Pierre-Yves Péneau

David Novo

Florent Bruguier

Gilles Sassatelli

Abdoulaye Gamatié

Performance and Energy Assessment of Last-Level Cache Replacement Policies

The Last-Level Cache (LLC) is a critical component of the memory hierarchy which has a direct impact on performance. Whenever a data requested by a processor core is not found in the cache a transaction to the main memory is initiated, which results in both performance and energy penalties. Decreasing LLC miss rate therefore lowers external memory transactions which is beneficial both power and performancewise. The cache replacement policy has a direct impact on the miss rate. It is responsible of data eviction of cache lines whenever the cache runs full. Thus, a good policy should evict data that will be re-used in a distant-future, and favour data that are likely to be accessed in the near-future. The most common cache replacement policy is the Least-Recently Used (LRU) strategy. It has been used for years and is cheaper in terms of hardware implementation. However, researchers have shown that LRU is not the most efficient policy from a performance point of view, and is further largely sub-optimal compared to the best theoretical strategy. In this paper, we analyze a number of cache replacement policies that have been proposed over the last decade and carry out evaluations reporting performance and energy.

I. INTRODUCTION

Computing systems are composed of several processing cores on the same chip and their performance is growing every year. In particular, this is a critical requirement in the High Performance Computing domain (HPC), where besteffort computation is highly desired. On the other hand, core computation capacity increases faster than memory capacity, leading to the well-known memory wall problem. It is widely accepted that the issue in such systems is no longer the computation but the memory access.

Therefore, an efficient memory system is key to providing compute systems with high performance capabilities. A key component of the memory hierarchy is the Last-Level Cache, or LLC. It is the last on-chip memory buffer where data can be stored before data exchanges reaches the off-chip levels, particularly the main memory. Data accesses that occur beyond the LLC are usually time and energy-consuming. Thus, an efficient LLC should decrease main memory access count. Such efficiency is directly related to the cache replacement policy of the LLC. This policy is responsible for deciding which cache line / data is to be evicted for other useful data to be cached. Thus, it is critical to suitably select which data must be discarded and which data should stay in a given cache level so as to maximize the amount of useful data in the cache.

In this paper, we review four popular cache replacement policies that have been proposed during the last decade in the literature. In order to support our analysis, we compare their performance against the classical Least-Recently Used (LRU) and the Random cache replacement policies. Furthermore, we evaluate the impact of each policy on the energy consumption of the LLC. To the best of our knowledge, no existing work provides a comprehensive comparison of these policies including energy consumption. In order to carry out our study, we use the ChampSim [START_REF]The ChampSim simulator[END_REF] simulator and the SPEC CPU2006 [START_REF] Henning | SPEC CPU2006 benchmark descriptions[END_REF] benchmark suite. The energy model for the LLC is obtained by using the popular CACTI tool [START_REF] Muralimanohar | CACTI 6.0: A tool to model large caches[END_REF].

The remainder of this paper is organized as follows: Section II presents the metrics used to assess the cache replacement policies; Section III reviews the considered policies; Section IV details the experimental setup used for their evaluation, then discusses the obtained results; and finally, Section V gives some concluding remarks.

II. BACKGROUND

A. Common metrics for replacement policy assessment

We review some commonly-used metrics for assessing the efficiency of CPUs and memory hierarchy. In the present paper, upper (or higher) cache levels correspond to those which are closer to the CPU. Thus, the L1 cache is the upper level and the main memory is a lower level in the memory hierarchy. A classical LLC cache organization supporting LRU is depicted by Figure 1, where the data structures used together with the LLC are illustrated. By default, the LRU cache replacement policy covers the components that are outside of the dashed box. The remaining components are used in the cache replacement policies presented later in this paper. In order to understand how the different policies affect the system performance, we introduce a few useful evaluation metrics.

One prominent issue in current compute systems lies is memory response time that cores must accommodate. This time has to be reduced as much as possible for performance reasons. A non negligible part of this time is due to misses in the LLC which often result in stalling of program execution until data is retrieved from the main memory. Thus, an efficient LLC helps reducing these expensive off-chip accesses such that execution can be resumed as early as possible.

A commonly used metric to assess the cache efficiency is the Miss Per Kilo Instructions metric, or MPKI, defined by formula [START_REF]The ChampSim simulator[END_REF]. When assessing a replacement policy, the number of executed instructions remains the same, but the number of misses varies. As mentioned previously, a good policy should reduce the number of cache misses. Thus, the MPKI should be decreased accordingly.

MPKI =

Total Miss Total instructions / 1000

Another common metric used to evaluate replacement policies is the Instruction Per Cycle (IPC), which describes the efficiency of the CPU. As the number of executed instructions should stay identical while evaluating different replacement policies, only the number of cycles required by the execution of the instructions may be different. This number relies on the memory efficiency since a non-negligible part of the execution time is wasted waiting for data, even more so when the fetch extends all the way to the external memory, which is the critical path. An efficient LLC policy should reduce the MPKI, which decreases the probability for a request to follow the critical path, and finally shortens the overall time required for the computation. If this objective is reached, the IPC increases, as expressed in the following equation:

IPC = Total Instructions Total Cycles (2)

B. Re-use distance and memory pattern

The replacement policy is responsible of block eviction on cache lines when they are full. Upon a miss, a block is selected as the victim and is discarded. Such a decision is based on heuristics that the LLC builds during the system execution. A heuristic has to predict accurately when a block will be reused in the future, also called the re-use distance. When the re-use distance is in the near future, the block should stay in the cache so as to avoid a miss. Otherwise, the block should have a high priority candidate whenever eviction is required.

Thus, the heuristic of cache replacement policy has to avoid wrong decisions as much as possible.

Application workloads generally have memory access patterns that have a varying impact on the efficiency of replacement policies. These patterns are divided into four categories [START_REF] Jaleel | High performance cache replacement using re-reference interval prediction (RRIP)[END_REF]: recency-friendly, trashing, streaming and mixed, respectively denoted by the formulas (3), ( 4), ( 5) and ( 6):

(a 1 , a 2 , ..., a k , a k-1 , a k-2 , ..., a 2 , a 1 ) N for any k

(3)

(a 1 , a 2 , ..., a k ) N for k > cache size (4) (a 1 , a 2 , a 3 , ..., a k ) for k = +∞ (5)
{(a 1 , ..., a k ) A P ε (a 1 , a 2 , ..., a k , a k+1 , ..., a m )} N for k< cache size, m> cache size, 0

< ε < 1, {A, N} ≥ 1 ( 6
)
With a recency-friendly pattern, blocks have a near-immediate usage. A trashing pattern leads to a cyclic access to K blocks that repeats N times. A streaming pattern occurs when blocks have no locality in their usage (also referred to as scan pattern). Finally, mixed patterns are a combination of the other three.

III. EVALUATED CACHE REPLACEMENT POLICIES

A. Least-Recently Used

The Least-Recently Used (LRU) cache replacement policy sorts the blocks of a line according to their usage over time. More precisely, they are ordered from 0 to N -1, where N is the cache associativity. Then, when a block B is used, it becomes the block number 0 and each block position between 1 and the last position of B is incremented by 1. Upon an eviction, the block N -1 becomes the victim. This heuristic predicts a short re-use distance for blocks that are recently used. Conversely, blocks with high position, i.e., near N, in the LRU list have a long distance re-use prediction. This policy works well with recency-friendly access pattern, and also with trashing pattern when the number K of blocks is less or equal than the number of blocks in the cache.

B. Static Re-Reference Interval Prediction

Proposed by Jaleel et al. [START_REF] Jaleel | High performance cache replacement using re-reference interval prediction (RRIP)[END_REF], the goal of the Static Re-Reference Interval Prediction (SRRIP) cache replacement policy is to predict the usage interval of blocks to avoid filling the cache with blocks that have a distant re-use interval. For this purpose, a Re-Reference Prediction Value (RRPV) is assigned to each block of the cache (see Figure 1). This value is coded on M bits, and varies between 0 and 2 M -1. When RRPV ≤ 1, the usage interval of the corresponding block is predicted as immediate. When RRPV = 2 M -1, the usage interval of the block is predicted as distant. Between these two cases, it is predicted as intermediate. On a block insertion, a RRPV value of 2 M -2 is assigned to the block. Upon an access, the SRRIP cache replacement sets RRPV to 0. Then, the block is predicted to be re-used in a near-immediate future. Upon an eviction, the SRRIP policy looks for a block such that RRPV = 2 M -1, i.e., a distant usage interval prediction. If no such a block exists in the line, each RPPV is incremented by 1 and the process is restarted until one RRPV at least reaches this requirement. The advantage of inserting with an intermediate usage interval prediction of 2 M -1 is to give more time to the SRRIP policy to learn the access pattern of a block, since its RRPV value is not enough for an eviction. The main disadvantage is that SRRIP can be polluted by blocks that have only two accesses, since it predicts a near-immediate usage interval on a hit. Thus, with SRRIP, a block that has been used only once is more likely to be evicted compared to a re-used block, even though this happened less recently.

This policy is scan-resistant (i.e., efficient when dealing with scan cache access pattern), since non temporal accesses do not pollute the LLC. It is also trash-resistant when the number k of blocks is less or equal to the cache size.

C. Dynamic Re-Reference Interval Prediction

When the usage interval of blocks is larger than the cache size, i.e., a trashing access pattern with a number of blocks k > associativity, the above SRRIP policy becomes inefficient and causes more misses. To prevent this situation, Jaleel et al. [START_REF] Jaleel | High performance cache replacement using re-reference interval prediction (RRIP)[END_REF] proposed the Dynamic Re-Reference Interval Prediction (DRRIP) cache replacement policy, as an extension of SRRIP.

Upon an insertion, the RRPV of a block is more likely to be set to 2 M -2. However, RRPV can be sometimes set to the maximum value 2 M -1. Then, for applications that have a trashing access patterns, each inserted block is evicted after its use and does not pollute the cache. Blocks that are reused are set to 0 and stay in the cache. To accurately detect the memory access pattern of an application, authors used the Set-Dueling techniques [START_REF] Qureshi | Adaptive insertion policies for high performance caching[END_REF] (see also Figure 1). They dedicated 32 lines of the LLC to the SRRIP policy only and 32 lines to the DRRIP policy only. A global counter, initialized to zero, is incremented or decremented by 1 respectively upon a cache miss on SRRIP or DRRIP dedicated cache lines. When the value of the counter is higher than 0, it means that the SRRIP policy leads to more cache misses than DDRIP. Conversely, when the counter is less or equal to 0, the DRRIP policy is less efficient. The cache uses the above global counter to dynamically adapt its replacement policy and to choose between SRRIP and DRRIP. This enables to select the most efficient replacement policy among the two when dealing with either scan or trashing pattern. One issue with DRRIP is the granularity of the prediction. According to the global counter, the entire cache has to follow the selected policy, i.e., the prediction is not at block granularity.

D. Signature-based Hit Prediction

One limitation of the SRRIP policy is that predictions made upon block insertion in the cache are static. The DRRIP policy addresses this issue by randomly alternating between two possibilities. This also relies on static choices upon a block insertion, and the learning phase is done after. The insertion strategy does not take into account the past activity of the cache. The Signature-based Hit Predictor (SHiP) policy [START_REF] Wu | SHiP: Signature-based hit predictor for high performance caching[END_REF] tackles this aspect by making predictions according to each block's access pattern. Authors added near the LLC a table of saturating counters, indexed by a hash called signature (see Figure 1). They evaluated signatures based on memory addresses, instruction sequences and program counters (PCs). Results show that the most efficient is the PC-based approach.

From a technical point of view, two fields are added in the cache meta-data: the signature and the outcome bit, both initialized to 0. Upon a block insertion, the corresponding signature is updated. Upon a hit on that block, the outcome bit is set to 1, meaning that since its insertion, this block has been re-used at least once. If a cache access results in a hit, the corresponding entry in the predictor is incremented by 1. In case of a cache miss, it is left unchanged. The entry is decremented when a block is evicted from the cache and its outcome bit is set to zero, i.e., this block has never been reused since it was inserted. Hence, the PCs that generate a lot of misses are identified and can be treated. When inserting a new block, the predictor is checked. When its signature points at an entry with the value of 0, the re-use interval is predicted as a distant interval. Any other value is considered as intermediate. Thus, the re-use interval prediction is dynamically predicted according to the PC responsible for the access.

Assigning a signature to each cache blocks is unrealistic for hardware resource requirement reason. Hence, the proposed implementation samples 64 lines of the LLC in order to train the predictor. The predictor is updated only upon a cache access to one of these 64 lines, while it is consulted on each access. Thus, a small portion of the cache activity steers the predictions over the entire cache.

This policy offers two main contributions. Firstly, the possibility to make predictions on application behavior thanks to program counters. Secondly, each block has its own signature, making the prediction more accurate than with SRRIP or DRRIP policies, which affect all cache blocks. The SHiP policy has been used to improve the SRRIP mechanism [START_REF] Wu | SHiP: Signature-based hit predictor for high performance caching[END_REF]. Instead of always predicting intermediate re-use interval, SHiP is applied to take decisions. This makes SRRIP more resistant to trashing pattern thanks to the signature-based prediction.

E. Hawkeye Approach

The Hawkeye policy [START_REF] Jain | Back to the future: leveraging Belady's algorithm for improved cache replacement[END_REF] is based on Belady's MIN [START_REF] Belady | A study of replacement algorithms for a virtual-storage computer[END_REF] algorithm. For a given set of cache accesses, MIN classifies blocks in two categories: hit blocks and miss blocks. This classification has been proved optimal [START_REF] Belady | On-line measurement of paging behavior by the multivalued min algorithm[END_REF], hence the number of misses found with MIN cannot be reduced.

Upon a block eviction in a cache, one can know if this is a good decision only after the victim block is later re-used. For example, given a cache with a capacity of 2 blocks per line and only 1 line containing B 0 and B 1 . B 1 is the victim block upon the insertion of a new block B 2 and the next access to the cache is B 1 . The MIN algorithm would detect that B 1 should not be evicted and would classify B 1 as a "hit block". Jain et al. [START_REF] Jain | Back to the future: leveraging Belady's algorithm for improved cache replacement[END_REF] proposed a mechanism called OPTgen that is able to reconstruct the MIN algorithm on past cache accesses (see Figure 1). Past accessed blocks are separated into two groups with the assumption that what happened is probably what will happen in the future. They verified this experimentally on the SPEC CPU2006 benchmark suite and found that MIN's decisions for a block remains the same 90% of the time on average during an entire execution. Then, they used MIN predictions on the past to guide future decisions.

The Hawkeye predictor uses the same sampling idea as that of Wu et al. [START_REF] Wu | SHiP: Signature-based hit predictor for high performance caching[END_REF]. A subset of the LLC is monitored in order to minimize the implementation hardware cost, while still having an accurate representation of the cache activity. The predictor is only trained when these lines are accessed. A table of saturating counters is used, which is indexed by PCs that are updated on each OPTgen prediction. When OPTgen predicts a cache hit, the corresponding entry is incremented by 1. When OPTgen predicts a cache miss, meaning that whatever is the victim choice, the next access to this block will result in a cache miss under optimal replacement policy, the corresponding entry is decremented by 1. Thus, such a block can be discarded and replaced by a useful data. Blocks within the cache are ordered according to their corresponding RRPV values. A high RRPV value corresponds to a high cache eviction priority, while the opposite is for low eviction priority. Blocks that are predicted to miss in the future are set to RRPV = 2 M -1, i.e., the maximum value. Those predicted to hit in the future are set to RRPV = 0. Upon a block insertion (i.e., a miss), if the block is predicted to hit in the future, each RRPV on the line are aged, i.e. incremented by 1. Blocks that are predicted to hit in the future can never reach the maximum RRPV value. Thus, those predicted to miss will always have the higher priority for eviction. If no such blocks exist, the block with the highest RRPV value is evicted.

This policy is efficient when dealing with scan, trash and mixed patterns. In fact, thanks to Belady's MIN algorithm which is able to accurately predict the future of blocks, each block behavior is detectable and can be anticipated. Thus, only useful cache blocks remain in the cache.

F. Summary

More generally, the data structures required by the above policies are summarized Figure 1. Rectangles with stripes in the LLC are the monitored lines. They are copied in the sampler, illustrated on the left of the LLC. The sampler contains meta-data about lines like tag or PCs. Upon an incoming request, the LLC is accessed and the sampler checks if the access is done on one of the sampled lines. If so, it is updated. The sampler is connected to the predictor. When the former is accessed, the later is updated. The Hawkeye policy requires an extra step between the sampler and the predictor called OPTgen. This step reproduces the MIN algorithm presented in Section III-E. The result is used to train the predictor. The predictor is used to change the value of the heuristics depicted by the three columns at the right of the LLC. The LRU updates the LRU bits only. SRRIP, DRRIP, SHiP and Hawkeye update the RRPV value. Set-Dueling mechanism is highlighted by the shaded rectangles. Rectangles in the LLC are the monitored lines and according to their activity, i.e hit or miss, the global counter (GC) is modified.

On the other hand, the reviewed cache replacement policies can be classified into two families: coarse-grained granularity versus fine-grained granularity. SHiP, SDBP and Hawkeye are considered as fine-grained prediction policies due to their usage of Program Counters that make prediction for each block. Conversely, LRU, SRRIP and DRRIP are considered as coarsegrained policies because they do not use an identification mechanism such as PC, and make a global prediction for the overall cache. The hardware implementation cost is however higher, even though sampling is used to reduce the storage overhead. Table I summarizes cache replacement policies w.r.t. their efficiency against the memory access patterns mentioned in this paper. Two features are also mentioned, i.e., Set-Dueling and Sampling, which enable to reduce the hardware overhead and to improve the accuracy in the implementation of the policies. We also report the hardware budget evaluated in the specific experimental setup used for our current study (thus, it may differ from the cost found in the corresponding literature). As an example, our SHiP implementation uses sampling on 256 lines while Wu et al. mentioned 64 lines.

IV. EXPERIMENTAL SETUP AND RESULTS

We evaluate the previous cache replacement policies by considering the ChampSim simulator [START_REF]The ChampSim simulator[END_REF] used during the Cache Replacement Championship at ISCA'17 conference. ChampSim models out-of-order (OoO) CPUs, a 3-level cache hierarchy and a main memory. This architecture is based on an Intel Core i7 system. More technical details are given in Table II. We executed 20 traces of the SPEC CPU2006 bench- [START_REF] Luk | Pin: building customized program analysis tools with dynamic instrumentation[END_REF]. Cores execute 1 billion instructions, with a warm-up period of 200 millions instructions. Average IPC is computed by applying a geometric mean on the IPCs measured for the 20 considered applications, as in previous work [START_REF] Jain | Back to the future: leveraging Belady's algorithm for improved cache replacement[END_REF]. As the Random cache replacement policy is not deterministic (unlike the other policies), it is run 10 times and the resulting geometric mean and average value for IPC

E dyn = (N read + N write ) × E access (7)
E stat = ExecTime ×W leak (8)
where E dyn and E stat respectively denote the dynamic and static energy; N read and N write are the total numbers of reads and writes on the LLC respectively; E access is the energy of either a read or a write on the LLC; ExecTime is the execution time and W leak represents the static power. Two architectures are addressed: a) monocore without prefetching and b) monocore with prefetching. Given these two architectures, we evaluate the performance and energy consumption of the LLC with the following policies: LRU, Random, SRRIP, DRRIP, SHIP and Hawkeye.

Figure 2a depicts IPC results with a monocore configuration without prefetching. On average, all policies outperform the LRU. The most efficient is Hawkeye, with an average speedup of 1.04. Applications that are memory-intensive such as lbm, mcf or soplex exhibit a greater speedup than others. The best performance improvement is achieved on the sphinx3 application. Random, SRRIP, DRRIP, SHiP and Hawkeye policies respectively perform a speedup of 1.22, 1.06, 1.31, 1.29 and 1.20. In addition, Hawkeye provides consistent results with no performance loss for any of the benchmarks, unlike others such as for zeusmp. Performance results on a singlecore configuration with prefetching w.r.t. LRU are shown in Figure 2b. Similar trends are observed, but the speedup is lower compared to the configuration without prefetching. Moreover, the Random policy no longer outperforms LRU.

The average speedup for the four considered policies varies between 1.01 (for SRRIP) and 1.03 (for Hawkeye).

However, the cache is more efficient when prefetching is activated. Indeed, Figure 3 shows that the average MPKI is drastically reduced by 48.2% in this case. This improvement affects the IPC, which is increased by 16% on average for all replacement policies. Thus, an important insight here is that prefetching has a positive effect on both MPKI and IPC. We also notice that this positive impact on IPC and MPKI is more visible on LRU compared to the other policies, as shown in Figure 3. Thus, cache replacement policies can make eviction decisions that are in conflict with the prefetcher, then benefit less from the positive effect of the latter.

IPC MPKI Energy

Improvement over no prefetch (%) One reason for this observation lies in the fact the prefetching system generates extra traffic in the memory hierarchy, which is hardly tractable for the prediction system that heavily relies on Program Counters. In such a situation it becomes challenging to predict block usage. Moreover, except Hawkeye, all policies analyzed in this paper do not take into account prefetched blocks. As a result, the LLC can be polluted by blocks that are wrongly prefetched, or prefetched blocks can be inadequately evicted due to lack of information. Yet, Figure 3 shows that SRRIP, DRRIP and SHiP benefit more from the prefetching than Hawkeye in terms of IPC despite its advanced mechanisms. This suggests that there is room for improvements about the adequate exploitation of prefetching with advanced cache replacement policy. The energy impact of cache replacement policies has been studied only marginally in the literature. Yet, this is an important topic that deserves a deeper understanding. These policies try to reduce the number of cache misses in general. Thus, the computation can be faster while the static energy can be reduced. Figure 4a shows the IPC improvement as a function of the energy improvement, w.r.t. LRU. A linear trend is observed: when the IPC is improved, the energy consumption is reduced accordingly. This can be explained by a faster computation reducing the static energy of the LLC.

Figure 4b shows that a prefetching system has less impact on the energy consumption, compared to a similar configuration without prefetching. Moreover, the linear trend observed previously is less pronounced. This is mainly due to two factors. Firstly, the prefetching is poorly (or not) managed by the covered replacement policies. As depicted in Figure 3, despite its advanced features, the Hawkeye policy benefits less from prefetching than the other policies in terms of energy reduction. Secondly, prefetching increases the dynamic energy of the cache since more memory access requests have to be processed. For example, it increases by 40% the amount of requests received by the LLC for the mcf application, with all policies. On the one hand, the average static energy, which represents 78.4% of the energy consumption, is reduced by 9.4% thanks to a faster execution. On the other hand, the average dynamic energy is increased by 37.2%. Finally, the total energy consumption is reduced by only 5.5%.

V. CONCLUSION AND FUTURE DIRECTION

In this paper, we reviewed four popular cache replacement policies proposed in the literature during the last decade. These policies are not currently implemented in real systems-onchip, as they require extra hardware which comes with an additional design complexity. We considered their implementation models within the ChampSim simulator for a crossevaluation. We assessed their impact on the performance and energy consumption of the LLC by measuring their induced Instructions Per Cycle, and their static and dynamic energy improvements. The SPEC CPU2006 benchmark suite has been considered in the experiments. While the obtained results confirmed the improvement of the evaluated policies over the commonly used LRU policy, an important gained insight is that their combination with the prefetching mechanism in modern architectures is very challenging for reaching improved performance. Moreover, the energy analysis shows that Hawkeye, the best replacement policy in the literature in terms of performance, benefits less from prefetching than other policies regarding energy improvement. Thus, future cache replacement policies should consider this issue in order to achieve the best performance/energy trade-off.

Fig. 1 :

 1 Fig. 1: Data structures for the LLC. Default configuration for LRU is outside the dashed box. SRRIP only uses RRPV; DRRIP uses RRPV, Set-Dueling and Global Counter; SHiP uses Sampler, Predictor and SRRIP structures; Hawkeye uses OPTgen in addition to SHiP structures.

Fig. 3 :

 3 Fig. 3: IPC, MPKI and energy improvement w.r.t. a configuration without prefetching.

Fig. 4 :

 4 Fig.4: IPC and energy improvement of evaluated policies over LRU for the 20 considered applications.

TABLE I :

 I Summary of reviewed cache replacement policies. Hardware budget corresponds to a 16-associative 2MB LLC.

TABLE II

 II

	: Experimental setup configuration
	mark suite. Traces are obtained by isolating a single region
	of interest of 1 billion instructions with SimPoint [10] and
	collected with Pin

ACKNOWLEDGEMENTS

This work has been funded by the French ANR agency under the grant ANR-15-CE25-0007-01, within the framework of the CONTINUUM project.


C Beck

N Keeley

A Diaz-Torres

Cluster Model for Near-barrier Fusion Induced by Weakly Bound and Halo Nuclei

Keywords: Fusion, breakup, transfer, weakly bound, halo, coupled-channels methods PACS: 25.60.Ge, 25.70.Jj, 25.70.Mn, 25.60.Dz, 24.10.Eq

The influence on the fusion process of coupling transfer/breakup channels is investigated for the medium weight 6,7 Li+ 59 Co systems in the vicinity of the Coulomb barrier. Coupling effects are discussed within a comparison of predictions of the Continuum Discretized Coupled-Channels model also applied to 6 He+ 59 Co a reaction induced by the borromean halo nucleus 6 He.

INTRODUCTION

In reactions induced by light weakly bound nuclei, the influence on the fusion process of coupling to collective degrees of freedom and to breakup (BU) and transfer channels is a key point for a deeper understanding of few-body systems in quantum dynamics [1,2,3,4,5,6]. Due to the very weak binding energies of halo nuclei, such as 6 He, a diffuse cloud of neutrons should lead to enhanced tunneling probabilities below the Coulomb barrier as compared to predictions of one-dimensional barrier penetration models [1,4]. This was understood in terms of the dynamical processes arising from strong couplings to collective inelastic excitations of the target and projectile [4]. However, in the case of reactions where at least one of the colliding nuclei has a sufficiently low binding energy for BU to become a competitive process, conflicting model predictions and experimental results were reported [4]. Recent experimental results with 6,8 He beams show that the halo of 6 He does not enhance the fusion probability, confirming the prominent role of neutron transfers in 6 He induced fusion reactions [7,8,9,10,11].

Excitation functions for sub-and near-barrier total -complete (CF) + incomplete (ICF) -fusion cross sections measured for the 6,7 Li+ 59 Co reactions [1] when compared to Continuum-Discretized Coupled-Channels (CDCC) calculations [2] indicate a small enhancement of total fusion (TF) for the more weakly bound 6 Li below the Coulomb barrier, with similar cross sections for both reactions at and above the barrier [2]. This result is consistent with BU -although with rather low cross sections even at incident energies larger than the Coulomb barrier [3] -being more competitive for the 6 Li+ 59 Co reaction.

In this contribution we present selected CDCC calculations for elastic scattering, TF, transfer (TR), and BU of weakly bound stable ( 6 Li considered as a α-d cluster) and radioactive ( 6 He as a α-2n cluster) light projectiles from a medium-mass target ( 59 Co). MeV [5]. The curves correspond to CDCC calculations [3] with (solid lines) or without (dashed lines) couplings with the continuum as discussed in the text.

FULL CDCC DESCRIPTION OF 6 LI+ 59 CO REACTION

In the present work, detailed CDCC calculations for the interaction of 6 Li on the medium-mass target 59 Co are applied in order to provide a simultaneous description of elastic scattering, fusion as well as breakup. Details of the calculations concerning the breakup space (number of partial waves, resonances energies and widths, maximum continuum energy cutoff, potentials, ...) have been given in previous publications [2,5].

The CDCC scheme is available in the general coupled channels code FRESCO [2]. Before investigating whether the proposed CDCC formalism can be also applied to halo structures such as the borromean nucleu 6 He, we present a complete description of the 6 Li → α+d cluster as a two-body object. In the fusion calculations the imaginary parts of the off-diagonal couplings were neglected, while the diagonal couplings included imaginary parts [2]. Otherwise full continuum couplings have been taken into account so as to reproduce the elastic scattering data [3,4]. We have used short-range imaginary fusion potentials for each fragment separately. This is equivalent to the use of incoming wave boundary conditions applied in previous CCFULL calculations [1].

Results of the comparison of the CDCC calculations for the elastic scattering with data of Ref. [3,4] are shown in Fig. 1 for 6 Li+ 59 Co at four different incident energies. The two different curves are the results of calculations performed with (solid lines) and without (dashed lines) 6 Li → α + d breakup couplings The agreement between the full calculations and data is very good. The effect of breakup on elastic scattering is illustrated by the difference between the one-channel calculations and the full CDCC results.

The total calculated BU cross sections were obtained by integrating contributions from the states in the continuum up to 8 MeV. They are found to be negligible fractions (between 3.7-9.7 %) of the total reaction cross sections and small compared with the TF cross sections (both the data [1] and two different sets of CDCC calculations [2,5]. This conclusion can be verified by the angular correlations for the sequential BU of 6 Li via the 2.18 MeV 3 + excited state [5] plotted in Fig. 2 at 41 MeV. Sequential BU via this state is the dominant contribution to the total 6 Li → α + d breakup cross section. Our CDCC cross section (22.5 mb) is, however, smaller than the measured value (45 ± 10

mb).

The total reaction cross sections obtained either from fits with Optical Model (OM) potentials [5] or CDCC calculations -both OM and "bare plus dynamic polarization" (DPP) potentials are shown in Fig. 3 for 6 Li whereas their equivalent values for 7 Li are given in [5] -confirm the observed small enhancement of TF cross section for the more weakly bound 6 Li nucleus at sub-barrier energies [1]. Fig. 3 illustrates how the surface strengths of the "bare plus DPP" potentials are in good agreement with OM potentials. Apparently, they exhibit the energy dependence characteristic of the "threshold anomaly" (TA), i.e. a rise in the strength of the real part as the incident energy is reduced towards the Coulomb barrier accompanied by a drop in that of the imaginary part. However, this conclusion largely rests on the values at 12 MeV incident energy, and as can be seen from the error bars, the potential strength in the nuclear surface is effectively not determined by the data due to its rather poor precision, a very wide range of values giving equally good fits to the data. The spread in values for the other energies, while much less than that at 12 MeV, is still such that we are unable to draw any concrete conclusions concerning the presence or absence of a TA in 6 Li+ 59 Co. Similar observations have been made for 7 Li+ 59 Co [5] and the question of the occurence a "BU threshold anomaly" remains widely open.

PREDICTIONS FOR 6 HE+ 59 CO FUSION AND CONCLUSIONS

Calculations applied to the two-neutron halo nucleus 6 He is much more complicated since 6 He breaks into three fragments (α+n+n) instead of two (α+d), and the CDCC method for two-nucleon halo nuclei has not yet been implemented in FRESCO [2]. A dineutron model is adopted for the 6 He+ 59 Co reaction [3,4,5]: i.e. we assume a two-body cluster structure of 6 He = 4 He+ 2 n with an α particle core coupled to a single particle representing a di-neutron ( 2 n) like cluster. Couplings to resonant (2 + , E ex = 0.826 MeV) and non-resonant continuum states (up to f-waves) are included. The fact that the dineutron is not an object with both fixed size and fixed energy (Heisenberg principle) might be a critical point in the present model. Results of the CDCC calculations for TF of the 6 He+ 59 Co system were compared to 4 He+ 59 Co and 6 Li+ 59 Co, used as basis reactions (see Refs. [3,4,5]). We observed that calculations with and without breakup give much larger TF cross sections for 6 He compared to 4 He and 6 Li. The inclusion of the couplings to the breakup channels notably increases the TF cross section for all energies. The predictions for the 59 Co target somewhat over predict the data published for other medium-mass targets such as 64 Zn [8] and 63,65 Cu [9]. Extended calculations are in progress to quantify the role of 1n-and 2n-transfer channels found to be significant in recent 6 He data [7,8,9,10,11].

The CDCC method [2] can be used to provide the almost complete theoretical description of all competing processes (TF, elastic scattering, TR and BR) in a consistent way. In this contribution we have shown that the 6 Li+ 59 Co reaction can be fairly well understood in this framework although CDCC does not separate CF from ICF. CDCC results for the 6 He+ 59 Co fusion process are also briefly discussed. A complete understanding of the reaction dynamics involving couplings to the BU and the neutron transfer channels will need high-intensity radioactive ion beams to permit measurements at deep sub-barrier energies and precise measurements of elastic scattering and yields leading to TR channels and to BU itself. The application of four-body (required for an accurate α-n-n description of 6 He) CDCC models still under current development [6] will then be highly desirable. It would be interesting to see what difference this more accurate model would have on the BU coupling effect on TF if applied to a fusion calculation in a similar manner to the calculations presented in this contribution.

FIGURE 1 .

 1 FIGURE 1. Elastic scattering for 6 Li+ 59 Co at (a) 30 MeV, (b) 26 MeV, (c) 18 MeV and (d) 12MeV[5]. The curves correspond to CDCC calculations[3] with (solid lines) or without (dashed lines) couplings with the continuum as discussed in the text.

FIGURE 2 .

 2 FIGURE 2. CDCC calculation for the angular distribution of the6 Li → α + d sequential breakup via the 2.18 MeV 3 + state of6 Li compared to data of6 Li+ 59 Co reaction at 41 MeV.

FIGURE 3 .

 3 FIGURE 3. Energy dependence of the real and imaginary parts of the "bare plus DPP" potentials as generated by the CDCC calculations (filled circles) and the best OM fits potentials (open circles) for the 6 Li+ 59 Co system at a radial distance of r = 9.5 fm.


Serge Florens

Priyanka Mohan

C Janani

T Gupta

R Narayanan

Magnetic fluctuations near the Mott transition towards a spin liquid state

Keywords: 71, 10, -w -Electronic structure: theories and models of condensed matter 71, 30, +h -Insulator-metal transitions 75, 10, Kt -Magnetic ordering: quantum spin liquids

Dynamics of magnetic moments near the Mott metal-insulator transition is investigated by a combined slave-rotor and Dynamical Mean-Field Theory solution of the Hubbard model with additional fully-frustrated random Heisenberg couplings. In the paramagnetic Mott state, the spinon decomposition allows to generate a Sachdev-Ye spin liquid in place of the collection of independent local moments that typically occurs in the absence of magnetic correlations. Cooling down into the spin-liquid phase, the onset of deviations from pure Curie behavior in the spin susceptibility is found to be correlated to the temperature scale at which the Mott transition lines experience a marked bending. We also demonstrate a weakening of the effective exchange energy upon approaching the Mott boundary from the Heisenberg limit, due to quantum fluctuations associated to zero and doubly occupied sites.

serge.florens@grenoble.cnrs.fr

Introduction. -The Mott metal-insulator transition, wherein electronic waves are localized by short-range electron-electron interactions (see [1] for a review), is one of the most complex phenomenon observed in strongly correlated electronic systems. Even though the appearance of a Mott gap is purely driven by the charge degrees of freedom, it is expected that magnetic fluctuations play a very crucial role in determining the true nature of this phase transition. In the paramagnetic Mott insulator, local moments are indeed well defined objects after their creation at high temperature (at a scale set by the local Coulomb interaction) and before their ultimate antiferromagnetic ordering at the Néel temperature, offering a window in which complex behavior of the spin excitations is yet to be clearly understood. Experimentally, the simplest situation in this respect occurs when the low-temperature magnetic ordering is first order, as in the case of Cr-doped V 2 O 3 . Since the magnetic correlations are expected to be weak in this case, many predictions can be made from a single-site approach like the Dynamical Mean Field Theory (DMFT) [2], where local moments are described as freely fluctu-ating from each other in the insulating state. As a consequence, the low-temperature metallic phase leads upon heating to an insulator, giving a Pomeranchuk-like effect where the entropy gain benefits to the state with magnetic degeneracy. Conversely, other classes of materials, such as the κ-organics, display a continuous magnetic transition into the Néel state, so that the localized spins will experience strong collective fluctuations. The most striking experimental consequence [3][4][5] lies in the progressive disappearance of the Pomeranchuk effect upon approaching the magnetic ordering temperature, so that the Mott transition lines in the pressure-temperature phase diagram bend at low temperature. These qualitative arguments have received recent confirmation from cluster DMFT calculations of the phase diagram of the two-dimensional Hubbard model on a frustrated lattice [6][7][8][9], that take into account nearest neighbor magnetic exchange leading to short-range singlet formation. However, the precise connection between the appearance of low-energy magnetic excitations and deviations from the phase diagram of the single-site approach has remained unclear.

Another challenging aspect of correlated fermion materials, that we wish to address here, concerns the interp-1 arXiv:1006.3620v3 [cond-mat.str-el] 12 Jul 2013 play with disorder, that can lead to a plethora of fascinating physics, such as Anderson-Mott transitions [10] and highly frustrated spin-liquid phases [11,12], for which extensions of Dynamical Mean Field ideas can also bring interesting light. In particular, introducing magnetic frustration (for instance due to random magnetic couplings) can have a profound influence on the nature of the Mott transition, by modifying the entropy of the magnetic Mott state, and also giving rise to slow spin dynamics [13] that can couple non-trivially to electronic excitations in the metallic phase as well [12,14,15]. The real nature of the Mott transition at zero temperature is believed infact to be strongly influenced by the type of magnetic correlations built by exchange interactions, so that the standard DMFT scenario [2] of a second order Brinkman-Rice transition may not apply anymore. However, due the complexity of the impurity models, which involve both fermionic and bosonic baths (see Eq. (2) below), such studies that combine Mott physics with random exchange have been rare [12]. One of our aim here is to provide a first attempt at a fully dynamical solution of this complex physical problem.

Methodology. -In order to investigate the interplay of Mott physics and magnetic fluctuations in a controlled setting, we consider the standard Hubbard model supplemented with long-range fully-frustrated and random Heisenberg exchange interactions, taking frustration as a necessary prerequisite for a non-trivial spin liquid ground state [16] on the insulating side of the Mott transition:

H = -t i j , σ d † i σ d j σ -µ i σ d † i σ d i σ +U i d † i↑ d i↑ d † i ↓ d i↓ + ij J ij S i . S j . (1)
Note that hopping parameters t run over adjacent sites of a yet arbitrary lattice, while the exchange terms J ij connect all the local spin variables

S i = σ,σ d † i σ τ σ,σ
2 d i σ in a random fashion (amplitude and sign). Also, in Eq. (1), µ is the chemical potential and U is the local on-site repulsion, which controls the degree of electronic correlations.

This non-trivial Hamiltonian greatly simplifies in the large coordination limit, where all the quantum dynamics can be exactly encapsulated by a single-site action:

S = β 0 dτ σ d † σ (∂ τ -µ)d σ + U d † ↑ d ↑ d † ↓ d ↓ (2)
+ β 0 dτ β 0 dτ ∆(τ -τ ) σ d † σ (τ )d σ (τ ) + β 0 dτ β 0 dτ Q(τ -τ ) S(τ ). S(τ ).
Here, two different baths that depend on imaginary-time τ have been introduced (β = 1/T is the inverse temperature): i) hopping processes from the local site into the lattice (and back) result in a retarded hybridization ∆(τ -τ ), which corresponds to the standard DMFT single site action [2]; ii) non-local magnetic coupling exactly amounts to a retarded spin interaction Q(ττ ), providing friction on the local moments, hence reducing their entropy. The latter term is easily obtained through a Gaussian averaging over the disordered exchange coupling J ij in Eq. ( 1), followed by a dynamical decoupling of a four-spin interaction, as shown in [11,17]. For a Bethe lattice, both baths are determined self-consistently, through the rather simple set of equations:

∆(τ ) = t 2 G d (τ ) (3)
Q(τ ) = J 2 χ(τ ). (4)
Here,

G d (τ ) ≡ -d σ (τ )d † σ (0)
is the local d-electron singleparticle Green's function, and χ(τ ) ≡ S(τ ). S(0) the local d-electron spin susceptibility. Free d-electrons (for U = 0) are endowed with a semi-circular local density of states with half-width D = 2t, while the degree of magnetic frustration is controlled by J, the width of the probability distribution used to perform the disorder average.

We now solve the still complicated quantum impurity problem (2), by laying recourse to a method that captures all the crucial physical ingredients at play. A simple possibility is to enlarge the spin symmetry from SU(2) to SU(N ) in the large N limit [START_REF] Auerbach | Interacting electrons and quantum magnetism[END_REF], from which the good emergent variables in the magnetic sector are known [11,12] to be spin-carrying zero-charge fermions (dubbed spinons) f † σ , with σ = 1 . . . N (we comment below on the validity of this approach for realistic fermion species with N = 2). Charge is then naturally encoded through a second auxiliary field, the phase φ dual to the local charge, resulting in the so-called slave-rotor representation [START_REF] Florens | [END_REF]20] of the physical electron:

d † σ = f † σ e iφ ≡ f † σ X.
Following [START_REF] Florens | [END_REF], we have written the phase as a complex bosonic field X constrained as |X(τ )| 2 = 1 through a Lagrange multiplier Λ(τ ), so that the action reads (particle-hole symmetry, i.e. µ = -U/2, is assumed here):

S = β 0 dτ |∂ τ X| 2 4U + Λ(τ )(|X| 2 -1) + σ f † σ ∂ τ f σ (5) + β 0 dτ β 0 dτ ∆(τ -τ ) σ f † σ (τ )X(τ )f σ (τ )X † (τ ) + β 0 dτ β 0 dτ Q(τ -τ ) N σ σ f † σ (τ )f σ (τ )f † σ (τ )f σ (τ ).
In the large N limit of [START_REF] Florens | [END_REF], the Green's functions for the auxiliary particles are readily obtained from Dyson equation in Matsubara frequency:

G -1 X (iν n ) = ν n 2 U + Λ -Σ X (iν n ) (6) G -1 f (iω n ) = iω n -Σ f (iω n ), (7)
where the associated self-energies read in imaginary time:

Σ X (τ ) = N ∆(τ )G f (τ ) (8) Σ f (τ ) = ∆(τ )G X (τ ) + Q(τ )G f (τ ). ( 9
)
Similarly, the parameter Λ in Eq. ( 6) is obtained by the average constraint G X (τ = 0) = 1. When re-expressed in terms of the emergent variables, the baths entering the self-consistency conditions Eqs. (3-4) can be simplified as:

∆(τ ) = t 2 G d (τ ) = t 2 G f (τ )G X (τ ) and Q(τ ) = J 2 χ(τ ) = J 2 G f (τ ) 2 .
Together with these self-consistent equations, the above self-energies form a non-linear set of integral equations that we solve numerically. We note that two important limits are naturally incorporated in this scheme: i) For J = 0 (standard Hubbard model), the spin bath Q(τ ) can be dropped, and we recover the slaverotor treatment [START_REF] Florens | [END_REF] of the usual DMFT equations, so that a Mott metal-insulator transition occurs at a critical value of U/t; ii) More interestingly at U = ∞, charge fluctuations are fully suppressed, as the gapped rotor propagator G X (τ ) can be discarded in Eq. ( 9). One then recovers the quantum spin liquid equations of Sachdev and Ye [11], where the exchange J is responsible for changing the high temperature Curie behavior of the spin susceptibility χ(τ ) 1/4 into the slow temporal decay χ(τ ) ∝ 1/|τ |. As a consequence, the temperature dependence of the static susceptibility χ(T ) = β 0 dτ χ(τ ) turns from a Curie law χ(T ) = 1/(4T ) at T J into a slower logarithmic divergence χ(T ) ∝ log(1/T ) at T J. This interesting absence of (spin glass) magnetic ordering at finite temperature, due to the enhanced quantum fluctuations brought by the large N fermionic representation of the spin, allows us to study the onset of magnetic fluctuations in the whole range of temperatures. Some comments are in order here on the validity of the large N limit for describing realistic single-band Heisenberg spin-liquid models at N = 2. Indeed, pioneer quantum Monte Carlo simulations [21] of the quantum spin glass model (in the U = ∞ limit) reported a very different class of solutions at finite temperature, showing a renormalized finite Curie constant, hence a still 1/T diverging spin susceptibility, instead of the log(1/T ) result obtained by Sachdev and Ye at large-N [11]. Further studies [22][23][24] demonstrated however that these paramagnetic solutions were actually metastable states below the freezing temperature. The correct low temperature physics at N = 2 is infact the one of a nearly quantum critical state (with tiny Edwards-Anderson order parameter q EA =0.06), where low-energy excitations are perfectly described by the large-N limit. In addition, we also note that the increase of quantum fluctuations occuring in our finite U Hubbard model has the tendency to enhance spin-liquid behavior, since we are far from the Heisenberg limit near the Mott boundary, as discussed previously [12,23]. This discussion thus certainly vindicates our chosen theoretical framework for describing spin-liquid effects. Finally, we also note that most recent studies on this general topic have focused on the more challenging finite-dimension Mott transition towards a spinon liquid [20,[25][26][27][28][29][30][31]. Our study aims rather at including explicitly the effect of random magnetic exchange terms, which are notoriously difficult to address on finite dimensional lattice.

In order to investigate the interplay of Coulomb repulsion U and exchange coupling J close to the Mott metalinsulator transition, one needs to analyze the relative stability of the various phases. This can be achieved from the knowledge of the total free energy (per site) which decomposes into two parts F tot = F imp + F lat , respectively related to the local impurity problem (2) solved by the rotor equations (8)(9) and to the original lattice model (1) via the self-consistency equations (3)(4). The former contribution can be obtained straightforwardly following the derivation of the slave-rotor equation as saddle-point equations [START_REF] Florens | [END_REF]:

F imp = -Λ + N β ωn [ln G f (iω n ) -iω n G f (iω n )] (10) + 1 β ωn [ln G X (iν n ) -(ω 2 n /U + Λ)G X (iν n )] -N β 0 dτ [∆(τ )G X (τ )G f (τ ) + 1 2 Q(τ )G f (τ ) 2 ].
Taken as a functional of the pseudoparticle Green's functions, the previous expression generates the expected Σ X and Σ f self-energies Eqs. (8-9) by differentiation with respect to G X and G f respectively. The lattice contribution, associated to the DMFT equations, can similarly be obtained from free energy functionals [START_REF]Lectures on the Physics of Highly Correlated Electron Systems VIII[END_REF]:

F lat = N 2 dτ ∆(τ )∆(τ ) t 2 + N 4 dτ Q(τ )Q(τ ) J 2 . ( 11
)
Minimization of F tot with respect to the dynamical fields ∆(τ ) and Q(τ ) reproduces indeed the DMFT selfconsistency condition Eqs. (3)(4).

Discussion of the results. -We are now equipped to investigate how magnetic correlations, controlled by the exchange term J, affect the Mott metal to insulator transition. To do so, we first construct the phase diagram in the (U ,T ) plane for fixed values of the magnetic exchange interaction J. At low temperatures this is done by monitoring the hysteresis in the local d-electron Green's function (G d (τ ) shows either slow Fermi liquid temporal decay or fast gapped behavior in the metallic and insulating states respectively). This gives rise to two boundary lines U c1 (T ) and U c2 (T ), such that a single metallic (resp. insulating) solution exists at a given temperature T for U < U c1 (T ) (resp. U > U c2 (T )). Furthermore, we determine from Eqs. (10)(11) the critical line U c (T ) at which the free energies of metallic and insulating branches coincide. The resulting phase diagram obtained for several values of J from J = 0 to J/D = 0.5 is displayed on Fig. 1 in the case of the spin-orbital index N = 3. We stress that the particular choice of N = 3 allows to match precisely the phase boundary of the usual single-band Hubbard model for canonical spin-1/2 fermions [START_REF] Florens | [END_REF], and is helpful to make quantitative comparisons when introducing other degrees of freedom (more orbitals for example) or new energy scales (random spin exchange as in the present work). We see from Fig. 1 that irrespective of J, there always exists an interaction range U c1 (T ) < U < U c2 (T ) bounding a coexistence region. With increasing temperature, the transition lines always merge onto a critical point that occurs at T c D/30 for J = 0, consistent with Refs. [2,[START_REF] Florens | [END_REF]. Interestingly, Fig. 1 shows that turning on the Heisenberg interaction J depresses both T c and U c , leading also to a shrinking of the co-existence region. This can be understood by an energetic stabilization of the Mott phase via magnetic exchange. Turning to the zero temperature limit, one recovers by the collapse of the U c (T ) and U c2 (T ) lines at J = 0 the continuous metalinsulator transition described by the Brinkman-Rice scenario [2,20]. In that case, the metal is the true ground state for all U < U c2 (T = 0). However, as seen in Fig. 1, the switching of J markedly separates the actual critical line U c (T ) from the two stability lines, so that the Mott transition becomes first order at zero temperature, in agreement with recent findings on models with shortrange magnetic correlations via cluster-DMFT [8].

U/D U c U c U c2 T c T /D U c1
Fig. 1: Metal-insulator transition phase diagram for several values of J = 0, 0.1, 0.2, 0.4, 0.5 (right to left). Labels of the various critical lines, as defined in the text, are explicited for J = 0. The absence of data for T /D < 0.003 is due to the simulations performed at finite temperature using Matsubara Green's functions, but the explored temperature range covers all the interesting physical regimes.

Next, we turn our attention to the orientation of the transition lines as the interaction between the localmoments is increased. At J = 0, the U c (T ) line is such that heating the correlated metal always leads to the Mott insulator, a Pomeranchuk-type effect that relates to the high entropy of the insulating phase. This feature persists even above the transition temperature: by determining at fixed T > T c the crossover interaction U c (T ) at which the d-electron Green's function (taken e.g. at τ = β/2) displays an inflection point, one distinguishes between a bad metal (i.e. an increasing resistivity with increasing temperature with non quadratic behavior) and a bad insulator (i.e. a non-gapped decreasing resistivity with increasing temperature), see [33][34][35][36]. The continuity of the critical U c (T ) line (determined from the free energy at T < T c ) into the crossover line U c (T ) (determined from the Green's function at T > T c ) serves as a vindication of our methodology. We also note that such a crossover at T > T c is seen experimentally in the magnetic and transport measurement of κ-BEDT salts [3,4]. Now, as J is increased one expects the entropy of the paramagnetic Mott state to be reduced by magnetic exchange, thus weakening the Pomeranchuk lineshape. This is indeed seen in Fig. 1 by a progressive leftward bending of the critical lines at increasing J, and consistent with results of Ref. [7][8][9]. A particularity of our model is the re-entrance of the Pomeranchuk effect at the lowest observable temperature, which we could attribute to the reduced (with respect to log(2) of free moments) yet non-zero entropy of the insulating spin liquid state [11,12]. We have checked also that increasing orbital degeneracy with larger N values tends to weaken all magnetic effects, recovering the standard J = 0 phase diagram.

A key point of our study concerns the connection between the exchange induced modifications of the phase diagram (which can be observed e.g. in transport experiments) and the anomalous spin-liquid dynamics. The spin susceptibility χ(τ ) in the absence of exchange J = 0 reduces trivially to a pure Curie law in the whole Mott state U > U c2 , as checked in Fig. 2. At finite J, Curie behavior is seen only at high temperatures such that T J, while departure from the independent local moment picture is observed by a slower logarithmic divergence at zero temperature. A crossover scale T sl characterizing the onset of the spin liquid state can be accurately determined from a fit of χ(T ) by the appealing form χ(T ) = 1/(4T sl ) ln(1 + T sl /T ) which follows the expected high and low temperature regimes (the agreement of this Ansatz with our data is excellent, see inset of Fig. 2 where scaling is demonstrated). This simple Ansatz allows us to extract precisely, for a given value of J, the U -dependence of the spin-spin correlation scale T sl , see Fig. 3, obtained for several J values. We note first that T sl saturates at very large U values to (half) the bare magnetic exchange constant J, an expected result since only spin dynamics plays a role upon freezing of the charge excitations, leaving J as the only relevant scale. More interesting is the net decrease of T sl when U is diminished towards the critical Mott boundary. This feature signals a weakening of the spin correlations in the vicinity of the Mott transition, brought by the finite probability of double and zero occupation of the d state (valence fluctuations in the particlehole symmetric regime). Quite intriguing is also that the spin susceptibility continues to display the characteristic signature of the spin liquid even in a regime where the ground state is metallic, close to the coexistence region, as seen by the fact that T sl remains finite near the Mott boundary (note that it is difficult to determine T sl precisely in the metallic phase, due to the existence of multiple energy scales). This result is reminiscent of a previous study by Parcollet and Georges [12], where the dopinginduced Mott transition (at U = ∞) within the fully random Heisenberg model was considered. These authors showed that the magnetic crossover scale T sl persists at small doping, leading to a slush state of incoherent metal with spin liquid correlations, similar to what we observe here at half-filling in the intermediate Coulomb interaction regime. Our last and perhaps most important prediction is the existence of a clear link between the spin liquid coherence scale T sl and the temperature where the bending of the metal-insulator crossover lines takes place, as manifest from Fig. 3. This crucial observation indeed shows that magnetic exchange is responsible for the quenching of the Pomeranchuk effect and the bending of the Mott transition lines, a result previously anticipated on physi-cal grounds.

Conclusion. -We have investigated the role played by fluctuating local moments coupled via frustrated longrange interactions near the Mott boundary. The metalinsulator transition was found to be modified by the Heisenberg exchange in favor of the insulating state and even to turn discontinuous at zero temperature. Studying the magnetic susceptibility, we were able to connect the bending of the critical lines to the appearance of collective spin correlations, with reduced effective exchange. These effects should be generic to other microscopic models and experimental systems near the Mott transition. One interesting open question is whether the quantum critical behavior of the large dimension Mott transition [37] persists to a quenching of magnetic moments induced by exchange interactions. Recent development of efficient bosonic impurity solvers [38][39][40] should allow such further explorations of this rich physics.

Fig. 2 :

 2 Fig.2: Spin susceptibility χ(T ) versus temperature for J = 0 and large U (circles, corresponding to uncoupled magnetic moments in the Mott phase), and for J = 0.2 and U/D = 1.8 (squares, corresponding to a spin-liquid Mott phase). A 1/T Curie behavior applies to the whole J = 0 curve (free moments), while a spin-liquid susceptibility Ansatz χ(T ) = (1/4T sl ) ln(1 + T sl /T ) describes the case of finite J = 0.2 exchange (full line). The inset demonstrates scaling of our numerical data for three different values of the exchange interaction J = 0.2, 0.4, 0.8 in the spin-liquid Mott phase.

Fig. 3 :

 3 Fig.3: Superposed plot of the U -dependent spin-liquid onset temperature T sl for several J = 0.1, 0.15, 0.2 (circles, bottom to top) with the corresponding metal-insulator phase diagrams (right to left). The merging of T sl with the temperature scale where the crossover lines bend demonstrate the crucial role of the magnetic fluctuations in the destruction of the Pomeranchuk effect.

We would like to acknowledge helpful discussions with V. DOBROSAVLJEVIC, S. R. HASSAN, R. MOESS-NER, and T. VOJTA.