**Quantum Phase Transitions**

Dimerised quantum magnets have emerged in the last decade as an ideal class of system for realising particularly clean quantum phase transitions (QPTs). This is a transition between different quantum ground states occurring at zero temperature and controlled by a non-thermal parameter, such as a pressure, doping or magnetic field. The nomenclature is explained in full in the seminal book of this name by Subir Sachdev.

Projects are listed here more or less in chronological order, so please skip down to read the latest news ...

Projects are listed here more or less in chronological order, so please skip down to read the latest news ...

**Altering Exchange Interactions**

Theoretically, one can always induce a transition by tuning the magnetic coupling constants [1], usually between disordered magnetic phases with a spin gap and ordered, gapless phases. We showed that this is more than a purely academic exercise by using it [2] to place poorly characterised compounds (

*e.g.*those for which single-crystal samples are unavailable) in the relevant magnetic phase diagram. More on these studies can be found under quantum magnetism and frustration. Some materials happen to be located close to an exchange-driven QPT [3], meaning that the energy scales relevant for physical phenomena are far smaller than the scales intrinsic to the system, and in these cases it is possible that (physical or chemical) pressure could indeed drive it through the transition. See quantum magnetism for more.**Field-Induced QPT**

Experimentally, the most transparent approach is to tune the magnetic field, and a number of quantum magnets exist where the excitation gap is small enough to close with a laboratory magnetic field. One is the organometallic, erstwhile spin-ladder system CuHpCl, which we studied both by semiclassical [4] and by microscopic [5] approaches. However, it was the three-dimensionally coupled dimer system TlCuCl3 which drew attention to the field, because i) it has a low critical field of 5.5 T and ii) it appears as large single crystals allowing accurate measurement of the magnetic excitations by inelastic neutron scattering (which has been performed at fields up to 14 T).

## We modelled TlCuCl3 using the bond-operator technique [6,7]. The crucial property of any theoretical

approach to the QPT is to use the same framework on both sides of the transition. The bond-operator formalism is naturally adapted to the quantum disordered phase of dimerised spin systems, and we developed it to give a unified, continuous description of the ordered side of the transition by a suitable choice of the ground state. While working on the complete story [7], we also discussed the interaction-tuned QPT and noted the following properties of the excitations.

- For field-induced quantum criticality, there is only one low-lying mode, which at the QPT becomes the sole massless phase mode (Goldstone mode) of the field-ordered magnetic regime. The dispersion at the band minimum remains quadratic at the critical field, with a linear component developing above this field [5-7]. Here the perturbation (the field) commutes with the (Heisenberg) Hamiltonian.
- For coupling-induced quantum criticality, at the QPT one has three massless, linearly dispersive spin waves; on the ordered side one expects only two spin waves (Goldstone modes), and a new type of dynamical excitation in the form of a low-lying but massive
*longitudinal*or*amplitude*mode of the ordered moment [3,7]. Here the perturbation does not commute with the Hamiltonian, which is why the mode mixed with the condensate is explicitly the one becoming massive.

**Magnon Bose-Einstein Condensation**

The magnetic field-driven QPT has been described by the notion of Bose-Einstein condensation (BEC) in the lowest-lying magnon mode, which in dimer systems may be represented by hard-core bosons. This description is valid in principle due to the

-- U(1) symmetry,

-- quadratic dispersion and

-- hard-core nature of the bosons (giving them a conserved number and conjugate chemical potential).

-- U(1) symmetry,

-- quadratic dispersion and

-- hard-core nature of the bosons (giving them a conserved number and conjugate chemical potential).

## We tested this in practice by using Stochastic Series Expansion QMC simulations [8], currently the

most advanced numerical technique available for large spin systems at low temperatures and close to QPTs, at which the spin gap vanishes (and characteristic lengths diverge). We showed that the BEC scenario is applicable, and indeed universal for all systems of the same geometry (

*i.e.*independent of coupling ratio). However, we found systematic deviations from the BEC description away from the transition, indicating that the concept is applicable only*at the QPT*.*Note, however, that the coupling-induced QPT is not a Bose-Einstein condensation, because the magnon dispersion is linear, not quadratic, at the QPT. While it can certainly be considered as a magnon condensation, it is not in the Bose-Einstein universality class.***The Pressure-Induced QPT**

In a major surprise, the pressure-induced phase transition in TlCuCl3 was found to occur at the shockingly low critical pressure of 1.07 kbar -- indicating both a very strong magnetoelastic coupling and a very special system. This led to what is indubitably the best piece of experimental work with which I have ever had the privilege to be associated: inelastic neutron scattering in a He gas pressure cell to map the triplet excitations across the QPT from 0 to 4 kbar. The result [9] was a clear and systematic evolution of the mode gaps which showed

- gap closure on the disordered side,
- a massless spin-wave on the ordered side,

## Properties of the longitudinal mode.

- a uniaxial anisotropy splitting off one mode on the disordered side which becomes a massive spin-wave on the ordered side and
**the clear and unequivocal emergence of the longitudinal mode**, its gap growing as the square root of the pressure.

*also grows as the square root of the pressure*. Thus as predicted by the theory of QPTs, where 3+1D is the upper critical dimension, the longitudinal mode is neither elementary nor overdamped, but

**is a critically damped excitation**.

The longitudinal mode is also a Higgs boson, which has been known in quasi-condensed matter (superconductors, CDWs, quantum magnets, cold atoms) in various guises for a lot longer than it has been in the Standard Model ... Our Higgs boson obviously has a much better lifetime than the one seen at LHC, and in fact it is by quite a long way the most clearly visible such longitudinal mode known to physics. There are various microscopic reasons for its good visibility in TlCuCl3 [9].

**Quantum Criticality **

We pushed these studies to finite temperatures to map out the behaviour of the system in the quantum critical regime. Here the dominant energy scale is neither the gap nor

*TN*but the temperature itself. We find [10] that quantum and thermal fluctuations have rather similar effects in destroying long-range order and opening a gap in the excitation spectrum, which is a quite remarkable result ... We also find- that quantum critical excitations are critically damped (their width scales with their energy),
- that the longitudinal "Higgs" mode of the ordered phase becomes overdamped at finite temperature,
- that a regime of classical criticality appears around the phase transition line TN(p).

**The Field-Induced QPT at Finite Temperature**

When both field and temperature are finite simultaneously, new effects enter which are not present if either is zero. In the theoretical analysis, these are manifest as magnetisation and triplet-correlation terms, and they lead to physical effects causing an observable deviation from field-induced Zeeman splitting. The full theoretical description of the disordered phase in anisotropic TlCuCl3 is complete but the analysis of the experiment is work in progress [11]. On the experimental front, the QPT is clearly of first-order type at finite temperatures, and this reflects the presence of magnetoelastic coupling terms. These are manifestly obvious from the existence of the pressure-induced QPT, but their inclusion in a consistent theory is a major step beyond the current level of sophistication.

**The Spin Ladder**

After a number of false hopes, including some of the materials mentioned on these pages, we now have a nearly-perfect spin ladder, the organometallic Cu system (C5H12N)2CuBr4 (bis-piperidinium copper bromide). Because the critical and saturation fields are respectively 7 and 14 T, one may now, without exaggeration, see "everything" there is to see in a spin ladder in this one material. We have done exactly this.

1) By combining a number of approaches, we made a comprehensive study of the thermodynamic response in the gapped, quantum disordered phase up to 7 T and in the gapless, field-induced Luttinger-liquid phase beyond this field, obtaining quantitatively accurate and "parameter-

1) By combining a number of approaches, we made a comprehensive study of the thermodynamic response in the gapped, quantum disordered phase up to 7 T and in the gapless, field-induced Luttinger-liquid phase beyond this field, obtaining quantitatively accurate and "parameter-

## free" fits of data from specific-heat and magnetocaloric effect (MCE) measurements [12]. [The

parameters are the ladder rung and leg exchange interactions and the

2) There is interladder coupling in any system, and here it leads to a field-induced three-dimensional order below 100 mK. We probed the phase boundary by MCE measurements and the phase itself by magnetic neutron diffration [13], using our results to calibrate numerical models for weakly coupled quantum systems.

*g*-factor, all extracted separately.]2) There is interladder coupling in any system, and here it leads to a field-induced three-dimensional order below 100 mK. We probed the phase boundary by MCE measurements and the phase itself by magnetic neutron diffration [13], using our results to calibrate numerical models for weakly coupled quantum systems.

## 3) The inelastic neutron scattering measurements of the dispersion in supercritical fields are

indubitably the second best piece of experimental work with which I have ever had the privilege to be associated. We saw [14] the magnon excitations of the disordered phase split continuously and be replaced by a Faddeev-type spinon continuum above the critical field. This field-induced fractionalisation is accompanied by a continuously tunable incommensuration across the critical (Luttinger-liquid) regime. The experimental results for all momenta, wavevectors and fields were fitted by numerical Bethe Ansatz calculations with one single scale factor for the overall intensity.

## 4) We have also presented [15] the complete theoretical description of the gapped (quantum disordered

and quantum critical) phase in a spin ladder at all fields and temperatures up to the Luttinger-liquid boundary. On taking into account the triplet-correlation and magnetisation terms mentioned above, the results are in quantitative agreement with numerics and experiment even at the phase boundary.

**Other Types of QPT**

In separate pieces of work, we have also studied the quantum phase transitions driven by changing the interaction parameter strengths in spin models where ring-exchange interactions [16], staggered magnetic fields [17] and single-ion anisotropy terms [18] compete with the Heisenberg interactions. These are discussed briefly under frustrated magnetism.

**QPTs in Electronic Systems**

Quantum phase transitions in the presence of charge degrees of freedom are more complex than in spin-only systems. One that we studied for a nanoscopic system was the topological (parity-inversion) transition in the ionic Hubbard model on a flux-threaded nanoring [19], which is reviewed here. We pursued our fascination with topological transitions in the -- admittedly very educational -- ionic Hubbard model to 3- and 4-site versions of the model [20], as reviewed here.

*Phase Diagram of the S = 1/2 Frustrated Coupled Ladder System*

B. Normand, K. Penc, M. Albrecht and F. Mila,

Phys. Rev. B**56**, R5736 (1997).*Magnetic Properties of the Coupled Ladder System MgV2O5*

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Phys. Rev. B**57**, R5005 (1998).*Dynamical Properties of an Antiferromagnet near the Quantum Critical Point: Application to LaCuO2.5*

B. Normand and T. M. Rice,

Phys. Rev. B**56**, 8760 (1997).*Nonlinear sigma-Model Treatment of Quantum Antiferromagnets in a Magnetic Field*

B. Normand, J. Kyriakidis and D. Loss,

Ann. Phys. (Leipzig)**9**, 133 (2000).*Quantum Antiferromagnets in a Magnetic Field*

B. Normand,

Acta Phys. Polonica B**31**, 3005 (2000).*Magnon Dispersion in the Field-Induced Magnetically Ordered Phase of TlCuCl3*

M. Matsumoto, B. Normand, T. M. Rice and M. Sigrist,

Phys. Rev. Lett.**89**, 077203 (2002).*Pressure- and Field-Induced Magnetic Quantum Phase Transitions in TlCuCl3*

M. Matsumoto, B. Normand, T. M. Rice and M. Sigrist,

Phys. Rev. B**69**, 054423 (2004).*Universal Scaling at Field-Induced Magnetic Phase Transitions*

O. Nohadani, S. Wessel, B. Normand and S. Haas,

Phys. Rev. B**69**, 220402 (2004).*Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl3*

Ch. Rüegg, B. Normand, M. Matsumoto, A. Furrer, D. McMorrow, K. Krämer, H.-U. Güdel, S. Gvasaliya, H. Mutka and M. Boehm,

Phys. Rev. Lett.**100**, 205701 (2008).*Quantum and Classical Criticality in a Dimerized Quantum Antiferromagnet*

P. Merchant, B. Normand, K. W. Krämer, M. Boehm, D. F. McMorrow and Ch. Rüegg,

Nature Physics**10**, 373 (2014).*Magnetic Field–Induced Quantum Critical Excitations and First-Order Classical Phase*

Transition in*TlCuCl3*

Ch. Rüegg, Ch. Niedermayer, A. Furrer, K. W. Krämer, H.-U. Güdel, H. Mutka and B. Normand,

unpublished.*Thermodynamics of the Spin Luttinger Liquid in a Model Ladder Material*

Ch. Rüegg, K. Kiefer, B. Thielemann, D. F. McMorrow, V. Zapf, B. Normand, M. B. Zvonarev, P. Bouillot, C. Kollath, T. Giamarchi, S. Capponi, D. Poilblanc, D. Biner and K. W. Krämer,

Phys. Rev. Lett.**101**, 247202 (2008).*Field-Controlled Magnetic Order in the Quantum Spin-Ladder System (Hpip)2CuBr4*

B. Thielemann, Ch. Rüegg, K. Kiefer, H. M. Ronnow, B. Normand, P. Bouillot, C. Kollath, E. Orignac, R. Citro, T. Giamarchi, A. M. Läuchli, D. Biner, K. W. Krämer, F. Wolff-Fabris, V. Zapf, M. Jaime, J. Stahn, N. B. Christensen, B. Grenier, D. F. McMorrow and J. Mesot,

Phys. Rev. B**79**, R020408 (2009).*Direct Observation of Magnon Fractionalization in the Quantum Spin Ladder*

B. Thielemann, Ch. Rüegg, H. M. Ronnow, J.-S. Caux, A. M. Läuchli, B. Normand, D. Biner, K. W. Krämer, H.-U. Güdel, J. Stahn, S. N. Gvasaliya, K. Habicht, K. Kiefer, M. Boehm, D. F. McMorrow and J. Mesot,

Phys. Rev. Lett.**102**, 107204 (2009).*Complete Bond-Operator Theory of the Two-Chain Spin Ladder*

B. Normand and Ch. Rüegg,

Phys. Rev. B**83**, 054415 (2011).*Phase Diagram of the Heisenberg Spin Ladder with Ring Exchange*

V. Gritsev, B. Normand and D. Baeriswyl,

Phys. Rev. B**69**, 094431 (2004).*Low-Energy Properties of Anisotropic Two-Dimensional Spin-1/2 Heisenberg Models in Staggered Magnetic Fields*

B. Xi, S. Hu, J. Z. Zhao, G. Su, B. Normand and X. Q. Wang,

Phys. Rev. B**84**, 134407 (2011).*Accurate Determination of the Gaussian Transition in Spin-1 Chains with Single-Ion Anisotropy*

S. Hu, B. Normand, X. Q. Wang and L. Yu,

Phys. Rev. B**84**, 220402 (2011).*Detection of Topological Transitions by Transport through Molecules and Nanodevices*

A. A. Aligia, K. Hallberg, B. Normand and A. P. Kampf,

Phys. Rev. Lett.**93**, 076801 (2004).*Quantum Phase Diagram of the Generalized Ionic Hubbard Model for ABn Chains*

M. E. Torio, A. A. Aligia, G. I. Japaridze and B. Normand,

Phys. Rev. B**73**, 115109 (2006).