- Email: [email protected]
BEC phase boundary in BaCuSi2O6
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462 www.elsevier.com/locate/jmmm
BEC phase boundary in BaCuSi2O6 Suchitra E. Sebastiana,,1, N. Harrisonb, C.D. Batistab, L. Balicasc, M. Jaimeb, P.A. Sharmab, N. Kawashimad, I.R. Fishera a
Department of Applied Physics, Geballe Laboratory for Advanced Materials, Stanford University, California 94305-4045, USA b MPA-NHMFL, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c National High Magnetic Field Laboratory, 1800 E. Paul Dirac Drive, Florida 32310, USA d Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Available online 7 November 2006
Abstract Torque magnetisation data are presented which probe the line of second order phase transitions approaching the Bose–Einstein condensation (BEC) quantum critical point (QCP) in BaCuSi2 O6 . Results reveal that as the temperature is lowered, the phase transition is increasingly dominated by quantum rather than thermal fluctuations near the QCP. r 2006 Elsevier B.V. All rights reserved. PACS: 75.50.y; 75.30.m Keywords: BEC; Quantum critical; BaCuSi2 O6
1. Introduction Critical power scaling behaviour of the Bose–Einstein condensation (BEC) quantum critical point (QCP) in the spin dimer compound BaCuSi2 O6 has been investigated in detail using torque magnetisation measurements [1,2]. Here, we present torque data on BaCuSi2 O6 , and discuss the nature of the measured finite temperature second order phase transition to the BEC ordered state. 2. Torque magnetisation Torque measurements on BaCuSi2 O6 were made between 30 mK and 1 K in a dilution refrigerator in a 33 T resistive magnet in the National High Magnetic Field Laboratory, Tallahassee. Fig. 1 shows the uniform magnetisation ðmz Þ obtained from the finite torque measured in an external magnetic field H applied at a small angle ðo10 Þ to the crystalline c-axis of BaCuSi2 O6 . Absolute values of mz / r (where r is the particle density) Corresponding author.
E-mail address: [email protected] (S.E. Sebastian). Current address: Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK. 1
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.394
were obtained by comparison with magnetisation data taken in pulsed fields [3]. Values of mz thus obtained are shown as a function of applied magnetic fields at different temperatures in Fig. 1. 3. Critical magnetic field At finite temperatures, the magnetic field H c at which an upturn in mz occurs indicates the critical field at which long range magnetic ordering takes place. At T ¼ 0, the ordering transition is at the QCP, which occurs at critical magnetic field H c1 . An accurate determination of H c is obtained from the position of a sharp feature in the second derivative of mz with respect to field. The following thermodynamic argument shows that this method of extracting the ordering transition is significantly more accurate and reliable than methods used in similar work on related compounds [4–7]. The non-analytic behaviour of the free energy F ðT; HÞ is dominated by a single relevant exponent. Hence, the singular behaviour of F ðT; HÞ when T and H are near a critical point on the line T c ðHÞ does not depend on the direction uðH H c ; T T c Þ of approach to the critical point, provided it is not tangent to the critical line [1].
ARTICLE IN PRESS S.E. Sebastian et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462
a
b 0.5
temperature of 190 mK in Fig. 1, and no other features are observed. 0.3
190 mK
e461
190 mK
0.3
mz (µB/Cu)
d2mz / dH2 (arb.units)
4. Quantum phase transition
0.0
0.2
0.1
0.0 20 22 24 26 28 30
0
µ0H (T)
5 10 15 20 25 30 µ0H (T)
c 60 0.056K
Of particular significance is the fact that the feature in the second derivative sharpens and grows as the temperature is reduced, approaching a divergence as T ! 0. This may be understood from the schematic phase diagram (after Ref. [9]) in Fig. 2. From Fig. 2, we observe that as the temperature is lowered, the region of phase space in the vicinity of the phase transition is dominated by quantum rather than thermal fluctuations. The feature observed in Fig. 3 is, therefore, seen to be weaker at higher temperatures where the transition is driven by thermal fluctuations, but approaches a divergence as the temperature is lowered, and the transition crosses over to a quantum phase transition. This is consistent with the singularity in q2 mz =qH 2 becoming a d-function at the QCP.
mz (10-3 µB/Cu)
0.113 0.230
40
5. Conclusion
0.400 0.525
Evidence from torque magnetisation confirms that the phase boundary of measured second order transitions approaches the quantum critical regime at experimentally accessible low temperatures. Results of power-law scaling analysis presented in Refs. [1,2], therefore, access the universal regime driven by quantum critical fluctuations.
0.700
20
0 20
21
22 µ0H (T)
23
24
Fig. 1. Uniform magnetisation ðmz Þ extracted from torque measured as a function of rising H applied close to the c-axis of BaCuSi2 O6 . A small background subtraction has been made to account for the cantilever response. (a) The second derivative of mz shown up to 32 T at 190 mK shows a sharp feature at the ordering transition. (b) The ordering transition and absence of other features in mz shown up to 32 T at 190 mK. (c) mz and ordering transitions for representative T, vertically shifted for clarity.
4.0 3.5 3.0
Tc (K)
2.5
We have q2 F =qu2 ¼ constant / ua
1.5
for
u ¼ T T c.
(1) 1.0
Choosing u ¼ H H c , we obtain q2 F =qH 2 ¼ qmz =qH / ðH H c Þa .
(2)
Since a 0:015 for a 3D XY-like transition [8], the second derivative of mz is divergent at the critical point: q2 mz =qH 2 / ðH H c Þ1a .
2.0
(3)
We thus obtain the ordering transitions from the feature in the second derivative of mz (shown for different temperatures in Fig. 3). Measured mz and its second derivative are shown up to higher magnetic fields for a
Driven by thermal fluctuations
0.5 0.0 22
24 µ0Hc1 QCP
26
28
30
32
µ0H (T)
Fig. 2. The measured phase boundary in BaCuSi2 O6 from Ref. [2], accompanied by a schematic tracing the width of the region driven by thermal fluctuations along the line of second order phase transitions approaching the QCP at H c1 (after Ref. [9]).
ARTICLE IN PRESS S.E. Sebastian et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462
e462
56 mK
0. 6 0. 4
0.4
0. 2
0.2
0. 0
0.0
d2mz / dH2 (arb. units)
-0.2 23.0
23.5
24.0 230 mK
0.6
-0.2 23.0
0. 4
0.2
0. 2
0.0
0. 0
23.5
24.0 525 mK
0.6
-0.2 23.0
0.4
0.2
0.2
0.0
0.0
23.5
24.0
-0.2 23.0
24.0 470 mK
23.5
24.0 700 mK
0.6
0.4
-0.2 23.0
23.5
0. 6
0.4
-0.2 23.0
96 mK
0.6
23.5
24.0
µ0 H (T)
Fig. 3. The second derivative of mz extracted from torque at various temperatures shows a sharp feature which grows with decreasing T, approaching divergence at T ¼ 0. The peak maximum defines the ordering transition, and the error is defined as the peak-tip width (i.e. 4–5 data points spaced at 0.005 T on either side of the maximum).
Acknowledgements This work is supported by the NSF, Florida State, and the DOE. I.R.F. acknowledges the Alfred P. Sloan Foundation and S.E.S. the Mustard Seed Foundation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
S.E. Sebastian, et al., Phys. Rev. B 72 (2005) 100404(R). S.E. Sebastian, et al., Nature 441 (2006) 617. M. Jaime, et al., Phys. Rev. Lett. 93 (2004) 087203. T. Nikuni, et al., Phys. Rev. Lett. 84 (2000) 5868. Y. Shindo, H. Tanaka, J. Phys. Soc. Japan 73 (2004) 2642. H. Tanaka, et al., J. Phys. Soc. Japan 70 (2001) 939. A. Oosawa, et al., Phys. Rev. B 63 (2001) 134416. M. Campostrini, et al., Phys. Rev. B 63 (2001) 214503. S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, 1999.
Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462 www.elsevier.com/locate/jmmm
BEC phase boundary in BaCuSi2O6 Suchitra E. Sebastiana,,1, N. Harrisonb, C.D. Batistab, L. Balicasc, M. Jaimeb, P.A. Sharmab, N. Kawashimad, I.R. Fishera a
Department of Applied Physics, Geballe Laboratory for Advanced Materials, Stanford University, California 94305-4045, USA b MPA-NHMFL, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c National High Magnetic Field Laboratory, 1800 E. Paul Dirac Drive, Florida 32310, USA d Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Available online 7 November 2006
Abstract Torque magnetisation data are presented which probe the line of second order phase transitions approaching the Bose–Einstein condensation (BEC) quantum critical point (QCP) in BaCuSi2 O6 . Results reveal that as the temperature is lowered, the phase transition is increasingly dominated by quantum rather than thermal fluctuations near the QCP. r 2006 Elsevier B.V. All rights reserved. PACS: 75.50.y; 75.30.m Keywords: BEC; Quantum critical; BaCuSi2 O6
1. Introduction Critical power scaling behaviour of the Bose–Einstein condensation (BEC) quantum critical point (QCP) in the spin dimer compound BaCuSi2 O6 has been investigated in detail using torque magnetisation measurements [1,2]. Here, we present torque data on BaCuSi2 O6 , and discuss the nature of the measured finite temperature second order phase transition to the BEC ordered state. 2. Torque magnetisation Torque measurements on BaCuSi2 O6 were made between 30 mK and 1 K in a dilution refrigerator in a 33 T resistive magnet in the National High Magnetic Field Laboratory, Tallahassee. Fig. 1 shows the uniform magnetisation ðmz Þ obtained from the finite torque measured in an external magnetic field H applied at a small angle ðo10 Þ to the crystalline c-axis of BaCuSi2 O6 . Absolute values of mz / r (where r is the particle density) Corresponding author.
E-mail address: [email protected] (S.E. Sebastian). Current address: Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK. 1
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.394
were obtained by comparison with magnetisation data taken in pulsed fields [3]. Values of mz thus obtained are shown as a function of applied magnetic fields at different temperatures in Fig. 1. 3. Critical magnetic field At finite temperatures, the magnetic field H c at which an upturn in mz occurs indicates the critical field at which long range magnetic ordering takes place. At T ¼ 0, the ordering transition is at the QCP, which occurs at critical magnetic field H c1 . An accurate determination of H c is obtained from the position of a sharp feature in the second derivative of mz with respect to field. The following thermodynamic argument shows that this method of extracting the ordering transition is significantly more accurate and reliable than methods used in similar work on related compounds [4–7]. The non-analytic behaviour of the free energy F ðT; HÞ is dominated by a single relevant exponent. Hence, the singular behaviour of F ðT; HÞ when T and H are near a critical point on the line T c ðHÞ does not depend on the direction uðH H c ; T T c Þ of approach to the critical point, provided it is not tangent to the critical line [1].
ARTICLE IN PRESS S.E. Sebastian et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462
a
b 0.5
temperature of 190 mK in Fig. 1, and no other features are observed. 0.3
190 mK
e461
190 mK
0.3
mz (µB/Cu)
d2mz / dH2 (arb.units)
4. Quantum phase transition
0.0
0.2
0.1
0.0 20 22 24 26 28 30
0
µ0H (T)
5 10 15 20 25 30 µ0H (T)
c 60 0.056K
Of particular significance is the fact that the feature in the second derivative sharpens and grows as the temperature is reduced, approaching a divergence as T ! 0. This may be understood from the schematic phase diagram (after Ref. [9]) in Fig. 2. From Fig. 2, we observe that as the temperature is lowered, the region of phase space in the vicinity of the phase transition is dominated by quantum rather than thermal fluctuations. The feature observed in Fig. 3 is, therefore, seen to be weaker at higher temperatures where the transition is driven by thermal fluctuations, but approaches a divergence as the temperature is lowered, and the transition crosses over to a quantum phase transition. This is consistent with the singularity in q2 mz =qH 2 becoming a d-function at the QCP.
mz (10-3 µB/Cu)
0.113 0.230
40
5. Conclusion
0.400 0.525
Evidence from torque magnetisation confirms that the phase boundary of measured second order transitions approaches the quantum critical regime at experimentally accessible low temperatures. Results of power-law scaling analysis presented in Refs. [1,2], therefore, access the universal regime driven by quantum critical fluctuations.
0.700
20
0 20
21
22 µ0H (T)
23
24
Fig. 1. Uniform magnetisation ðmz Þ extracted from torque measured as a function of rising H applied close to the c-axis of BaCuSi2 O6 . A small background subtraction has been made to account for the cantilever response. (a) The second derivative of mz shown up to 32 T at 190 mK shows a sharp feature at the ordering transition. (b) The ordering transition and absence of other features in mz shown up to 32 T at 190 mK. (c) mz and ordering transitions for representative T, vertically shifted for clarity.
4.0 3.5 3.0
Tc (K)
2.5
We have q2 F =qu2 ¼ constant / ua
1.5
for
u ¼ T T c.
(1) 1.0
Choosing u ¼ H H c , we obtain q2 F =qH 2 ¼ qmz =qH / ðH H c Þa .
(2)
Since a 0:015 for a 3D XY-like transition [8], the second derivative of mz is divergent at the critical point: q2 mz =qH 2 / ðH H c Þ1a .
2.0
(3)
We thus obtain the ordering transitions from the feature in the second derivative of mz (shown for different temperatures in Fig. 3). Measured mz and its second derivative are shown up to higher magnetic fields for a
Driven by thermal fluctuations
0.5 0.0 22
24 µ0Hc1 QCP
26
28
30
32
µ0H (T)
Fig. 2. The measured phase boundary in BaCuSi2 O6 from Ref. [2], accompanied by a schematic tracing the width of the region driven by thermal fluctuations along the line of second order phase transitions approaching the QCP at H c1 (after Ref. [9]).
ARTICLE IN PRESS S.E. Sebastian et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e460–e462
e462
56 mK
0. 6 0. 4
0.4
0. 2
0.2
0. 0
0.0
d2mz / dH2 (arb. units)
-0.2 23.0
23.5
24.0 230 mK
0.6
-0.2 23.0
0. 4
0.2
0. 2
0.0
0. 0
23.5
24.0 525 mK
0.6
-0.2 23.0
0.4
0.2
0.2
0.0
0.0
23.5
24.0
-0.2 23.0
24.0 470 mK
23.5
24.0 700 mK
0.6
0.4
-0.2 23.0
23.5
0. 6
0.4
-0.2 23.0
96 mK
0.6
23.5
24.0
µ0 H (T)
Fig. 3. The second derivative of mz extracted from torque at various temperatures shows a sharp feature which grows with decreasing T, approaching divergence at T ¼ 0. The peak maximum defines the ordering transition, and the error is defined as the peak-tip width (i.e. 4–5 data points spaced at 0.005 T on either side of the maximum).
Acknowledgements This work is supported by the NSF, Florida State, and the DOE. I.R.F. acknowledges the Alfred P. Sloan Foundation and S.E.S. the Mustard Seed Foundation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
S.E. Sebastian, et al., Phys. Rev. B 72 (2005) 100404(R). S.E. Sebastian, et al., Nature 441 (2006) 617. M. Jaime, et al., Phys. Rev. Lett. 93 (2004) 087203. T. Nikuni, et al., Phys. Rev. Lett. 84 (2000) 5868. Y. Shindo, H. Tanaka, J. Phys. Soc. Japan 73 (2004) 2642. H. Tanaka, et al., J. Phys. Soc. Japan 70 (2001) 939. A. Oosawa, et al., Phys. Rev. B 63 (2001) 134416. M. Campostrini, et al., Phys. Rev. B 63 (2001) 214503. S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, 1999.