Rukhsan Ul Haq

Renormalization Group Methods for Quantum Many-body Hamiltonians

Renormalization group methods become natural for quantum many-body systems where there is an emergence of new energy scales like as Kondo scale in case of Kondo model. It was the Numerical Renormalization Group which gave the full quantitative understanding of the strong coupling fixed point of Kondo model and also of Anderson Impurity model. However, it was Andersons poor man scaling method which gave the analytical insights into the scaling behaviour of quantum impurity models. That makes poor man scaling method very important, and hence it has been used extensively in carrying out renormalization studies of quantum impurity models. This method has been generalized and also applied to other quantum many-body Hamiltonians. In this paper, we give a detailed presentation of poor man scaling method and apply it to study the scaling behaviour of Kondo model and Anderson impurity model as well. Though there are many reviews of perturbative renormalization of Kondo model, in this paper we have also presented a detailed discussion about scaling analysis of Anderson impurity model. We also present another method called flow equation method which is a generalization of poor man scaling method and has recently been applied to study the quantum many-body dynamics both in equilibrium and out-of equilibrium. We also present a numerical procedure to extract Kondo scale from flow equations of Kondo model. The importance of this procedure is that it can be applied to calculate the Kondo scale for situations where poor mans scaling method can not be applied.

Introduction

Strong correlation leads to the renormalization of the bare quantities of the electrons in real systems. Landau's Fermi liquid theory, which has been a corner stone of condensed matter physics, is based on the renormalization of the free fermions due to weak correlation.

The effective description of Fermi liquid theory is in terms of "quasiparticles" which have got dressed due to renormalization effects. Similarly, all the bare properties of free electrons like mass, charge and lifetime also get renormalized, and it is these renormalized quantities which are physical rather than the bare quantities. At the level of quantum many-body Hamiltonian, the model parameters get renormalized, and as the energy scales of the model are changed, the parameters undergo renormalization flows. In the field of strongly correlated electron systems, renormalization group methods were first applied by Anderson 1,[START_REF] Anderson | Basic Notions of Condensed Matter Physics[END_REF] to get an understanding of Kondo model for which perturbation methods had given divergent results. Hence renormalization method played a very important role in understanding the quantum many-body physics of Kondo effect. Later on Wilson 3 came up with an even more robust method of renormalization called Numerical renormalization group(NRG) which helped eventually to solve what had been called "Kondo problem".

Renormalization methods become natural when we deal with many-body systems where there is an interplay between various degrees of freedom at different scales present in the system. The logarithmic divergence in the perturbation theory is a symptom of the many energy scales. The divergent integral takes the form dE E and as the energy scale goes to the infrared limit, there is a divergence. Another feature of these systems where renormalization group becomes important is the lack of an energy scale or in other words scale invariance.

Critical phenomena are the prime examples where there is no energy scale (hence scale invariance) and that is why understanding phase transitions necessitated scaling and renormalization group methods [START_REF] Kadanoff | Statistical Physics[END_REF][START_REF] Cardy | Scaling and Renormalization[END_REF][START_REF] Zinn-Justin | Phase transitions and Renormalization Group[END_REF] . In strongly correlated electron systems, Kondo physics was the first such phenomenon which exhibited scale invariance and hence the universality [START_REF] Anderson | Basic Notions of Condensed Matter Physics[END_REF][START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF] .

Renormalization group methods capture these scale invariant aspects of the systems by calculating the scaling equations and the scaling invariants of the quantum many-body system.

Renormalization group method finds extensive application in quantum field theory, statistical physics, condensed matter physics, non-linear dynamics [START_REF] Mccomb | Renormalization Methods: A guide for Beginners[END_REF] . By now, there are many different methods to do the renormalization group study in condensed matter physics which include, most importantly poor man scaling 7,[START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF] , flow equation method [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF] , functional renormalization group [START_REF] Kopietz | Schutz Introduction to the Functional Renormalization Group[END_REF] , numerical renormalization group [START_REF] Bulla | [END_REF] and density matrix renormalization group 17 . However, the underlying philosophy in all the methods is similar. One identifies high energy(ultraviolet) and low energy scales in the system. Then the high energy states are integrated out iteratively. This leads to a series of effective Hamiltonians which scale towards a fixed point where the scaling flow stops. However, a fixed point can be stable, unstable or marginal depending on what happens to the scaling flow under a perturbation. This gives the fixed point structure of the Hamiltonian and determines the scaling behaviour.

A Hamiltonian flows towards stable fixed point. The scaling flow of a given Hamiltonian can be written as a differential equation, often called beta function which can be solved to calculate the scaling invariants for the given model.

In this paper, we will discuss renormalization methods for the quantum impurity models which include Anderson Impurity model and Kondo model. In quantum impurity models, renormalization group methods have played a very important role. In these models, there are two main energy scales associated with charge and spin fluctuations. The energy scale associated with the charge fluctuations is higher as compared to that of spin fluctuations which survive down to the low energy scales. Schrieffer-Wolff transformation [START_REF] Schrieffer | [END_REF] is a unitary transformation which is used to project out the high energy real charge excitations and it generates an effective Hamiltonian which has spin fluctuations alone. The scaling procedure can be continued on the effective Hamiltonian. That we will do for Kondo model by applying Anderson's poor man scaling method of perturbative renormalization. We will extract the Kondo scale from the beta function of Kondo model. Then we will carry out scaling analysis of Anderson impurity model directly without integrating out the charge excitations.

Next, we will introduce another renormalization method which is an extension of Poor man scaling method. This method is called flow equation renormalization method. We will apply this method to Kondo model and show how we capture aspects of scaling behaviour of Kondo model which do not get captured in poor man scaling method. We will especially solve the flow equations for the Kondo spin operator and from them calculate dynamic spin susceptibility for Kondo model.

dg dlnD = -g 2 (1)
where J is Kondo coupling constant, g = Jρ 0 is dimensionless Kondo coupling constant and ρ 0 is the density of states at Fermi level. Though this differential equation looks very common and simple, its solution is very interesting and gave very important physical insights into Kondo physics.

g(D) = g 0 1 -g 0 ln( D 0 D ) (2)
Here D is the running band-width and D 0 is the initial band-width. The solution shows that as the band width is decreased and hence the high energy states are projected out, Kondo coupling constant grows and in fact it diverges logarithmically as the band-width approaches zero. This is the well-known Kondo divergence which plagues perturbative expansions and is symptomatic of the the formation of Kondo singlet. Later on, it was confirmed that Kondo coupling actually diverges when Kondo effect takes place and consequently Kondo singlet is formed. We can define Kondo scale as the energy scale at which flow of coupling enters strong coupling regime. So Kondo scale is the energy scale at which coupling constant diverges, and perturbation expansion breaks down. Kondo scale itself is a scaling invariant and hence does not depend on band-width.

Poor Man's Scaling of Kondo model

In this section, we will carry out the poor man's scaling analysis of Kondo model 1 . In literature, this method has been called Poor man's scaling and poor man scaling method, so we will use these names interchangeably. Poor man scaling method being very important method has been discussed by many authors including [7][START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF][START_REF] Phillips | Advanced Solid State Physics[END_REF][START_REF] Solyom | [END_REF][11] . Particularly the last two authors have done a critical study of this method. We will mainly follow 8 and 9 in this section.

First, we will show how Anderson arrived at renormalized interaction using the T matrix and projection operator algebra. Then we will calculate the scaling equation for the Kondo couplings by considering the renormalization effects on various spin scattering processes.

Since Kondo effect involves spin scattering, so one introduces the T matrix which incorporates the scattering effects due to interactions.

T (ω) = V int + V int G 0 (ω)T (ω) (3)
G 0 (ω) = 1 ω-H 0 is the resolvent operator corresponding to H 0 = kσ k c † kσ c kσ for the conduction electrons. For the Kondo model V int is given by:

V int = J ± 2 (S + s -+ S -s + ) + J z S z s z (4)
To project out the high energy states from the conduction band we need to introduce a projection operator P δD which projects onto the states which have at least one particle in the range(D-δD,D) or one hole in the range(-D,-D+δD). (1-P δD ) is the projection operator for orthogonal subspace. Using the properties of the projection operators, T matrix can be written as:

T = V int + V int (1 -P δD )G 0 T + V int P δD G 0 T (5)
Substituting for T matrix in third term we get:

T = V int + V int (1 -P δD )G 0 T + V int P δD G 0 V int + V int P δD G 0 V int (1 -P δD )G 0 T + V int P δD G 0 V int P δD G 0 T (6)
Re-arranging the terms, T matrix can be written as:

T = V int + V int P δD G 0 V int + (V int + V int P δD G 0 V int )(1 -P δD )G 0 T + V int P δD G 0 V int P δD G 0 T (7)
We can define V int = V int +V int P δD G 0 V int as the renormalized interaction and after neglecting the last term, T matrix takes the form.

T = V int + V int (1 -P δD )G 0 T (8)
T matrix is of the same form as in equation 3, however the interaction has got renormalized.

Projecting out the high energy states renormalizes the interaction while keeping the form of T matrix same. ∆V int = V int -V int gives the renormalized interaction:

∆V int = V int P δD G 0 V int ∆V int = k 2 σ 2 k 1 σ 1 kσ c † k 2 σ 2 (τ ) σ 2 σ c kσ .s d P δD ω -H 0 c † kσ (τ ) σσ 1 c k 1 σ 1 (9)
We have used the Abrikosov's pseudofermion representation for conduction electrons.

S z = 1 2 (c † k↑ c k↑ -c † k↓ c k↓ ) S + = c † k↑ c k↓ S -= c † k↓ c k↑ (10)
This is how renormalized interaction was derived by Anderson 1 . Though Anderson arrived at the renormalized interation using T matrix and the fact that it should remain invariant under scaling transformation, we can obtain the renormalized interaction using effective Hamiltonian theory [START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF]20 . This method was later on used by Haldane 18 and Jefferson 19 to do the scaling analysis of asymmetric Anderson impurity model and by Kuramoto 20 for the renormalization of multi-channel Kondo models. The details of this method can be found in [START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF][START_REF] Phillips | Advanced Solid State Physics[END_REF] . In this paper, we will apply the effective Hamiltonian way of doing the poor man scaling analysis of quantum impurity models. The effective Hamiltonian formulation of poor man scaling is an iterative way of generating effective Hamiltonians in which using the projection operators, the high energy states are being projected out and the resulting effective Hamiltonian acts on smaller Hilbert space. The renormalization effects of the projected out states gets incorporated in the coupling constants. So unlike Anderson's original formulation in terms of T matrix, where the original problem is being mapped to a simpler problem keeping the physics(T matrix) invariant, in the effective Hamiltonian method, one maps the original Hamiltonian to low energy effective Hamiltonian which has smaller Hilbert space and is simpler than the original Hamiltonian, the physics is kept invariant by incorporating the effects of the high energy states on the coupling constants of the low energy effective Hamiltonian. The effective Hamiltonian can be calculated as 19,20 :

H ef f ( D) = (1 -P δD )H(1 -P δD ) + (1 -P δD )HP δD + (1 -P δD )HP δD 1 E -P δD HP δD P δD H(1 -P δD ) (11)
The renormalized interaction is given by the third term. When we substitute Kondo interaction for H V in above equation, we arrive at the renormalized interaction calculated using T matrix approach. To carry out the poor man scaling analysis of Kondo model, we need to consider the renormalization effects of all the spin scattering processes. Spin conserving scattering processes will renormalize longitudinal Kondo coupling while as the transverse Kondo coupling will get renormalized due to spin flip(transverse) scattering processes.

Renormalization of J z

In this case only those scattering processes will contribute which are spin conserving and hence involve two spin flip scattering.

J + J - q S -c † k ↑ c q↓ 1 ω -H 0 q S + c † q ↓ c k↑ (12)
The sum on q over the intermediate states is restricted within δD from the top of the band.

Because the band edge states are originally unoccupied, we get c q c † q = δ qq . Using the commutator [H 0 , c † q ↓ c k↑ ] = ( qk )c † q ↓ c k↑ , Equation 12 simplifies to

J + J -δDρ 0 S -S + c † k ↑ c k↑ (E -q -k -H 0 ) -1 (13)
Setting q = D and using the relation S -S + = h 2 2 -hS z , we arrive at

J + J -δDρ 0 h 2 ( 1 2 - S z h )c † k ↑ c k↑ (E -D + k ) -1 (14)
Similar contribution when the holes are involved in scattering is given below.

J + J -δDρ 0 h 2 ( 1 2 + S z h )c k↑ c † k ↑ (E -D -k ) -1 (15)
Here we have set q = -D and S + S -= h 2 2 + hS z . Similar contribution from the down spin electrons and holes are:

J + J -δDρ 0 h 2 ( 1 2 + S z h )c † k ↓ c k↓ 1 E -D + k (16)
J + J -δDρ 0 h 2 ( 1 2 - S z h )c k↓ c † k ↓ 1 E -D -k (17)
Summing the contributions from all these terms and comparing with the original Hamiltonian we obtain the renormalized longitudinal coupling.

δJ z = -J -J + ρ 0 δD 1 E -D + 1 E -D (18)
Renormalized transverse Kondo coupling can be obtained in a similar manner by considering all spin-flip scattering processes.

δJ ± = 1 2 J ± J z ρ 0 δD 1 E -D + 1 E -D (19)
The renormalized Kondo couplings are energy dependent which shows that the renormalization leads to retardation of effective interaction. For low energy excitations close to Fermi level, E dependence can be neglected compared to D and k , k can be set to zero. In this case the the scaling equations for anisotropic Kondo model become:

dJ z dlnD = -2ρ 0 J + J - ( 20
)
dJ ± dlnD = -2ρ 0 J z J ± (21)
Dividing these two scaling equations and integrating gives the scaling trajectories which are hyberbolic curves.

J 2 z -J 2 ± = κ (22)
J z always increases as the band-width is decreased. In case of ferromagnetic models, J z < 0 and |J z | > J ± , the above equation shows that J ± vanishes along the scaling flow. While as for antiferromagnetic case, Kondo coupling grows and leads to divergence for the perturbative renormalization methods. For the antiferromagnetic case, scaling equation can be solved for

J ± = J z dJ dlnD = -J 2 ρ 0 (23)
J(D) = J(D 0 ) 1 + ρ 0 J(D 0 ) ln D D 0 (24)
Scaling trajectories are characterised by a scaling invariant which also defines the Kondo scale T k .

De

-1 2Jρ 0 = De -1 2 Jρ 0 = k B T k ( 25
)
Kondo temperature is the only energy scale present in the Kondo regime, and thermodynamic quantities depend only on this scale and hence show universality.

Scaling analysis of Anderson model

In this section, we will extend perturbative renormalization to 19 . Though both papers had a similar motivation of understanding the mixed valence regime of asymmetric SIAM, in 18 the scaling invariants and scaling trajectories of the model parameters have been calculated.

Perturbative renormalization was done till second order where hybridization was not found to get renormalized. Renormalization of impurity energy levels was also calculated. In 19 third order perturbative renormalization of the model was done, and scaling equation was also obtained for hybridization.

In this section, we will apply poor man scaling method to study the scaling behaviour of Anderson impurity model following [START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF]18 . We will calculate the scaling equations for the model parameters and also the renormalization of impurity energy levels. We begin with a calculation of the renormalization of the impurity energy levels. This will allow us to extract the scaling equations for the orbital energy, d and the Hubbard U . As mentioned in the previous section, projecting out the high lying conduction band states yields an effective interaction as follows:

H v ( D) = (1 -P δD )H v P δD 1 E -P δD H c P δD -H loc P δD H v (1 -P δD ) (26)
where H c is the Hamiltonian for conduction electrons and H loc is the the Hamiltonian for the impurity. Since there are real charge fluctuations involved in this model, the natural choice for representing impurity operators is through Hubbard operators. These are generalized projection operators and satisfy the following superalgebra:

[X pq , X lm ] = δ ql X pm -δ mp X lq (27)
The usual fermionic operators can also be represented in X operator representation as:

d † σ = X σ:0 + ηX 2:σ (28)
Where η = ∓ for for down and up spin respectively. In the new representation, the hy-bridization term becomes:

H v = kσ V k c † kσ (X 0:σ + ηX σ:2 ) + V * k X σ:0 + ηX 2:σ )c kσ ) (29)
This implies that

H 01 = kσ V k c † kσ X 0:σ (30)
H 21 = kσ V k ηX 2:σ c kσ (31)
Similarly the local part of the Hamiltonian can be written in Hubbard operator representation:

H loc = σ d n dσ + U n d↑ n d↓ (32) = E 0 X 00 + σ E 1 X σσ + E 2 X 22 (33)
where E 1 , E 2 , E 3 are the energies of empty, singly occupied and doubly occupied enegy impurity states.

The contribution from the term H 10 1

E-QH 0 Q H 01 is q V q (X σ:0 + ηX 2:σ )c qσ 1 E -P δD H c P δD -H loc q V * q c † qσ (X 0:σ + ηX σ:2 ) = q V q V * q (X σ:0 X 0:σ 1 E -P δD H c P δD + d -q ) + q V q V * q (X 2:σ X σ:2 1 E -P δD H c P δD + d + U -q ) (34) = q V q V * q 1 E -q + d X σ:σ + q V q V * q 1 E -q + d + U X 2:2 = q V q V * q 1 -D + d X σ:σ + 1 -D + d + U X 2:2 (35)
The contribution from the term

H 12 1 E-QH 0 Q H 21 q V q c qσ (X σ:0 + ηX 2:σ ) 1 E -P δD H c P δD -H loc q V * q c † qσ (X 0:σ + ηX σ:2 ) = q V q V * q X σ:0 X 0:σ 1 E -P δD H c P δD -d -U -q + q V q V * q X 2:σ X σ:2 1 E -P δD H c P δD -d -U + q = q V q V * q 1 -D -d -U X σ:σ + 1 -D + d + U X 2:2 (36)
The contribution of the term

H 12 1 E-QH 0 Q H 01 q V * q c † qσ (X 0:σ + ηX σ:2 ) 1 E -P δD H c P δD q V q (X σ:0 + ηX 2:σ )c qσ ) = q V * q V q X 0:σ X σ:0 1 -q -d + X σ:2 X 2:σ 1 -q -d -U (37) = q V q V * q (X 0:0 1 -q -d + q | V q | 2 X σ:σ 1 -q -d -U = q V q V * q X 0:0 1 -D -d + X σ:σ 1 -D -d -U (38)
Summing up all the contributions from all the terms and comparing them with the local Hamiltonian given in equation 33, we get the following equations for the renormalization of the quantum impurity energy levels:

E 0 = E 0 - 2 π | δD | D + d (39)
E 1 = E 1 - | δD | π 1 D -d + 1 D + d + U (40) E 2 = E 2 - 2 | δD | π 1 D -d -U (41)
where E 0 , E 1 , E 2 are the energies of empty, singly occupied and doubly occupied impurity electron states. Given these renormalized energies, the scaling equations for the interaction strength and the orbital energy, may be obtained

through d = E 1 -E 0 and U = E 2 -2E 1 + E 0 8 as: dU dD = - 2∆ π 1 D -d + 1 D + d + U - 1 D -d -U - 1 D + d (42) d d dD = - ∆ π 2 D + d - 1 D -d + 1 D + d + U (43)

Renormalization of Hybridization

The hybridization does not get renormalized at the second order level and hence is taken as a scaling invariant for the Anderson impurity model 18 . However at third order, hybridization does get renormalized and Jefferson has calculated the corresponding scaling equations 19 .

In this section,following 19 , we will calculate the scaling equations for hybridization. Here also we will continue to keep doubly occupied state decoupled and hence there will be no contributions of the processes to/from that state. The third order contributions to the effective Hamiltonian are given by:

H v ( D) = (1 -P δD )H v α G α H v G α H v (1 -P α δD ) (44) -(1 -P δD )H v α G α ( α G α H v (1 -P α δD )H v (1 -P α δD )
where G α = P δD Eα-H 0 is the projected resolvent and α, α are the indices for the degenerate states. Since we have excluded the doubly occupied state, the first term in the above equation will not contribute. So the second term is the only third order contribution to the effective Hamiltonian. To get the scaling equation, we need to calculate this term for our model.

We will see that the two terms of the hybridization (c † kσ d σ and its Hermitian conjugate) get renormalized in different ways so we write them as follows and find the scaling equations separately for them.

H v = kσ V k1 c † kσ d σ + V k2 d † σ c kσ (45)
H 3 ef f = kk qσσ αα -V k2 V k1 V k2 Q δD d † σ c k σ G α G α c † k σ d σ Q α δD d † σ c kσ Q α δD (46) -V k1 V k2 V k1 Q δD c † kσ d σ G α G α d † σ c kσ Q α δD c † kσ Q α δD = - kqσ V k2 V k1 V k2 d † c qσ c † qσ d σ d † σ c kσ (-q + d )( q -k ) - qkσ V k1 V k2 V k1 c † qσ d σ d † σ c qσ c † kσ d σ (-q -k )(-q -d ) (47)
Once again using the fact there are no particles/holes in high energy states and summing over the intermediate states, we get the renormalized hybridization expressions as follows:

V 2 -V 0 = -ρ 0 V 1 V 2 2 δD (D -d )(D -k ) (48) V 1 -V 0 = -2ρ 0 V 2 V 2 1 δD (D + d )(D + k ) ( 49
)
The scaling equations for the hybridization can then be written from the above equations:

dV 2 dD = ρ 0 V 1 V 2 2 (D -d )(D -k ) ( 50
)
dV 1 dD = 2ρ 0 V 2 V 2 1 (D + d )(D + k ) (51)
In the mixed valent regime, and close to the Fermi level, we can choose k = 0, d = 0. If we divide the two scaling equations for hybridizations and integrate, we find that V 1 and V 2 are

related as V 1 = V 2 2
V 0 . This relation implies that V 2 and V 1 renormalize in exactly the same way in this limit ( d = 0 or | d | D). It also needs to be noted that we have ignored the momentum dependence of the hybridization amplitudes which is physically reasonable in this regime because of the closeness to Fermi level. So the scaling equation for V 2 becomes:

dV 2 dD = V 2 2 V 1 ρ 0 D 2 (52)
Solving for V 2 we arrive at the following equation which was obtained by Jefferson 19 .

V 2 = V 0 1 + ρV 2 0 4D 0 D 0 D -1 -1/3 (53)
As can be seen from the above equation, the hybridization becomes weaker under the scaling flow. Re-arranging the terms we arrive at following scaling invariant for Anderson impurity model. At second order, hybridization does not have scaling flow and hence is a scaling invaraint. At third oder hybridization gets renormalized in such a manner that the below given expression remains a scaling invariant. Though the scaling equation was obtained in 19 , scaling invaraint as given below was not derived there.

1 V 3 2 - ρ 0 4V 0 D = 1 V 3 0 - ρ 0 4V 0 D 0 (54)

Flow Equation Renormalization Method

In the previous sections we have used poor man scaling method, which is a perturbative renormalization scheme to understand the scaling behaviour of quantum impurity models.

The perturbative renormalization methods are very important in getting analytical insights into renormalization of quantum many-body Hamiltonians. However due to their perturbative nature they break down in the strong coupling regime. So it becomes natural to ask whether there are methods which can be used to study the renormalization flows of the models which exhibit strong coupling physics. In this section we will introduce one such method which has recently found extensive applications in quantum many-body physics.

This method is called flow equation renormalization group (FERG) method or flow equation method (FEM) for short [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF] . We will present the formalism of this method and apply it to study the renormalization flow of Kondo model which exhibits strong coupling physics.

This method is an extension of poor man scaling method as it can be applied to situations where perturbative methods either fail or need to be modified. One very important example where flow equation method describes the physics in a natural manner is non-equilibrium quantum many-body dynamics [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF][24][25][26] . Perturbative methods fail here by their very construction as they depend on projecting out the high energy states which can not be done in the non-equilibrium situation.

Flow equation renormalization method is a non-perturbative method and hence is able to capture the strong coupling regime of Kondo model 23 and other models like Sine-Gordon model [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF] . Hence FEM captures the full crossover from weak-coupling to strong coupling fixed points. While poor man scaling is applicable only in weak coupling and breaks down precisely at the strong coupling fixed point where the coupling constant diverges. Flow equation method also integrates very well with bosonization. In fact, FERG is similar to bosonization in the sense that it is a non-perturbative method to calculate effective Hamiltonians of quantum many-body systems. So mathematically it gives us a quadratic form for a fermionic system whose Hamiltonian also has quartic interaction terms. Bosonization uses Kac-Moody algebras(current algebras) [START_REF] Gogolin | Tsvelik Bosonization and Strongly Correlated Systems[END_REF] to write down the Hamiltonian, and when we do the same procedure in FERG, we end up having closed form solutions to flow equations. The flow equations for the Kondo model written in bosonized form, have closed form solutions and Kondo coupling constant flows to Toulouse point which turns out to be the strong coupling fixed point.

In flow equation method we can also study the renormalization of observables and calculate correlation and response functions. This is one of the distinct advantages of flow equation method over perturbative methods like poor man scaling in which we can only study the renormalization flows of the coupling constants. In flow equation method, there is an expansion of observables which is very similar to operator product expansion(OPE).

However, this expansion of observables has to satisfy mathematical consistency and physical plausibility conditions. The expansion has to conserve canonical commutation(bosons)or anti-commutation(fermions) relations. Similarly, the expansion has to respect the sum rules as applicable to correlation and response functions. Since flow equation method is more general method than poor man scaling so the scaling equations of latter can be recovered in a limit called infrared parametrization in which case momentum/energy is restricted close to Fermi level.

Recently there have been many new applications and developments of flow equation method. It has been used to calculate the entanglement entropy for quantum many-body systems, and hence a new direction called "flow equation holography" [START_REF] Kehrein | Flow Equation Holography[END_REF] has opened up which looks very promising for bringing holographic methods to quantum many-body systems. In a similar vein, flow equation method has been integrated with tensor network renormalization methods [START_REF] Serkan Sahin | [END_REF] and the corresponding scheme has been called "Entanglement Continuous unitary transformation"(e-CUT for short). Since flow equation method brings the quantum manybody Hamiltonian in the diagonal or block-diagonal form, so it was used 30 with density matrix renormalization group and applied in quantum chemistry 31 .

Formalism

Flow equation method is Hamiltonian renormalization method which was introduced by Rather it removes the interaction matrix elements and hence attains the diagonalization.

H(l) = U (l)H(0)U † (l) ( 55
)
where l is the continous parameter. H(0) is the original Hamiltonian for l = 0

dH dl = [η(l), H(l)] (56)
where η(l) = dU (l) dl U † . One can see that η(l) is an anti-hermitian operator.

η † (l) = U (l) dU † (l) dl = - dU dl U † (l) = -η(l) (57)
Here comes the important contribution of Franz Wegner 21 who gave the method to calculate the canonical generator. Wegner's generator η is given by the commutator of diagonal and off-diagonal terms of the Hamiltonian.

η(l) = [H 0 (l), H v (l)] (58)
Wegner's generator removes the off-diagonal matrix elements and hence in the limit l -→ ∞,

Hamiltonian is either diagonal or band-diagonal.

dH(l) dl = [η(l), H(l)] ( 59
)
dH dl ij = (ηH) ij -(Hη) ij (60) = k ( i -k )h ik h kj -( k -j )h kj h ik (61) = k ( i + j -2 k )h ik h kj (62)
So the evolution of matrix elements is given by:

dh ij dl = k ( i + j -2 k )h ik h kj (63)
From this equation we can write down the flow of diagonal and off-diagonal elements:

d i dl = k 2( i -k )h ik h kj dV ij dl = k ( i + j -2 k )h ik h kj (64)
One of the very important properties of Wegner's generator is that it leads to the decay of the off-diagonal elements. That can bee shown by using the invariance of trace under unitary transformations.

T r(H

2 ) = ik h ik h ki = k 2 k + ik,i =k h ik h ki (65) d dl T r(H 2 ) = ik d dl (h ik h ki ) = 0 (66)
Invariance of trace under the unitary flow leads to following relation between the flow and diagonal and off-diagonal elements in Equation 64.

k d dl 2 k = - ik,k =i d dl (h ik h ki ) (67) k d 2 k dl = k 2 k d k dl (68) d dl k 2 k = 2 i,k ( i -k ) 2 | v ik | 2 ≥ 0 ( 69
)
One can see from the last equation that the off-diagonal elements decrease proportional to the energy differences between the two levels. The interaction elements with largest energy differences vanish first and this process continues until only diagonal elements remain.

Thus, Wegner's choice of the generator is appropriate to calculate the flow equations of

Hamiltonians. There are other choices of the generator for flow equations like as the one by Glazek and Wilson 22 , in this paper, we will always use Wegner's choice of generator.

Flow Equation Method treatment of Kondo Model

In this section we will carry out flow equation renormalization of Kondo model following [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF] .

We write Kondo model as:

H(l) = H 0 + H int (l) (70)
where

H 0 = t,α t c † tα c tα (71)
H int (l) = t t J t t (l) : S.s t t : (72)
Here t and t' are general indices which in this case represent momenta. One should note that normal ordering prescription which is used for quantum many-body systems has been incorporated. The conduction electron spin desnity is given by:

s t t = α,β c † t α σ αβ 2 c tβ (73)
First we calculate the generator for flow equations of Kondo model.

η 1 (l) = [H 0 , H int (l)] (74)
η 1 (l) = t t η 1 t t (: S.s t t :) (75)
where η 1 t t (l) = ( tt )J t t (l). The commutator of generator with the diagonal part of the Hamiltonian is

[η 1 (l), H 0 ] = - t t ( t -t ) 2 J t t (l) : S.s t t : (76)
To calculate the commutator of the generator with the interaction term needs working out some algebra which has been done in following.

C 1 = [η 1 (l), H int (l)] (77)
= [η 1 (l) : S.s t t :,

u u J u u (l) : S.s u u :] (78)
B 1 = [: S.s t t :, : S.s u u :]

= i 8 ijk ijk σ i αβ σ j µν S k (: c † t α c tβ :: c † u µ c uν : + : c † u µ c uν :: c † t α c tβ :) + 3 16 α (δ tu : c † t α c uα : -δ t u : c † u α c tα :) + 3 8 δ tu δ t u (n(t ) -n(t)) (79)
After doing the normal ordering one gets

B 1 = i : S.(s t t × s u u ) : + : S.s t u : δ tu (n(t) -1/2)-: S.s u t : δ t u (n(t ) -1/2) + 3 16 α (δ tu : c † t α c uα : -δ t u : c † u α c tα :) + 3 8 δ tu δ t u (n(t ) -n(t)) (80)
Fermi functions arise due to the normal ordering prescription which is used in flow equation method. Plugging in the expression of B 1 in the the commutator C 1 we obtain:

C 1 = i t,t ,u,u ( t -t )J t t J u u : S.(s t t × s u u ) : + t ,t,v ( t + t -2 v )J t v J vt (n(v) -1/2) : S.s t t : + 3 16 t ,t,v,α ( t + t -2 v )J t v J vt : c † t α c tα : + 2 × 3 16 t,v (2 t -2 v )J tv J vt n(t) (81)
The flow equation for Kondo coupling till first loop is:

dJ t t dl = -( t -t ) 2 J t t + v ( t + t -2 v )J t v J vt (n(v) -1/2) (82)
Unlike poor man scaling where Kondo coupling is momentum independent and hence there is only one scaling equation for isotropic Kondo model, the flow equation given above is actually a system of equations corresponding to different momenta. The set of equations in non-linear and coupled. Hence analytical solution is not possible except in some special cases. Infrared limit is one such special case and we will now discuss that in next section.

Infrared parametrization

To get the scaling behaviour of the coupling constant at energies close to Fermi level(infrared limit) we will use infrared parametrization and calculate the beta function to the leading order. We will extract the Kondo scale from the flow equation which we obtain after infrared parametrization.

J t t (l) = J IR N (l)e -l( t -t) 2 (83)
where J IR is the Kondo coupling close to Fermi level(t = t = 0). Substituting it in the flow equation one gets:

dJ IR dl = -2J 2 IR 1 N ν ν e -2l 2 ν (n( ν ) - 1 2 ) (84)
= -2J 2 IR d ρ( ) e -2l 2 (n( ) - 1 2 ) (85)
To proceed further, we need to specify density of states. We take the constant density of states with conduction band width 2D ( ∈ [-D, D]) and set the temperature to be zero.

That makes electron occupation function a step function.

dg dl = g 2 2l (1 -e -2lD 2 ) ( 86
)
Where g = ρJ IR is dimensionless Kondo coupling. For l ≤ D -2 flow is negligible.

dg dl = g 2 2l ( 87
)
We can obtain the poor man scaling equation by noting that RG scale Λ is related to flow parameter as Λ = l -1 2 and dl = -2ldlnΛ

dg dlnΛ = -g 2 (88)
This is one-loop scaling equation for isotropic Kondo model. This is the scaling equation which we obtained from poor man scaling analysis of isotropic Kondo model.

Extraction of Kondo Scale

Flow equation can be solved for running coupling constant.

g g 0 dg g 2 = D -2 D -2 0 dl 2l 1 g - 1 g 0 = ln( D D 0 ) g = 1 1 g 0 -ln( D 0 D ) (89)
Energy scale can not be continuously decreased, and there is special energy scale at which coupling constant diverges. This scale is called Kondo scale T k and physically corresponds to the Kondo singlet formation.

T k ∼ D 0 e -1 g 0 (90)
We have seen that in infrared limit, flow equations recover poor man scaling equations.

Infrared parametrization of flow equations is important for another reason as well. As

shown above we can extract Kondo scale from flow equations in this limit by using the definition of Kondo scale as the scale at which flow equations diverge in infrared limit, as used in poor man scaling. Based on this idea, we have come up with a numerical procedure for extraction of Kondo scale from the numerical solution of flow equations as well. We will describe the details of this numerical procedure in next section.

Numerical Solution of Flow equations of Kondo Model

Flow equations are coupled non-linear differential equations. They generally can not be solved analytically except in some special limits e.g. in the infrared limit when the momentum of the coupling constants is restricted to be close to Fermi level. In this case, flow equations reproduce the results of the conventional scaling methods like poor man scaling. Flow equations can be solved by numerical methods like as Runga-Kutta methods.

The number of flow equations to be solved scales as O(N 2 ) where N is the number of energy states of the conduction band in case of Kondo model. Since flow equations are renormalization flows and hence they meet many different energy scales during the unitary flow and hence become stiff also. In this section, we solve the flow equations of Kondo model numerically. We have used DOPRI5 Fortran subroutine as the solver. DOPRI5 is based on fifth order Runga-Kutta method. As we have seen above that in the infrared limit, However, the main significance of our numerical procedure is that we can extract Kondo scale from the numerical solution of flow equations. From the solution of flow equation in infrared limit as given in Equation 89, we find that T k = 1 √ lc where l c is the flow parameter value where Kondo coupling diverges. In the inset is plotted the Kondo scale which we have extracted from our procedure, versus inverse Kondo coupling and as can be seen that it a straight line. We also found that slope of this straight line is 2(within numerical error) as expected from the expression ln T k = ln D -2 1 J IR . Our procedure can be used to extract Kondo scale for the cases where poor man scaling can not be applied. In chapter 6, we have applied this procedure to extract Kondo scale for Majorana-Kondo model.

In right panel of Figure 1, we see that as temperature is increased, there is softening of Kondo divergence which is because of going away from Kondo temperature and also there are incoherent thermal fluctuations which dominate over Kondo spin fluctuations.

Flow equation for Observables

In the flow equation method, we can calculate the renormalization flows of observables as well. The way observables and their time evolution is calculated in FEM is quite different than that of conventional many-body methods. Solving the Heisenberg equations of motion for observables is easier in the flow equation method. The reason for that is the in FEM Hamiltonian is brought in diagonal(or block-diagonal) form and then evaluating the dynamics with quadratic Hamiltonian becomes easier. While as to evaluate the dynamics with the original Hamiltonian is very difficult. In this section, we will present the formalism for calculating the renormalization flows of the observables. And then we will apply this formalism to calculate the dynamical spin susceptibility of Kondo model. Zero Temperature expectation value of observable O is given by:

O GS = Ψ GS |O|Ψ GS ( 91
)
|Ψ GS is the ground state of the full interacting Hamiltonian.

H|Ψ GS = E GS |Ψ GS (
O GS = ΨGS | Õ| ΨGS (94) Õ = U (l → ∞)OU † (l → ∞) (95)
The observable O satisfies the flow equation

dO(l) dl = [η(l), O(l)] (96)
So under the unitary flow the observable O becomes a linear combination of infinitely many operators.

O(l → ∞) = a t a (O)T a (97)
The coefficients t a depend on the observable. Since H is in the diagonal form we get

H, T a = Ω a T a (98)
So we can write the expectation value as:

O GS = a t a (O) ΨGS |T a | ΨGS (99) C(t 1 , t 2 ) = O 1 (t 1 )O 2 (t 2 ) (100)
The operators are in the Heisenberg representation

O(t i ) = e iHt i Oe -iHt i (101)
Transforming to the diagonal basis:

C(t 1 , t 2 ) = ΨGS | Õ1 (t 1 ) Õ2 (t 2 )| ΨGS (102) = ΨGS |e i Ht 1 Õ1 e -i H(t 1 -t 2 ) Õ2 e -i Ht 2 | ΨGS (103) = ΨGS | Õ1 e i( H-E GS )(t 2 -t 1 ) Õ2 | ΨGS (104)
Employing the expansion for transformed observables:

C(t 1 , t 2 ) = ΨGS | a 1 t a 1 T a 1 e i( H-E GS )(t 2 -t 1 ) a 2 t a 2 T a 2 | ΨGS (105) = ΨGS | a 1 t a 1 a 2 t a 2 e -iΩa 2 (t 1 -t 2 ) ΨGS |T a 1 T a 2 | ΨGS (106)
Similarly one can also write down the (retarded) Greens function.

G kk (τ ) = -iΘ(τ ) a 1 a 2 t a 1 t a 2 ΨGS |{T a 1 , T † a 2 }| ΨGS (107)
Taking the Fourier transform one gets

G kk (ω) = a 1 a 2 t a 1 (k)t * a 2 (k ) ΨGS |{T a 1 , T † a 2 }| ΨGS ω -Ω a 2 + i (108)

Finite Temperature Formalism

In this section we will extend the formalism of previous section for non-zero temperature.

The expectation value of observable O at finite temperature is given by:

O = T r(ρO) = 1 Z n n|e -βH O|n ( 109
)
where ρ is the density matrix and Z is the partition function of the system. Using the fact that trace is invariant under cyclic permutation we insert the unitary transformation and get: In this section we will calculate the flow equation for Kondo impurity spin operator and then calculate the dynamic spin susceptibility from the numerical solution of the flow equations. dS a (l) dl = [η l , S a (l)] (114)

We will use the one loop generator, given in equation 75 to evaluate the above commutator.

We use the following ansatz for spin operator:

Using the above formalism we can calculate spin-spin correlation function.

C(t) = 1 2 S z (0), S z (t) (118)
Plugging the ansatz of spin operator in equation 113 and Fourier transforming to frequency domain, we obtain:

C(ω) = - π Z(β)
The quantity which we are interested in and which we have calculated is imaginary part of the dynamic spin susceptibility χ(ω). Fluctuation-dissipation theorm relates χ(ω) to the spin-spin correlation function C(ω) calculated above.

χ(ω) = tanh ω 2T C(ω) (121)
Spin susceptibility is plotted in Figure 2. As we decrease Kondo coupling and hence Kondo effect becomes weaker, spin susceptibility gets enhanced. We want to point out that we obtained these curves with out any broadening which has been done in [START_REF] Kehrein | The Flow Equation Approach to Many Particle Systems[END_REF]23 .

Summary

In this paper, we have given discussion about the renormalization group methods for quantum impurity system which exhibit quantum many-body physics like the Kondo effect.

Anderson's poor man scaling method has been discussed and then applied to calculate the scaling equations for Kondo coupling. Solving the scaling equation we have calculated Kondo scale. We have also carried out scaling analysis of Anderson impurity model. Then we have discussed about flow equation renormalization method and shown how it is an extension of poor man scaling method. After presenting the formalism of this method we have done the flow equation renormalization of Kondo model and showed how in infrared limit, we recover poor man scaling results. We also presented our numerical procedure to extract Kondo scale from the numerical solutions of flow equations of Kondo model. Our numerical procedure can be applied to extract Kondo scale for cases where poor man scaling can not be applied.

One distinct advantage of FERG is that we can also study the renormalization behaviour of observables and hence calculate correlation and response functions. For the Kondo model, we have calculated the flow equation for spin observable and then solving those equations numerically, we obtain dynamic spin susceptibility. Within flow equation method we can calculate flow equations at finite temperature as well and they give reliable behaviour for temperatures above Kondo scale.

2 .

 2 Renormalization group methods for quantum impurity systems Perturbation theory of quantum impurity Hamiltonians leads to logarithmic divergences. This logarithmic dependence comes from the scaling properties of these models. The low energy physics, which is also the strong coupling regime of these models, does not depend on the bare parameters only. Strong coupling physics depends on renormalized parameters which have the renormalization effects of high energy states as well. These renormalized quantities which determine the low energy physics are the scaling invariants of the Hamiltonian. In the process of scaling, though the parameters of the Hamiltonian get renormalized, however, physics remains same. The scaling transformations not only renormalize the bare parameters of the model but also produce retardation effects and generate new terms as well. So the scaling transformation is an iterative map which starting with a given Hamiltonian generates a series of effective Hamiltonians as the high energy states are projected out, and the parameters of the effective Hamiltonians keep getting renormalized. All these effective Hamiltonians lie on scaling trajectories which are determined by scaling invariants of the Hamiltonian. The iterative scaling transformations can be written as differential equations for the coupling constants of the model. Kondo model is one of the celebrated models which shows this kind of scaling behaviour. For Kondo model, Anderson based on his poor man's scaling analysis calculated the flow equation for Kondo coupling.

Wegner 21 and

 21 Glazek and Wilson 22 . It uses unitary transformations to bring the Hamiltonian in a diagonal or band diagonal form. It does not integrate out the states as in Wilsonian RG.

FIG. 1 .

 1 FIG.1. Kondo coupling has been plotted versus flow parameter. In the left panel, the effect of increasing initial value of Kondo coupling can be seen. In right panel, temperature T has been varied.

  92) Since flow equation method basically diagonalizes the Hamiltonian so we can write H| ΨGS = E GS ΨGS (93) where | Ψ GS = U † (l)|Ψ GS Now comes the important idea: In FEM the observables also get transformed under the unitary flow. So to calculate the expectation value of the observable we use the transformed basis.

S

  a (l) = h(l)S a + i u u γ u u (l) : (S × s u u ) a :(115)

2

 2 tt (l → ∞)γ u u (l → ∞) × n| : (S × s t t ) z :: (S × s u u ) z : |n × (δ(ωu + u ) + δ(ω + uu )) u+ω, u (l → ∞) × (n( u )(1 -n( u + ω)(1 -n( u )

FIG. 2 .

 2 FIG. 2. Spin susceptibility of isotropic Kondo model is plotted as function of frequency. Kondo coupling J k is increasing from top to bottom. β = 1 T = 78.0 has been used .

  and hence parameter regimes apart from the Kondo regime which corresponds to the strong coupling fixed point of Anderson impurity model. As shown in 13 , Kondo model is related to Anderson impurity model via Schrieffer-Wolff transformation. Poor man scaling analysis of SIAM was first done by Haldane 18 and Jefferson

Anderson impurity model where valence fluctuations are also present in addition to spin fluctuations. The local Hilbert space of this model is larger, and it also has bigger parameter space as compared to the fixed points

  Using the expansion for the observable we get the final expression for the expectation value of the observable at finite temperature. -β n n|T a |n (111) Here Z is the partition function with transformed Hamiltonian. Similarly the equations for correlation functions can also be generalized to finite temperature. a 1 (O 1 )t a 2 (O 2 )e -βEn-iΩa 2 (t 1 -t 2 ) n|T a 1 T a 2 |n (113)

				1 Z	n	n|e -β H Õ|n	(110)
			O =	1 Z	a	t a (O)
			C(t 1 , t 2 ) =	1 Z	T r(ρO 1 (t 1 )O 2 (t 2 ))	(112)
	Using the operator expansion one gets
	C(t 1 , t 2 ) =	1 Z	n a 1 a 2

n e t 5.8. Spin Dynamics

  ( tt )J t t γ tt n(t )(1 -n(t))

	The flow equations for the coefficients are
	dh dl	=	t t	(116)
	dγ t t dl	= h( t -t )J t t
	-	1 4

u (( uu )J t u γ t u )(1 -2n(u)))


Jean-Rémi Lapaire
email: jrlapaire@u-bordeaux-montaigne.fr

The choreography of time : metaphor, gesture and construal

His current research is focused on the physicality of speech in relation to embodied social cognition. He has designed and tested multimodal learning environments where students are invited to use their sensorimotor abilities to engage in dynamic acts of observation and reenactment as they analyze human communication systems. He has built multidisciplinary partnerships with professional choreographers and dance theory specialists to explore the choreography of speech, i.e. how speakers use patterned moves to shape and display meanings in space.

Introduction

As speakers physically engage in communication, they move. As they move, they produce rich kinaesthetic imagery that opens a window onto thought [START_REF] Mcneill | Hand and Mind[END_REF][START_REF] Mcneill | Gesture and Thought[END_REF]; GOLDIN-MEADOW; ALIBALI 2013). Some of this imagery relates to time: it is remarkably patterned, intrinsically metaphoric, occasionally metonymic, and overwhelmingly spatial. The body parts that are characteristically involved-"hand, index finger, head or gaze" (CALBRIS 2011, p. 131)-can be identified with relative confidence, together with the general orientation system that governs kinetic action in the temporal domain.

Speakers unconsciously stage bodily displays of their experience and understanding of time. Their performance is based on a genuine "choreography of time" that determines the figures they trace and their occupation of conceptual space. The choreography may be observed, studied and eventually enhanced to create new embodied approaches to cognitive grammar (LAPAIRE 2011b). But the shift from spontaneous to controlled conceptual action is not so simple, as the present study reveals.

Gesture and the verbal-spatial expression of time

Gestural action routinely occurs along with verbal expressions of time. The highly patterned movements are not mere additions to lexical and grammatical constructions but essential co-articulators of time reference. Gestures are "purposeful constructive actions" (KENDON 2004, p. 359) which make a vital contribution to the shaping of temporal meanings. Their symbolic properties are fully integrated with the semiotics of temporality. The reference to time may be direct (1.1) or indirect (1.2.).

Movement and orientation in kinetic expressions of time

Temporal experience is commonly expressed in spatial terms. A significant number of verbs (go, come, follow, continue, stretch…), prepositions (at, in, on, to, from), adjectives (short, long, continuous…) or adverbs (far away, back, forward) are indifferently used to express spatial or temporal relations. Building on previous observations by WHORF (1956, p. 155) andLAKOFF &JOHNSON (1980, pp. 41-45), CALBRIS (1990, p. 85) remarks:

The spatial expression of time is overwhelmingly evident on the verbal level. The notions of length or duration, path, interruption, localization, point of departure, point of arrival, limit, interval, distance, sequel or prolongation, posteriority or anteriority are expressed identically in space and time.

The spatial construal of time is also reflected in the kinetic activity manifested by speakers as they locate events or indicate duration in gesture space. In her landmark study of the role of gesture in the expression of time, CALBRIS (1990, p. 87) spells out the basic rules for temporal localization in European languages:

-"the future is placed in the direction of walking or writing, that is forward or to the right." The chin may be "lifted forward to designate a date in the future," or "a leap of the hand or forefinger" might occur, especially for predictions, developments or postponements.

-"the recent past" is "situated directly behind the speaker" and typically expressed with "the thumb or head turned over the shoulder." The movements are usually "small and quick" (p. 87). To locate the more distant past, the head may be "raised high and backwards," more slowly, suggesting that things are "far behind".

Figure 1 -Gestural localization of time relative to the speaker's body (CALBRIS 1990, p. 88) CALBRIS does not explore the "geometry of gaze direction" (TODOROVIĆ 2006, p. 3550) but her drawings and diagrams suggest that a change in both head and gaze orientation characteristically accompanies the shift from the present moment to a past situation. This is especially true of narrative contexts, where speakers are seen redirecting their gaze away from the central or frontal axis, with eyes looking up to the left1 .

Figure 2 -Looking up and away (CALBRIS 2011, p. 129) It is important to note that events are not systematically localized relative to speaker's here and now. Some other moment of reference may be chosen and established manually. Once the temporal landmark has been set up, hand movements indicate what occurred "beforehand" (anteriority) or what is "forthcoming" (posteriority). The head is sometimes involved too, especially when a contrast needs to be established between "before" and "after".

Figure 3 -Gestural localization of time respective to a given moment (CALBRIS 1990, p. 90) Duration is the second major temporal dimension open to gestural expression. The spatial construal of temporal distance is visually explicit: a "span" or "stretch of time" is measured by allowing the dominant hand to travel from left to right (or in front of oneself) between invisible time limits (CALBRIS 1990, p. 90). Different hand configurations may be used to mark off boundaries whenever these need to be highlighted: "two palms held in the sagittal plane, facing each other"; "the thumb and the index finger held apart" (for brief time intervals); "a hand folded with the backs of the fingers forming a frontal panel that moves forwards" are common instances (CALBRIS 2011, p. 136).

The expression of duration is not confined to the simple and direct measurement of time intervals. Such notions as continuity, succession, repetition, progressive or regressive unfolding (CALBRIS 1990, p. 92) are commonly expressed by repeated curved movements (small vertical loops or circles). A sense of duration may be conveyed straightforwardly or indirectly, through an indication of origin, transformation or iteration.

Indirect reference to time

Gestures typically function and signify at different levels. Their contribution to utterance meaning is multi-layered. The layers may be self-contained or interdependent. In Figure 5 below, reference to time is metonymically implied via some related element of meaning. Akram Khan2 uses a combined eye, head, and thumb movement over his shoulder to point to the theater space where rehearsals have been taking place in the last two weeks. In so doing, he directly refers to a location placed behind him (spatial deixis), while indirectly referring to the stage work previously done there (temporal deixis). The form, orientation and dynamics of the gestural action are congruent with both meanings. In Figure 6, another British choreographer, Wayne McGregor3 , is seen telling the audience about the loving support he got from his family when he decided to become a professional dancer: "My parents were very up for me going. They absolutely encouraged me to take risks, to go, to try, to try." The utterance of "encouraged", "go", "try" and "try" is matched with a succession of two-handed gestures: palms facing each other, with a swift forward projection. As the bilateral hand moves are repeated, slight variations in hand configuration occur. Remarkably, the speaker does slightly lean forward and down while stressing "encouraged," thus indicating that success lay ahead and ambition was a driving force egging him on. All three actions involve metaphoric motion towards a goal. The "purpose metaphor"(LAKOFF; JOHNSON 1999, p. 190), in which desired events are imaginatively placed in front of the speaker, combines with an "event-for-time metonymy" (p. 154) and a "space-time metaphor" (p. 159): success is construed as a craved-for state, out there, in front of the desiring subject. "They absolutely ENCOURAGED me to take risks

Further extensions to choreographic thinking

The co-speech movements examined so far may be regarded as spontaneous manifestations of construal operations. Speakers physically enact the spatialization of time-a complex cognitive mechanism. Scenes of "visual thinking" are choreographed in which "concepts take shape" (ARNHEIM 1969, p. 116). Visible kinetic form is given to invisible representational processes, perceptual imagery and mental imagery meet and merge.

What ordinary speakers unconsciously do-"thinking in the medium of gesture" and "manu-facturing" meanings (STREECK 2009, p. 151)-may be put to conscious use for teaching and learning purposes. Through language, we all acquire the ability to create homologies between bodily moves and conceptualizations. We learn to manually shape, locate and connect entities; we tirelessly practice "the poetry of motion" in "imaginary spaces" (WHORF 1956, p. 155). This remarkable semiotic and aesthetic empowerment means that teaching strategies can be developed that use movement to develop plastic analogies between sensorimotor activity and cognitive mechanisms (LINDGREN; JOHNSON-GLENBERG 2013). In the following section we discuss the use of kinesthesia and spatial consciousness to introduce the semiotics and cognitive semantics of time reference in English.

Moved by time : the Grammar in motion study

Grammar in Motion (LAPAIRE; MASSE 2006) was conceived to explain "how grammar works" by turning meanings, construals and pragmatic configurations into kinetic action. 4 professional dancers were filmed performing metaphoric scenes of conceptualization (LAPAIRE 2011a). As they engaged physically in a series of symbolic interactions and manipulations, images of abstract concepts and semantic configurations were generated. The short scenes were designed to be physical enactments of metaphoric statements like "blocking (barring, obstructing, standing in) the way" for deontic can't constructions: -In narrative or argumentative situations, all speakers produce "gestures of the abstract" that create "homologies" between ideas and objects, mental activity and sensorimotor experience (MCNEILL 1992, p. 145). The homologies are congruent with those existing between meanings and substances, change and motion, purposes and destinations, knowing and seeing at the verbal level (LAKOFF; [START_REF] Lakoff | Metaphors We Live By[END_REF][START_REF] Lakoff | Philosophy in the Flesh. The Embodied Mind and its Challenge to Western Thought[END_REF].

Metaphors, including metaphoric gestures, provide us with the power to think of the abstract in concrete terms -in images of space, form, and movement that are not just concrete images but that become abstract concepts." (MCNEILL 1992, p. 178) -All speakers use "multimodal metaphors" in their speech. The metaphors are often "deployed" in gesture and speech at the same time (MÜLLER 2007, p. 113-14).

-Gesture forms are patterned. Lines, loops, curves, areas and boundaries all have conventional schematic structure.

Two Grammar in Motion episodes were used for the present study. The first, entitled "Time" (Appendix 1), featured a performer "standing in the "here and now" of current consciousness, then turning round to "look back" and (mentally) "travel back into the past." The performance was clearly an artistic creation, not an imitation of what speakers actually do in real life when referring to time. Yet, using head movements, gaze activity and hand gestures to localize past or future time with respect to the present moment was little more than an extrapolation of the authentic semiotic mechanisms operating in natural language (cf. 1.1.). The presentation closed with the performer "looking ahead" into the future: The second video used in the study was entitled "Ago and the past." Ago allows speakers to "travel back in time" while "counting their steps". The size of the steps and the distance covered may be short (e.g. a moment ago) or long (e.g. ages ago), definite (e.g. ten days ago) or indefinite (e.g. some time ago). As shown in Figure 10 below, metaphoric motion through symbolic space may involve:

-the entire bodily frame, with a strong engagement of the feet and legs as steps are visibly taken down the alley of time;

-the hands only, as the dancer faces back and produces a manual simulation of steps being taken down memory lane. Screen captures were printed that summed up the main points in each episode. The tasks were the following:

-identify the pictures ("What is this gestural performance about?"); -explain the scenes ("What do the moves stand for?"); -evaluate the approach ("Does this help? Please give your sincere opinion").

Our first hypothesis was that the video on "Time" would be easier to process because the gesture sequence rested on a simple front-back orientational metaphor (LAKOFF;JOHNSON 1980, p.14). This kind of metaphor is very much alive in everyday language6 and continuously fed by the spontaneous kinetic activity that accompanies "temporal localization" in every day speech (CALBRIS 2011, p. 134). It was also conjectured that the aesthetics and phenomenological quality of the "Time" performance would make a more lasting impression on the students' minds than the leaps and bounds used to illustrate "ago." The results of the study confirmed our prediction. Our second hypothesis was that embodiment-or "the enactment of knowledge and concepts through the activity of our bodies" (LINDGREN; JOHNSON-GLENBERG 2013, p. 445)-would create "meaningful connections" between movements and concepts. The "smart moves" (HANNAFORD 2005) would stick in everyone's mind and act as cues to access essential information about the constructions and cognitive mechanisms. But a more complex picture emerged from the study. Although the process of retrieval was largely successful with the "Time" visuals (97%), only a third of the students (34 %) produced detailed descriptions of the scene. Unsurprisingly, the remarks were confined to "time orientation" and the "space-time metaphor": "the future is ahead of us" and "the past is behind us" (LAKOFF; JOHNSON 1999, pp. 152-159). Scant attention was paid to the sensorimotor events linked to the cognitive acts of reminiscing and predicting: "turning around" and "looking back" (for remembering); "reaching out" (for attempting); "bringing back" (for retrieving and displaying); "bending forwards" and "looking ahead" (for anticipating and foreseeing). Bodily action was given more prominence in accounts of "Ago and the past" because the jumps made by the performers were conspicuous enough and could hardly go unnoticed. But students scored relatively low on basic identification tasks (40%) and no reference was ever made to gaze activity, head and hand movements, when measuring the distance between the present moment and a past situation: Only a fraction (6 %) of the students discussed the interconnection between motion, sensation and understanding. The role of the body as a medium-a "living form" through which meaning is simultaneously "conveyed" and allowed to "emerge" by itself (SIMPSON STERN; HENDERSON 2010, p. 136)-was not even hinted at. A likely explanation is that the students never engaged physically in the kinaesthetic experience they were asked to report on. As one of them insightfully remarked, "it would have made much more sense to do the moves, not just watch them 7 ". In ordinary spoken interaction, the listeners are exposed to forms of gesture symbolism-concrete or abstract, iconic or metaphoric-that they themselves perform when speaking. Production and reception alternate, so everyone has some practical knowledge of the semiotic system used and communicative strategies applied. But the expressive style of Grammar in Motion (LAPAIRE; MASSE 2006) was not resonant with anyone's gestural style. The visual metaphors made sense but were too far removed from conventional ways of using movements as "material carriers" of meanings (MCNEILL 2005, p. 58).

Because testing raw reception and basic understanding was deemed a priority, students were simply presented with the "smart moves". Favourable conditions for them to respond and engage physically were not created. Feedback was limited and participants could not build on their own kinaesthetic intelligence to explore "embodied, enactive forms" created "to organize and represent content" (STREECK 2009, p. 162). Their position was largely one of passive "spectatorship" (PUSTIANAZ 2016) in a nonparticipatory performance, with little or no access to performativity-"the making of signs and scenes" (SCHECHNER 2003, p. 327). This is why future studies should be careful to design movements that resemble natural co-speech gestures more. Also, greater viewer-involvement should be sought, probably by adopting an immersive workshop format (as successfully tried elsewhere): Our third and last hypothesis was that designing a kinetic system that "creates images of abstractions" (MCNEILL 1992, p. 145) would be a welcome innovation in the disembodied world of theoretical linguistics. Students, we believed, would find the phraseology poetic, the choreographed movements compelling, and the sensuous beauty of the moving bodies alluring. But once again, a more contrasted picture emerged from the study, although the expectation seemed to be confirmed at first. The videos were met with spontaneous manifestations of surprise and amusement. Occasional giggles or sneers soon gave way to cheers, and scattered pockets of resistance eventually dissolved. Within weeks, a fascination for "visual grammar" had developed-or so it seemed. Looks of disappointment would show on some faces when screenings of Grammar in Motion had to be postponed. However, the written reports produced for the study sent out a different signal (Appendix 2). The practical usefulness of the approach was challenged by over a quarter of the respondents. Adding to the disappointment, the expected learning gains were not considered as decisive as had been wishfully anticipated.

Figure 16 -Student rating of the visuo-kinetic method By and large, the form and aesthetics of the movements were positively assessed. The gestures were praised for their "beauty", "fluidity" and "plasticity." Most importantly, the moves were seen as giving "visible shape to the invisible mechanisms of grammar" thus creating "a new form of grammatical imagery." Yet, some reservations were also expressed, although more marginally, by those unable to perform the recognition tasks: "weird," "confusing," "strange and disturbing."

The contribution of the method to the successful comprehension of language functioning was generally acknowledged: "clear", "concise", "accurate", "relevant." A clear majority of the respondents expressed support for an approach that facilitates understanding by giving visible shape and spatial location to abstract notions: "an instant connection [is established] between grammatical meaning and a visual concept, thus avoiding the need to translate or learn technical definitions." But what constituted a "motivating" or "stimulating way of learning" for many (55%), also "added a layer of complexity" and caused "unnecessary disturbance" in some (25%). A small but articulate group of respondents (20%) expressed mixed feelings. The system is "clever" but functions in a paradoxical way, they remarked: it makes things "look simpler" while "adding a layer of complexity." Adding to the ambivalence, the creations are of unequal worth and inspiration: "some topics are very nicely handled, others are made trickier to grasp." Lastly, students were found investigating the limitations of the kinaesthetic approach. A harshly critical comment mentioned "complete aversion to this kind of stuff," while another judged the innovation "pointless" and "gimmicky." Both came from students with low scores on identification and description tasks. Elsewhere the criticism was more subdued: some practical-minded respondents found "no real benefit" in watching the videos since they "already knew the rules." Driven by similar utilitarian considerations, other respondents found the visuals positively enlightening but of questionable value to improve actual language performance: "Enjoyable but won't help you speak or write better"; "Makes you see things and understand things but remains very abstract. At the end of the day, what you will be judged by is whether you use the forms correctly or not, in real life situations." Some students showed deeper critical insight by challenging the durability of conceptual and perceptual memory. The imagery, they complained, "fades away too fast" and the concepts are hard to recollect. The visuo-kinetic arrangements, they pointed out, may strike viewers as beautiful and meaningful "in the instant" but that does not necessarily mean that the mental imagery will last for long. Even if some suitable degree of entrenchment is achieved, "you (still) need to put words back onto the pictures if you want them to mean anything at all." So there is no doing away with words.

Concluding remarks

"Gestural meaning-making and understanding" (STREECK 2009, p. 3) play a central role in the expression of time relations. Sensory reality is given to fundamental construal operations and "basic cognitive abilities" (LANGACKER 2008, p. 34) are revealed in action. Thus, the ability to locate, connect, oppose, limit, measure, orient, establish reference points is displayed and enacted as speakers engage in the verbal-gestural evocation of time. As suggested in the second half of this research paper, "smart moves" (HANNAFORD 2005) may be designed and successfully used as "cognitive artifacts" (STREECK 2009, p. 173) in teaching-learning settings, if they tap into the existing semiotic resources of co-speech gesticulation. For this to happen, "conceptual acts" (STREECK 2009, p. 160) must be observed and studied, as they spontaneously occur in everyday communicative interaction. But above all, symbolic actions, cognitive actions and bodily actions must be integrated into a single framework for exploring and understanding, where learners are simultaneously cast in the roles of viewers, cognizers and movers, and all eventually find themselves "emboldened by embodiment" (LINDGREN; JOHNSON-GLENBERG 2013, p. 445).

Figure 4 -

 4 Figure 4 -Tracing curved lines to express the course of time(CALBRIS 2011, p. 138)

Figure 5 -

 5 Figure 5 -Pointing back: spatiotemporal deixis "We're behind, as usual. But IT'S GREAT TO HAVE THE THEATRE."

Figure 6 -

 6 Figure 6 -Leaning forward towards the promising times ahead "They absolutely ENCOURAGED me to take risks

Figure 7 -

 7 Figure 7 -Deontic can't "Blocking the way" on an "action path"

Figure 8 -

 8 Figure 8 -The past

Figure 9 -

 9 Figure 9 -The future.

Figure 10 -

 10 Figure 10 -"Travelling back in time with ago"

Figure 11 -

 11 Figure 11 -TIME: PAST, PRESENT, FUTURE Topic identification (97% success rate)

Figure 13 -

 13 Figure 13 -Recapturing the past "Turning around" -"Looking back" "Reaching out" -"Bringing back"

Figure 14 -

 14 Figure 14 -Retrospective measurement of temporal distance between past and present (e.g. 10 years ago)

Figure 15 -

 15 Figure 15 -Multimodal linguistics seminar (2015)Exploring exclamatory structures Variations on Laugh and laugh in the face of adversity!

More research attention is needed here, using eye-tracking technology to monitor the eye activity that accompanies "visions of the past" in speech.

Interview with Akram Khan, Vertical Road Rehearsals at Curve Theatre, Week 2, Leicester, Sept 2010.

Wayne McGregor: A choreographer's creative process in real time, TEDGlobal 2012, Jun 2012.

Introduction to English linguistics : grammar and lexicology, semester 1. Department of anglo-american studies, Faculty of Modern Languages, Université Bordeaux Montaigne, France. Period of instruction : Autumn semester 2014. Total number of students : 100.

The

videos that the students were tested on were (1) "Time" (2) "Ago" (3) "Verbal base + affix" (4) "Tag questions" (5) "Past participles" (6) "Countable vs. uncountable nouns." 6 E.g. Back in the 1920s; Ten years behind; Looking forward to meeting you.

See Appendix -Negative ratings.

Appendix 1: Grammar in Motion -"Time" The French voiceover script and its English adaptation "Le temps, je ne le vois pas mais je le sens. Là, sous mes pieds, sous mes yeux, autour de moi, le present: here and now. Et là, dans mon dos, le passé. En anglais: back. Back in the past. J'y retourne, je le revois, parfois. Plus loin, là-bas, devant moi, le future! There, in front of me. J'y vais ou il vient vers moi. Quand je le devine, c'est comme si je le voyais déjà! "Time is invisible, yet I can feel it. I can sense the present under my feet and before my eyes, here and now, surrounding me. The past is behind. I can turn round and travel back in time to see what happened. The future lies ahead of me! I can travel forward in time, or wait for times yet to come. Anything I predict, I can see looming on the horizon, out there, somewhere."

Appendix 2: student assessment Learning gains and practical usefulness of Grammar in Motion

Open-answer question n°3: "Does this kind of approach help? Please give your sincere opinion" Sample of student responses Positive ratings FORM AND AESTHETICS: "Gives visible shape to the invisible mechanisms of grammar"; "A vivid illustration of abstraction"; "Creates a new form of grammatical imagery"; "Beautiful"; "Plastic and fluid" UNDERSTANDING: "Clear and accurate"; "Short and to the point"; "Helps us understand better", "Enhances understanding"; "Understanding made easier"; "Stimulating"; "A fun way to learn"; "Uses powerful imagery to visualize abstract grammatical concepts"; "Sets up concepts in space"; "Helps understand a concept through a concrete analogue"; "Promotes a better understanding of grammar"; "Establishes an instant connection between grammatical meaning and a visual concept, thus avoiding the need to translate or learn technical definitions."

Negative ratings

FORM AND AESTHETICS: "Weird"; "Personifying grammar rules in this way is somewhat strange and disturbing" UNDERSTANDING: "I'm not sure I understood the video on AGO"; "Adds a layer of complexity"; "Causes unnecessary disturbance. I much prefer the usual way of explaining things." RELEVANCE, PRACTICAL USEFULNESS: "Pointless"; "I have no problem making sense of the movements and comments but they don't add much to my understanding of grammatical issues. To be perfectly honest, I don't see the point and am rather averse to all this stuff"; "No real benefit: I already knew about time and tense. And using ago is not a problem for me." LIMITATIONS: "It would have made much more sense to do the moves, not just watch them"; "Everything looks fine while you watch, but the imagery quickly fades away."

Mixed feelings

SIMPLE AND COMPLEX: "Makes things both simpler and more complex"; "Some topics are very nicely handled, others are made trickier to grasp"; "Facilitates understanding in some places, but may add complexity elsewhere"; "Clever and special but also confusing in its own way." LIMITED GAINS: "Inspiring, but then you need to put words back onto the pictures if you want them to mean anything at all. So, at the end of the day, what's the gain? ; "Enjoyable but it won't help you speak or write better"; "Innovative and thought-provoking but you need to spell out rules again in order to bridge the gap between the smart moves and the grammatical rules"; "Makes you see things and understand things but remains very abstract. At the end of the day, what you will be judged by is whether you use the forms correctly or not, in real life situations."