to an increase in as compared to the axially symmetric case, see Fig. 2d. This trend can be clearly seen in the calculated -curves from Ref. J. Sirker, A. Weisse and O.P.
Sushkov (2005), where the minimum in that is usually taken as a criterion to define is shifted towards higher temperatures as soon as . If we again take = 0.01 meV we obtain a shift in of the order of + 0.1 K, which has to be compared to the result of the more precise HF calculations with + 0.5 K J. Sirker, A. Weisse and O.P.
Sushkov (2005).

As a consequence of the anisotropy term the original invariance of with respect to a transformation is lifted. The maximum variation in the potential-energy per dimer along the ellipsoid contour with local minima in the radial -direction, see Fig. 1, is , or per ground-state triplon. This determines the order-of-magnitude of an anisotropy gap that can be calculated to = J. Sirker, A. Weisse and O.P.
Sushkov (2005); us .

Excitations above the ground state with energies below will clearly not show the typical gapless Goldstone-like behavior as expected for an axially symmetric system. For we calculate with meV A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); J. Sirker, A. Weisse and O.P.
Sushkov (2005) a gap 0.3 meV for = 14 T along , which is somewhat below what has been resolved in inelastic-neutron scattering measurements Ch. Rüegg, N. Cavadini, A. Furrer, H.-U.G del, K.
Kr mer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch (2003). For excitation energies larger than the presence of an anisotropy gap may remain unnoticed, see Fig. 2e.

In such a gap may arise from a pre-existing anisotropy that is already present in = 0 V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004). In the following we argue, however, that even a perfectly axially symmetric magnetic system is unstable towards a spontaneous violation of this symmetry at , which inevitably leads to the formation of a small anisotropy gap above of real materials.

### iii.2 Instability of the condensate towards violation of axial symmetry

A striking fact in our analysis is that the minimum potential energy per dimer is smaller with a nonzero than in an analogous axially symmetric system with = 0. This means that a distortion of the original crystal symmetry may spontaneously occur at together with an increase of , provided that the total energy, including both magnetic and crystal-lattice contributions, is lowered along with this distortion. This argument is so general that it should be applicable to all insulating spin systems that are supposed to show a Bose-Einstein condensation of magnetic bosonic quasiparticles. As we do not not make any specific assumptions on the microscopic arrangement of the spin-carrying atoms, one cannot make any more precise, universal statement about the details of the resulting lattice distortion.

The gain in potential energy per dimer upon condensation in combination with this simultaneous distortion is , see Fig. 3a. If the critical field is approached from below with increasing, the parameter may therefore jump discontinuously from zero to its optimum value either at or at a transition field with , while with decreasing the transition can take place at a different field in the same magnetic-field interval, see grey shaded area in Figs. 3a and b. In an ideal situation with a perfect axial symmetry in , the critical field corresponds to a ”normal-state” value with that is determined only by the gap energy, while in the condensate phase the effective critical field is , lowered by with respect to due to the increase in . If the transition occurs at a transition field that is strictly larger than , one will observe at corresponding small discontinuities in (, see Fig. 3a), (, see Fig. 2c) and (), which would qualify the transition as of weakly first order with a maximum observable hysteresis width . The occurrence of hysteretic effects in a real material may depend, however, on further conditions that are not considered here, such as material-quality issues or the relevance of possible quantum fluctuations at = 0 to the order of the phase transition.

Even if a non-zero evolves continuously above the critical field as a function of along with a continuous structural distortion with no detectable hysteresis, one should still be able to distinguish between the normal-state extracted from the experimental data taken below , and a from corresponding data taken well above , respectively, to obtain an estimate for from the resulting difference . In any case, if , one expects a finite energy difference at the transition field between the lowest-triplon state and the singlet states, respectively, see Fig. 3b.

A possible pre-existing anisotropy that may already be present at = 0 (which is likely to be the case in ) can be easily included in this formalism by identifying with a renormalized ”normal-state” value that already contains this pre-existing anisotropy. Any additional that may develop together with the lattice distortion around this critical field will somewhat change the value of the anisotropy gap . The difference , however, and possible discontinuities in , and are determined by the additional alone.

It is remarkable that electron-spin resonance (ESR) data taken on do indeed show a clearly gapped behavior at and above A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004), as we sketched in Fig. 3b. Moreover, the ESR frequencies due to the lowest triplon gap in the ”normal” phase for extrapolate to zero at a somewhat larger (by T) than the square-root like gap that we attribute to in the condensate phase and that can be fitted nicely with a 0.016 meV, see energy-level scheme in Fig. 3b. Our scenario may also be an explanation for the observed abrupt changes in the Cl quadrupole shift O. Vyaselev, M. Takigawa, A. Vasiliev, A. Oosawa and
H. Tanaka (2004) that has been interpreted as an indication of a weakly-first order lattice deformation, as well as for the pronounced hysteretic behavior (with a T) of the observed peaks in the sound-attenuation data of at E. Ya. Sherman, P. Lemmens, B. Busse, A. Oosawa and H.
Tanaka (2003). These observations may indicate that of is indeed larger for than well below this value.

The present picture may also account for the first-order like features that have been seen in the X-ray data of the spin-ladder compound J. E. Lorenzo, K. Katsumata, Y. Narumi, S. Shimomura,
Y. Tanaka, M. Hagiwara, H. Mayaffre, C. Berthier, O. Piovesana, T. Ishikawa
and H. Kitamura (2004) and in the magnetocaloric effect of the axially symmetric system V.S. Zapf, D. Zocco, B.R. Hansen, M. Jaime, N.
Harrison, C.D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda and A.
Paduan-Filho (2006) at the respective magnetic phase transitions. It is also not unreasonable to assume that the observed gap feature in the ESR-data of this latter compound at = 8 T S. A. Zvyagin, J. Wosnitza, A. K. Kolezhuk, V. S.
Zapf, M. Jaime, A. Paduan-Filho, V. N. Glazkov, S. S. Sosin and A. I.
Smirnov (2008) is also related to a possible lattice distortion.

Such a spontaneous distortion arising from the interplay between the emerging magnetic Bose-Einstein condensate and its host crystal that lowers the total energy, with a tendency to increase or even create an anisotropy perpendicular to the external magnetic field above even in a perfectly axially symmetric system, is rather unique and is reminiscent of the spin-Peierls instability in one dimensional magnetic chains. A spin-Peierls-type scenario has indeed been suggested to explain the
NMR data of the spin-ladder compound around its critical magnetic field R. Calemczuk, J. Riera, D. Poilblanc, J.-P.
Boucher, G. Chaboussant, L. Lévy and O. Piovesana (1999); H. Mayaffre, M. Horvatic , C. Berthier, M.-H.
Julien, P. Ségransan, L. Lévy and O. Piovesana (2000). This instability should be a universal feature of magnetic BEC systems at their magnetic phase transition, and it is not expected to occur in axially symmetric Bose gases composed of real particles such as superfluid or atomic condensates, where the condensate cannot create an axial-symmetry breaking term by itself.

In a microscopic picture, the tendency of a magnetic condensate to spontaneously
violate the axial symmetry can be interpreted as a natural consequence of the transverse magnetic ordering that develops in the condensate phase and that locks to the crystal lattice due to unavoidable magnetoelastic coupling T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000); H. Tanaka, A. Oosawa, T. Kato, H. Uekusa, Y. Ohashi,
K. Kakurai and A. Hoser (2001); M. Matsumoto and M. Sigrist (2005). As soon as the transverse magnetic moments point to a specific, energetically preferred crystal direction, the phase that is associated with the angle between these moments and the crystal axes T. Giamarchi, Ch. Rüegg and
O. Tcherniyshyov (2008) is indeed fixed (in with an angle to the -axis H. Tanaka, A. Oosawa, T. Kato, H. Uekusa, Y. Ohashi,
K. Kakurai and A. Hoser (2001)), and the magnetic analogue to a supercurrent velocity (where is the effective mass of a triplon) is naturally zero for excitation energies below the anisotropic gap . This gap covers an excitation-frequency range that is crucial for experiments that rely on the existence of a long-lived phase coherent condensate D. Snoke (2006), such as the detection of a long-lived spin supercurrent as in A. S. Borovik-Romanov, Yu M. Bunkov, V. V.
Dmitrieva and Yu M. Mukharskiy (1987); H. Tanaka, F. Yamada, T. Ono, T. Sakakibara, Y.
Uwatoko, A. Oosawa, K. Kakurai and K. Goto (2007), of macroscopic second-sound like oscillations as observed in superfluid V.P. Peshkov (1944) or of stable vortex-like structures as they have been observed both in superfluid and in atomic condensates P. Rosenbusch, V. Bretin and J.
Dalibard (2002). To achieve a lifetime of the order of seconds for a phase coherent condensate, a corresponding anisotropy gap may not exceed a few femto electron volts.

In the absence of a Goldstone mode (for which all values of the phase have to be energeticaly equivalent), the quantity in the zero-frequency limit does no longer represent a condensate fraction, but it is rather related to the order parameter characterizing the antiferromagnetic state I. Affleck (1991).

## Iv Conclusions

We have analyzed the spontaneous symmetry breaking in a Bose gas of magnetic bosonic quasiparticles in insulating spin systems based on simple functional methods in the classical limit. Our results reproduce several results from earlier HF approximated computations T. Nikuni, M. Oshikawa, A. Oosawa and H.
Tanaka (2000); J. Sirker, A. Weisse and O.P.
Sushkov (2005); J. Sirker, A. Weisse and O.P.
Sushkov (2004), and various experimental findings in such as the occurrence and the magnitude of an anisotropy gap A.K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A.
Oosawa (2004); V.N. Glazkov, A.I. Smirnov, H. Tanaka and A.
Oosawa (2004) and a weakly first-order like behavior at the magnetic phase transition O. Vyaselev, M. Takigawa, A. Vasiliev, A. Oosawa and
H. Tanaka (2004); E. Ya. Sherman, P. Lemmens, B. Busse, A. Oosawa and H.
Tanaka (2003) can be explained. On the basis of an energetic argument we expect that all magnetic BEC systems in insulating spin systems are intrinsically unstable towards a spontaneous anisotropic distortion perpendicular to the external magnetic field, which leads to the formation of an anisotropy gap that is seriously limiting the lifetime of a phase coherent condensate.

We thank to O. Sushkov, A. K. Kolezhuk and E. Ya. Sherman for stimulating discussion. This work was partly supported by the Schweizerische Nationalfonds zur Förderung der wissenschaftlichen Forschung, Grant. No. 20-111653.

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